Repository: metamath/set.mm
Pull request #4801: Claim 2 AKS Theorem 6.1
Author: login: metakunt
Created: 2025-05-01T20:33:53Z | Merged: 2025-05-06T13:55:31Z

The most disgusting proof so far.  ad7antr is used.
I had to split it up just to get a clean context so that autocomplete will work.

I hope I didn't to a mistake as last time, but this one was truly difficult to keep the head straight.


Diff:
diff --git a/changes-set.txt b/changes-set.txt
index e58f788913..8089ccba24 100644
--- a/changes-set.txt
+++ b/changes-set.txt
@@ -84,7 +84,11 @@ make a github issue.)
 
 DONE:
 Date      Old       New         Notes
- 2-May-25 bj-sbnf   sbnf        Moved from BJ's mathbox to main set.mm
+05-May-25 idomdomd  [same]      Moved from TA's mathbox to main set.mm
+05-May-25 idomringd [same]      Moved from TA's mathbox to main set.mm
+02-May-25 bj-sbnf   sbnf        Moved from BJ's mathbox to main set.mm
+01-May-25 rnexd     [same]      Moved from TA's mathbox to main set.mm
+01-May-25 imaexd    [same]      Moved from TA's mathbox to main set.mm
 30-Apr-25 idomrootle [same]     Moved from SO's mathbox to main set.mm
 30-Apr-25 ply1ascl0 [same]      Moved from TA's mathbox to main set.mm
 30-Apr-25 hashf1dmcdm [same]    Moved from BT's mathbox to main set.mm
diff --git a/set.mm b/set.mm
index 68b8cfeb30..a8c55f95a5 100644
--- a/set.mm
+++ b/set.mm
@@ -77800,6 +77800,19 @@ Axiom of Replacement (unlike ~ resfunexg ).  (Contributed by NM,
       ( cvv wcel cima imaexg ax-mp ) ADEABFDECABDGH $.
   $}
 
+  ${
+    rnexd.1 $e |- ( ph -> A e. V ) $.
+    $( The range of a set is a set.  Deduction version of ~ rnexd .
+       (Contributed by Thierry Arnoux, 14-Feb-2025.) $)
+    rnexd $p |- ( ph -> ran A e. _V ) $=
+      ( wcel crn cvv rnexg syl ) ABCEBFGEDBCHI $.
+
+    $( The image of a set is a set.  Deduction version of ~ imaexg .
+       (Contributed by Thierry Arnoux, 14-Feb-2025.) $)
+    imaexd $p |- ( ph -> ( A " B ) e. _V ) $=
+      ( wcel cima cvv imaexg syl ) ABDFBCGHFEBCDIJ $.
+  $}
+
   ${
     $d x y A $.  $d x y R $.  $d x V $.
     $( Any set relation is set-like.  (Contributed by Mario Carneiro,
@@ -273138,6 +273151,20 @@ regarding the multiplication ( ~ rngqiprnglin ) via ~ rngqiprnglinlem1
   isidom $p |- ( R e. IDomn <-> ( R e. CRing /\ R e. Domn ) ) $=
     ( ccrg cdomn cidom df-idom elin2 ) ABCDEF $.
 
+  ${
+    idomringd.1 $e |- ( ph -> R e. IDomn ) $.
+    $( An integral domain is a domain.  (Contributed by Thierry Arnoux,
+       22-Mar-2025.) $)
+    idomdomd $p |- ( ph -> R e. Domn ) $=
+      ( ccrg cdomn cidom cin df-idom eleqtrdi elin2d ) ADEBABFDEGCHIJ $.
+
+    $( An integral domain is a ring.  (Contributed by Thierry Arnoux,
+       22-Mar-2025.) $)
+    idomringd $p |- ( ph -> R e. Ring ) $=
+      ( ccrg cdomn cidom cin df-idom eleqtrdi elin1d crngringd ) ABADEBABFDEGCH
+      IJK $.
+  $}
+
   $( A field is an integral domain.  (Contributed by Mario Carneiro,
      29-Mar-2015.)  (Proof shortened by SN, 11-Nov-2024.) $)
   fldidom $p |- ( R e. Field -> R e. IDomn ) $=
@@ -500497,19 +500524,6 @@ its graph has a given second element (that is, function value).
       DERUTUNSGJEDFUDUCUEUFVBFUKUMNZLUOVAVDFBCDTUGFUKUMUHUIUPUQUSTUJ $.
   $}
 
-  ${
-    rnexd.1 $e |- ( ph -> A e. V ) $.
-    $( The range of a set is a set.  Deduction version of ~ rnexd .
-       (Contributed by Thierry Arnoux, 14-Feb-2025.) $)
-    rnexd $p |- ( ph -> ran A e. _V ) $=
-      ( wcel crn cvv rnexg syl ) ABCEBFGEDBCHI $.
-
-    $( The image of a set is a set.  Deduction version of ~ imaexg .
-       (Contributed by Thierry Arnoux, 14-Feb-2025.) $)
-    imaexd $p |- ( ph -> ( A " B ) e. _V ) $=
-      ( wcel cima cvv imaexg syl ) ABDFBCGHFEBCDIJ $.
-  $}
-
 
 $(
 -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
@@ -510153,20 +510167,6 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
 -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
 $)
 
-  ${
-    idomringd.1 $e |- ( ph -> R e. IDomn ) $.
-    $( An integral domain is a domain.  (Contributed by Thierry Arnoux,
-       22-Mar-2025.) $)
-    idomdomd $p |- ( ph -> R e. Domn ) $=
-      ( ccrg cdomn cidom cin df-idom eleqtrdi elin2d ) ADEBABFDEGCHIJ $.
-
-    $( An integral domain is a ring.  (Contributed by Thierry Arnoux,
-       22-Mar-2025.) $)
-    idomringd $p |- ( ph -> R e. Ring ) $=
-      ( ccrg cdomn cidom cin df-idom eleqtrdi elin1d crngringd ) ABADEBABFDEGCH
-      IJK $.
-  $}
-
   ${
     $d .0. a b c $.  $d .x. a b c $.  $d B a b c $.  $d R a b c $.
     $d X a b c $.  $d Y b c $.  $d Z c $.
@@ -677236,6 +677236,863 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
+      aks6d1c2p3.7 $e |- ( ph -> I = ( p E q ) ) $.
+      aks6d1c2p3.8 $e |- ( ph -> I e. NN0 ) $.
+      $( Lemma for ~ aks6d1c2 to simplify context.  (Contributed by metakunt,
+         1-May-2025.) $)
+      aks6d1c2lem3 $p |- ( ph -> ( J ( .g ` ( mulGrp ` K ) ) ( (
+      ( eval1 ` K ) ` ( G ` s ) ) ` M ) ) = ( I ( .g ` ( mulGrp ` K ) ) ( (
+        ( eval1 ` K ) ` ( G ` s ) ) ` M ) ) ) $=
+        ( vv cv cfv ce1 cmgp cmg co cexp cdiv cmul cn0 cvv cmpo wceq a1i simprl
+        wa oveq2d simprr oveq12d cc0 cfz wcel elfznn0 ovexd ovmpod eqtrd oveq1d
+        syl oveq2 fveq2d eqeq12d wbr wral syl2anc eqid aks6d1c1rh aks6d1c1p1rcl
+        mpbid cbs cn simpld aks6d1c1p1 rspcdva eqcomd w3a nnnn0d 3jca eqtr2d wf
+        cprimroots fveq2 cmap wb nn0ex elmapg cpl1 simprd caddc cplusg cmnd crg
+        ccrg fldcrngd crngring ringmgp nn0mulcld c0g wi ccmn crngmgp isprimroot
+        cdvds biimpd mpd simp1d mulgnn0dir mulgnn0ass simp2d mulgnn0cld mndrid
+        mulgnn0z ) AUEUHUKWMZUBWNZUFWOWNZWNZWNZUFWPWNZWQWNZWRGULWMZWSWRZUIGWTWR
+        ZRWMZWSWRZXAWRZUVRUVTWRZUDUVRUVTWRZAUEUWFUVRUVTAUEUWAUWDSWRUWFWGAQUPUWA
+        UWDXBXBGQWMZWSWRZUWCUPWMZWSWRZXAWRZUWFSXCSQUPXBXBUWMXDXEAVHXFZAUWIUWAXE
+        ZUWKUWDXEZXHXHZUWJUWBUWLUWEXAUWQUWIUWAGWSAUWOUWPXGXIUWQUWKUWDUWCWSAUWOU
+        WPXJXIXKAUWAXLEXMWRZXNUWAXBXNWEUWAEXOXTZAUWDUWRXNUWDXBXNWFUWDEXOXTZAUWB
+        UWEXAXPXQXRZXSAUWHGUNWMZWSWRZUWCUMWMZWSWRZXAWRZUVRUVTWRZUWGAUDUXFUVRUVT
+        AUDUXBUXDSWRUXFWJAQUPUXBUXDXBXBUWMUXFSXCUWNAUWIUXBXEZUWKUXDXEZXHXHZUWJU
+        XCUWLUXEXAUXJUWIUXBGWSAUXHUXIXGXIUXJUWKUXDUWCWSAUXHUXIXJXIXKAUXBUWRXNUX
+        BXBXNWHUXBEXOXTZAUXDUWRXNUXDXBXNWIUXDEXOXTZAUXCUXEXAXPXQXRZXSAUXGUXFUHU
+        VTWRZUVQWNZUWGAUXFCWMZUVQWNZUVTWRZUXFUXPUVTWRZUVQWNZXEZUXGUXOXECUVSIUUB
+        WRZUHUXPUHXEZUXRUXGUXTUXOUYCUXQUVRUXFUVTUXPUHUVQUUCZXIUYCUXSUXNUVQUXPUH
+        UXFUVTYAYBYCAUXFUVOHYDUYACUYBYEABCDGHIUXBLMNPUXFUVNUBUFUXDUIUOUQURUSUTV
+        AVBVCVDAUVNXBXLDXMWRZUUDWRXNZUYEXBUVNUUAZWDAXBXCXNZUYEXCXNUYFUYGUUEUYHA
+        UUFXFAXLDXMXPXBUYEUVNXCXCUUGYFYJZVFVGUXKUXLUXFYGVJVKYHZACUFUUHWNYKWNZUV
+        THILMUXFUVTUVOUVSUVPUQAUXFYLXNZUVOUYKXNZACUYKUVTHILMUXFUVTUVOUVSUVPUQUY
+        JYIZUUIZAUYLUYMUYNYMYNYJVLYOAUXOUWFUHUVTWRZUVQWNZUWGAUXNUYPUVQAUXNUDUHU
+        VTWRZUYPAUXFUDUHUVTAUDUXFUXMYPXSAUYPUEUHUVTWRZUYRAUWFUEUHUVTAUEUWFUXAYP
+        XSAUYSUDKIXAWRZUUJWRZUHUVTWRZUYRAUEVUAUHUVTWCXSAVUBUYRUYTUHUVTWRZUVSUUK
+        WNZWRZUYRAUVSUULXNZUDXBXNZUYTXBXNZUHUVSYKWNZXNZYQVUBVUEXEAUFUUMXNZVUFAU
+        FUUNXNZVUKAUFUSUUOZUFUUPXTUFUVSUVSYGZUUQXTZAVUGVUHVUJWKAKIAKWBYRZAIVAYR
+        ZUURAVUJIUHUVTWRZUVSUUSWNZXEZWLWMZUHUVTWRVUSXEIVVAUVDYDUUTWLXBYEZAUHUYB
+        XNZVUJVUTVVBYQZVLAVVCVVDAUVSUVTIUHWLAVULUVSUVAXNVUMUFUVSVUNUVBXTVUQUVTY
+        GZUVCUVEUVFZUVGZYSVUIVUDUVTUVSUDUYTUHVUIYGZVVEVUDYGZUVHYFAVUEUYRVUSVUDW
+        RZUYRAVUCVUSUYRVUDAVUCKVURUVTWRZVUSAVUFKXBXNZIXBXNZVUJYQVUCVVKXEVUOAVVL
+        VVMVUJVUPVUQVVGYSVUIUVTUVSKIUHVVHVVEUVIYFAVVKKVUSUVTWRZVUSAVURVUSKUVTAV
+        UJVUTVVBVVFUVJXIAVUFVVLVVNVUSXEVUOVUPVUIUVTUVSKVUSVVHVVEVUSYGZUVMYFXRXR
+        XIAVUFUYRVUIXNVVJUYRXEVUOAVUIUVTUVSUDUHVVHVVEVUOWKVVGUVKVUIVUDUVSUYRVUS
+        VVHVVIVVOUVLYFXRXRXRYTXRYBAUWGUYQAUWFUXQUVTWRZUWFUXPUVTWRZUVQWNZXEZUWGU
+        YQXECUYBUHUYCVVPUWGVVRUYQUYCUXQUVRUWFUVTUYDXIUYCVVQUYPUVQUXPUHUWFUVTYAY
+        BYCAUWFUVOHYDVVSCUYBYEABCDGHIUWALMNPUWFUVNUBUFUWDUIUOUQURUSUTVAVBVCVDUY
+        IVFVGUWSUWTUWFYGVJVKYHZACUYKUVTHILMUWFUVTUVOUVSUVPUQUYOAUWFYLXNUYMACUYK
+        UVTHILMUWFUVTUVOUVSUVPUQVVTYIYMYNYJVLYOYPXRXRYTXR $.
+    $}
+
+    ${
+      $d .~ a $.  $d A a o p q r s $.  $d A g i o p q r s $.  $d A h s $.
+      $d A k l o p q r s $.  $d A o p q r s x $.  $d B a o p q r $.
+      $d B g i o p q r $.  $d B k l o p q r $.  $d B o p q r x $.
+      $d E a o p q r $.  $d E g i o p q r $.  $d E k l o p q r $.
+      $d E o p q r x $.  $d G e f o p q r y $.  $d G h $.  $d H s $.
+      $d I a o p q r $.  $d I g i o p q r $.  $d I k l o p q r $.
+      $d I o p q r x $.  $d J a o p q r $.  $d J g i o p q r $.
+      $d J k l o p q r $.  $d J o p q r x $.  $d K a o p q r s $.
+      $d K e f o p q r s y $.  $d K g i o p q r s $.  $d K h s $.  $d K v $.
+      $d K o p q r s x $.  $d M h $.  $d M o p q r y $.  $d M v $.
+      $d N a o r $.  $d N e f o r y $.  $d N k l o r $.  $d N o r x $.
+      $d P e f y $.  $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.
+      $d R x $.  $d S s $.  $d a o p ph q r s $.  $d g i o p ph q r s $.
+      $d h ph s $.  $d k l o p ph q r s $.  $d ph v $.  $d ph q r s x $.
+      $( Claim 2 of Theorem 6.1 AKS, Preparation for injectivity proof.
+         (Contributed by metakunt, 1-May-2025.) $)
+      aks6d1c2lem4 $p |- ( ph -> ( # ` ( H " ( NN0 ^m ( 0 ... A ) ) ) ) <_
+       ( N ^ B ) ) $=
+        ( vs vv vr vo vp vq cn0 cc0 cfz co cima chash cfv c0g cexp wcel cvv cle
+        wbr fvexd syl cv wa wceq a1i fveq2d cbs eqid adantr simpr nnnn0d mpd cn
+        cmul c1 wf adantl simprd eqeltrd oveq2d wss nn0expcld cz wne wb syl3anc
+        mpbid jca elnn0z sylibr wrex nn0red cr syl2anc r19.29vva ad7antr eqcomd
+        eqtrd reexpcld letrd recnd eqbrtrd cmap ce1 csn cfn cnvexg imaexd fmptd
+        ccnv nfv ffnd fnfund cpl1 csg fveq1d ccrg fldcrngd cmpt fvmptd cmgp cmg
+        cdvds wi wral cprimroots w3a crngmgp isprimroot biimpd simp1d eleqtrrdi
+        ccmn mgpbas cdiv cfield cprime cgcd elmapi 0nn0 cascl cplusg aks6d1c1rh
+        cv1 czrh aks6d1c1p1rcl fveval1fvcl ply1crng cmnmndd cmpo simprl oveq12d
+        crs simprr fz0ssnn0 sselda sseli ovexd prmnn nn0zd dvdsval2 nnred nnrpd
+        ovmpod nnne0d nn0ge0d divge0d nn0mulcld cxp eleqtrd aks6d1c2p1 c0 csqrt
+        wfn cuz cfl czn hashscontpowcl resqrtcld flcld sqrtge0d 0zd sylib eluz1
+        flge ax-mp fzn0 ssxpb mpbird ovelimab crg crngringd vr1cl eqcomi eleq2i
+        xpnz mulgnn0cld evl1vard evl1expd caddc ad6antr simp-6r simp-5r simp-4r
+        clt simpllr simplr ad3antrrr ad2antrr aks6d1c2lem3 ad4antr 3eqtrd eqidd
+        0z evl1subd cgrp crnggrpd grpsubid elsng wfun cdm grpsubcl eqeltrid crh
+        cpws evl1rhm rhmf ffvelcdmda pwsbas eleqtrrd elmapd fnfun fndm fvimacnv
+        ex ffn funimassd ssexd cdg1 cnzr cdr isfld biimpi simpld drngnzr deg1pw
+        3brtr3d deg1sub cidom fldidom deg1nn0clb hashbnd hashcl hashss redivcld
+        fta1g remulcld 0red 1red prmgt1 expge0d nnge1d elfzuz3 leexp2ad mullidd
+        0le1 ltled nnzd dvdsle lemuldivd lemul12ad divcan2d oveq1d eqtr2d leidd
+        mulexpd ) AUBWEWFDWGWHZUUAWHZWIZWJWKZJUEUUBWKZWKZUUHZUEWLWKZUUCZWIZWJWK
+        ZUHEWMWHZAVXRAVXQUUDWNZVXRWEWNAVXQWOWNVYEWEWNZVXRVYEWPWQZVYGAVXQVYDWOAV
+        YAVYCWOAVXTWOWNVYAWOWNAJVXSWRVXTWOUUEWSUUFZAVSVXPVYDUBAVSUUIAVXPUBAVXPW
+        OUBAOVXPUGOWTZUAWKZVXSWKZWKZWOUBAVYKVXPWNXAUGVYMWRVHUUGUUJUUKAVSWTZVXPW
+        NZXAZVYOUBWKZVXTWKZVYCWNZVYRVYDWNZVYQVYTVYSVYBXBZVYQVYSVYRUDUISWHZUCUIS
+        WHZUEUULWKZUUMWKZWHZVXSWKZWKZVYBVYQVYRVXTWUHVYQJWUGVXSJWUGXBZVYQVPXCXDU
+        UNVYQWUIVYRWUDVXSWKWKZWUKUEUUMWKZWHZVYBVYQWUGWUEXEWKZWNZWUIWUMXBVYQUEXE
+        WKZWULWUEUEWUNWUCWUFWUDVXSWUKWUKVYRVXSXFZWUEXFZWUPXFZWUNXFZAUEUUOWNZVYP
+        AUEUNUUPZXGZVYQVYRUGVYOUAWKZVXSWKZWKZWUPVYQOVYOVYNWVFVXPUBWOUBOVXPVYNUU
+        QXBVYQVHXCVYQVYKVYOXBZXAZUGVYMWVEWVHVYLWVDVXSWVHVYKVYOUAVYQWVGXHXDXDUUN
+        AVYPXHZVYQUGWVEWRUURZVYQWUPWUEUEWUNWVDVXSUGWUQWURWUSWUTWVCAUGWUPWNVYPAU
+        GUEUUSWKZXEWKZWUPAUGWVLWNZIUGWVKUUTWKZWHWVKWLWKZXBZVTWTZUGWVNWHWVOXBIWV
+        QUVAWQUVBVTWEUVCZAUGWVKIUVDWHWNZWVMWVPWVRUVEZVGAWVSWVTAWVKWVNIUGVTAWVAW
+        VKUVKWNWVBUEWVKWVKXFZUVFWSAIUPXIWVNXFZUVGUVHXJUVIWUPUEWVKWWAWUSUVLUVJXG
+        VYQGWFWMWHUHGUVMWHZWFWMWHXLWHZXKWNWVDWUNWNVYQCWUNWVNHILMWWDWVNWVDWVKVXS
+        ULVYQBCDGHIWFLMNPWWDVYOUAUEWFUHUJULUMAUEUVNWNZVYPUNXGAGUVOWNZVYPUOXGAIX
+        KWNZVYPUPXGAUHXKWNZVYPUQXGAGUHUVAWQZVYPURXGAUHIUVPWHXMXBZVYPUSXGVYPVXOW
+        EVYOXNAVYOWEVXOUVQXOVAADWEWNZVYPVBXGWFWEWNVYQUVRXCZWWLWWDXFAUHUEUWBWKUJ
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+        XVCWXKWXMXVPVUGVXNVXLWXNVYFYWGVXMYTYRYTXGYTXVSYMYTYTYTYRYR $.
+    $}
+  $}
+
+  ${
+    $d A f $.  $d A x y $.  $d B f $.  $d F f $.  $d F x y $.
+    hashnexinj.1 $e |- ( ph -> A e. Fin ) $.
+    hashnexinj.2 $e |- ( ph -> B e. Fin ) $.
+    hashnexinj.3 $e |- ( ph -> ( # ` B ) < ( # ` A ) ) $.
+    hashnexinj.4 $e |- ( ph -> F : A --> B ) $.
+    $( If the number of elements of the domain are greater than the number of
+       elements in a codomain, then there are two different values that map to
+       the same.  (Contributed by metakunt, 2-May-2025.) $)
+    hashnexinj $p |- ( ph -> E. x e. A E. y e. A
+     ( ( F ` x ) = ( F ` y ) /\ x =/= y ) ) $=
+      ( vf cfv wa wn cfn wcel syl notbid mpd wral cv wceq wne wrex wf1 wal cdom
+      wf wex wbr chash cle clt cn0 hashcl nn0red ltnled mpbid wb hashdom biimpd
+      syl2anc wi brdomg alnex sylibr cmap elmapdd f1eq1 spcgv dff13 iman anbi2i
+      co df-ne xchbinxr 2ralbii ralnex2 bitri a1i mpnanrd notnotrd ) ABUAZFLCUA
+      ZFLUBZWCWDUCZMZCDUDBDUDZADEFUHZWHNZJADEFUEZNZWIWJMZNZADEKUAZUEZNZKUFZWLAW
+      PKUIZNZWRADEUGUJZNZWTADUKLZEUKLZULUJZNZXBAXDXCUMUJXFIAXDXCAXDAEOPZXDUNPHE
+      UOQUPAXCADOPZXCUNPGDUOQUPUQURAXFXBAXEXAAXHXGXEXAUSGHDEOUTVBRVASAXGXBWTVCH
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+      KDEWOFVIRVJQSAWLWNAWKWMWKWMUSAWKWIWEWCWDUBZVCZCDTBDTZMWMBCDEFVKXLWJWIXLWG
+      NZCDTBDTWJXKXMBCDDXKWEXJNZMWGWEXJVLWFXNWEWCWDVOVMVPVQWGBCDDVRVSVMVSVTRVAS
+      WAWB $.
+  $}
+
+  ${
+    $d A w x y z $.  $d F w x y z $.  $d ph x y $.
+    hashnexinjle.1 $e |- ( ph -> A e. Fin ) $.
+    hashnexinjle.2 $e |- ( ph -> B e. Fin ) $.
+    hashnexinjle.3 $e |- ( ph -> ( # ` B ) < ( # ` A ) ) $.
+    hashnexinjle.4 $e |- ( ph -> F : A --> B ) $.
+    hashnexinjle.5 $e |- ( ph -> A C_ RR ) $.
+    $( If the number of elements of the domain are greater than the number of
+       elements in a codomain, then there are two different values that map to
+       the same.  Also we introduce a one sided inequality to simplify a
+       duplicateable proof.  (Contributed by metakunt, 2-May-2025.) $)
+    hashnexinjle $p |- ( ph -> E. x e. A E. y e. A
+     ( ( F ` x ) = ( F ` y ) /\ x < y ) ) $=
+      ( vw vz cv cfv wceq clt wbr wa wrex simpr wb fveq2 eqeq2d anbi12d fveqeq2
+      breq2 breq1 cbvrex2vw a1i biimpd imp sylib rexcom hashnexinj wcel simplrl
+      wne wo jca orcd eqcomd olcd simprr simpl simprl sselda syl adantr lttri2d
+      cr mpbid mpjaodan ex reximdvva r19.43 rexbii mpd ) ABNZFOZCNZFOZPZVSWAQRZ
+      SZCDTZBDTZWGWBVTPZWAVSQRZSZCDTZBDTZAWGUAAWLSZWEBDTCDTZWGWMLNZFOZMNZFOZPZW
+      OWQQRZSZLDTMDTZWNAWLXBAWLXBWLXBUBAWJXAWBWRPZWAWQQRZSBCMLDDVSWQPZWHXCWIXDX
+      EVTWRWBVSWQFUCUDVSWQWAQUGUEWAWOPXCWSXDWTWAWOWRFUFWAWOWQQUHUEUIUJUKULXAWEW
+      PWBPZWOWAQRZSMLCBDDWQWAPZWSXFWTXGXHWRWBWPWQWAFUCUDWQWAWOQUGUEWOVSPXFWCXGW
+      DWOVSWBFUFWOVSWAQUHUEUIUMWECBDDUNUMAWCVSWAURZSZCDTBDTZWGWLUSZABCDEFGHIJUO
+      AXKXLAXKSZWFWKUSZBDTZXLXMWEWJUSZCDTZBDTZXOAXKXRAXJXPBCDDAVSDUPZWADUPZSZSZ
+      XJXPYBXJSZWDXPWIYCWDSZWEWJYDWCWDYBWCXIWDUQYCWDUAUTVAYCWISZWJWEYEWHWIYEVTW
+      BYBWCXIWIUQVBYCWIUAUTVCYCXIWDWIUSYBWCXIVDYCVSWAYBVSVKUPZXJYBAXSSYFYBAXSAY
+      AVEZAXSXTVFUTADVKVSKVGVHVIYBWAVKUPZXJYBAXTSYHYBAXTYGAXSXTVDUTADVKWAKVGVHV
+      IVJVLVMVNVOULXQXNBDWEWJCDVPVQUMWFWKBDVPUMVNVRVM $.
+  $}
+
+  ${
+    $d .~ a $.  $d A a b c d $.  $d A b c d g i $.  $d A b c d h $.
+    $d A b c d k l $.  $d A b c d x $.  $d B a b c d $.  $d B b c d g i $.
+    $d B b c d k l $.  $d B u w $.  $d B b c d x $.  $d C a b c d $.
+    $d C b c d g i $.  $d C b c d h $.  $d C b c d k l $.  $d C b c s t $.
+    $d C b c d x $.  $d E a $.  $d E g i $.  $d E k l $.  $d E o $.  $d E t $.
+    $d E u w $.  $d E x $.  $d G e f y $.  $d G h $.  $d H b c d $.  $d K a $.
+    $d K e f y $.  $d K g i $.  $d K h $.  $d K x $.  $d L b c d t $.
+    $d L b c s t $.  $d M h $.  $d M y $.  $d N a b c d $.  $d N e f y $.
+    $d N b c d k l $.  $d N b c d x $.  $d P e f y $.  $d P k l $.  $d P x $.
+    $d R a d $.  $d R e f y $.  $d R d g i $.  $d R d h $.  $d R d k l $.
+    $d R d x $.  $d a b c d ph $.  $d g i ph $.  $d h ph $.  $d k l ph $.
+    $d o ph $.  $d ph s t $.  $d ph w $.  $d ph x $.
+    aks6d1c2a.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2a.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2a.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2a.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2a.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2a.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2a.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2a.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2a.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2a.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2a.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2a.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2a.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2a.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2a.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2a.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2a.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2a.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2a.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2a.20 $e |- ( ph -> ( Q e. Prime /\ Q || N /\ P =/= Q ) ) $.
+    $( Claim 2 of Theorem 6.1 of ~ https://www3.nd.edu/andyp/notes/AKS.pdf
+       (Contributed by metakunt, 2-May-2025.) $)
+    aks6d1c2 $p |- ( ph -> ( # ` ( H " ( NN0 ^m ( 0 ... A ) ) ) ) <_
+       ( N ^ B ) ) $=
+      ( vb vc vd vt vs vo vu vw cv clt wbr cmul co caddc wceq cn wa cn0 cc0 cfz
+      wrex cima chash cfv cle wcel simprl jca simprr cmgp cmg ad5antr cprime c1
+      cdvds wf cbs cmpt simpllr eqid simplr simpr imp syl cz cvv nfcv a1i fvexd
+      fveq2d fvmptd eqcomd eqtrd wb adantr wss cxp nn0red nn0ge0d syl2anc mpbid
+      wne cr sylibr mpbird wf1 sstrd sseq1d ad2antrr syl3anc recnd cfn breqtrrd
+      crn cmap cexp simpl cv1 cpl1 csg cfield cgcd czrh cascl cplusg cprimroots
+      wral simp-5r simp-4r aks6d1c2lem4 ex rexlimdva cmin fveq2 cbvmpt fz0ssnn0
+      crs nnnn0d cuz csqrt cfl czn hashscontpowcl resqrtcld flcld sqrtge0d flge
+      c0 0zd elnn0z eleq1d nn0zd eluz fzn0 xpnz biimpi imass2 nfv simp1d simp2d
+      simp3d aks6d1c2p2 f1f ffnd fnfund ffvelcdmda funimassd sseldd nnzd zndvds
+      ssxpb zsubcld divides biimpd mpd cdiv nnred resubcld nnrpd posdifd rpgt0d
+      crp rpred divgt0d zred mulcomd gt0ne0d divmuld breqtrd cc eqeltrd subaddd
+      elnnz reximssdv wfun fzfid xpfi imafi czring crh crg ccrg zncrng crngring
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+      AFWBYIWWEWBYIYBAVVQXNYMVVRVVSVVT $.
+  $}
+
+  ${
+    $d B x $.  $d D x $.
+    $( Special case related to ~ rspsbc .  (Contributed by metakunt,
+       5-May-2025.) $)
+    rspcsbnea $p |- ( ( A e. B /\ A. x e. B C =/= D ) ->
+     [_ A / x ]_ C =/= D ) $=
+      ( wcel wne wral csb wsbc rspsbc wceq wn wb df-ne sbcbii a1i sbcng sbceq1g
+      bitrd notbid biidd necon3bbid sylibd imp ) BCFZDEGZACHZABDIZEGZUFUHUGABJZ
+      UJUGABCKUFUKUIELZMZUJUFUKDELZMZABJZUMUKUPNUFUGUOABDEOPQUFUPUNABJZMUMUNABC
+      RUFUQULABDECSUATTUFULUIEUFULUBUCTUDUE $.
+  $}
+
+  ${
+    $d .^ x y $.  $d A x y $.  $d N x $.  $d R x y $.  $d ph x y $.
+    idomnnzpownz.1 $e |- ( ph -> R e. IDomn ) $.
+    idomnnzpownz.2 $e |- ( ph -> A e. ( Base ` R ) ) $.
+    idomnnzpownz.3 $e |- ( ph -> A =/= ( 0g ` R ) ) $.
+    idomnnzpownz.4 $e |- ( ph -> N e. NN0 ) $.
+    idomnnzpownz.5 $e |- .^ = ( .g ` ( mulGrp ` R ) ) $.
+    $( A non-zero power in an integral domain is non-zero.  (Contributed by
+       metakunt, 5-May-2025.) $)
+    idomnnzpownz $p |- ( ph -> ( N .^ A ) =/= ( 0g ` R ) ) $=
+      ( wcel wa co cfv wne wceq oveq1 eqid syl adantr vx vy cn0 c0g ancli cv c1
+      cc0 caddc neeq1d cur cmgp cbs eleqtrdi mulg0 ringidval eqtr4di cnzr cidom
+      mgpbas cdomn ccrg isidom simprbi domnnzr nzrnz eqnetrd cmnd crg idomringd
+      cplusg ringmgp simplr ad2antrr mulgnn0p1 syl3anc cmulr mgpplusg a1i oveqd
+      eqcomd mulgnn0cl eqcomi eleqtrd simpr jca domnmuln0 nn0indd ) AAEUCKZLEBD
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+      BXKJWBVPXDXFPXTXFXDXIWCVSWDXSWRWEWFXSYMWRAYMXRAYKYLGHWFTTXFCYGWQBWKXHYIXQ
+      WGVPVGVGWHS $.
+  $}
+
+  ${
+    $d A m y z $.  $d A x y z $.  $d G m y z $.  $d G x y z $.  $d N m n y z $.
+    $d N n x y z $.  $d R m n y z $.  $d R n x y z $.  $d m n ph y z $.
+    $d ph x y z $.
+    idomnnzgmulnz.1 $e |- G = ( mulGrp ` R ) $.
+    idomnnzgmulnz.2 $e |- ( ph -> R e. IDomn ) $.
+    idomnnzgmulnz.3 $e |- ( ph -> N e. Fin ) $.
+    idomnnzgmulnz.4 $e |- ( ( ph /\ n e. N ) -> A e. ( Base ` R ) ) $.
+    idomnnzgmulnz.5 $e |- ( ( ph /\ n e. N ) -> A =/= ( 0g ` R ) ) $.
+    $( A finite product of non-zero elements in an integral domain is non-zero.
+       (Contributed by metakunt, 5-May-2025.) $)
+    idomnnzgmulnz $p |- ( ph -> ( G gsum ( n e. N |-> A ) )
+    =/= ( 0g ` R ) ) $=
+      ( vm cmpt cgsu co cfv wne wceq wcel adantr vx vy vz cv c0g csn cun mpteq1
+      c0 oveq2d neeq1d mpt0 a1i eqid gsum0 eqtrd cur ringidval cidom cnzr cdomn
+      eqcomi ccrg isidom simprbi domnnzr syl nzrnz 3syl eqnetrd wss cdif wa csb
+      cplusg nfcv nfcsb1v csbeq1a cbvmpt oveq2i cbs ccmn simplbi crngmgp simprl
+      cfn ssfid wral ad2antrr simpr sseldd ralrimiva ad3antrrr rspcsbela mgpbas
+      syl2anc eleqtrd eldifi adantl wn eldifn csbeq1 gsumunsn gsummptcl equcoms
+      eqcomd jca rspcsbnea cmulr mgpplusg domnmuln0 syl3anc ex findcard2d ) AED
+      UAUDZBMZNOZCUEPZQEDUIBMZNOZXRQEDUBUDZBMZNOZXRQZEDYAUCUDZUFUGZBMZNOZXRQZED
+      FBMZNOZXRQUAUBUCFXOUIRZXQXTXRYLXPXSENDXOUIBUHUJUKXOYARZXQYCXRYMXPYBENDXOY
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+      EUEPZXRAXTEUINOZYPAXSUIENXSUIRADBULUMUJYQYPRAEYPYPUNUOUMUPAYPCUQPZXRYPYRR
+      AYRYPCYREGYRUNZURVBUMACUSSZCUTSZYRXRQHYTCVASZUUAYTCVCSZUUBCVDZVEZCVFVGCYR
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+      QUUKUVSUMWQDUULYEBXBXCUPUUKUUBUUOUVKSZUUOXRQZVMUWDUUPXRQZVMUURXRQUUJUUBYD
+      AUUBUUIAYTUUBHUUEVGTTUUKUWFUWGUUKUVKLEYAUUMUVSUVFUVHUUKUVNLYAUVPWLXDUUKUU
+      OYCXRUUKUUNYBENUUNYBRUUKLDYAUUMBUVBUVAUULDUDRBUUMBUUMRDLUVCXEXFVSUMUJUUJY
+      DWJVJXGUUKUWDUWHUWEUUJUWHYDUUJUVTBXRQZDFWHZUWHUWAAUWJUUIAUWIDFKWLTDYEFBXR
+      XHWPTXGUVKCUUQUUOUUPXRUVRCXIPZUUQCUWKEGUWKUNXJVBUUFXKXLVJXMIXN $.
+  $}
+
+  ${
+    $d .^ x y $.  $d N x $.  $d R x y $.  $d ph x y $.
+    ringexp0nn.1 $e |- ( ph -> R e. Ring ) $.
+    ringexp0nn.2 $e |- ( ph -> N e. NN ) $.
+    ringexp0nn.3 $e |- .^ = ( .g ` ( mulGrp ` R ) ) $.
+    $( Zero to the power of a positive integer is zero.  (Contributed by
+       metakunt, 5-May-2025.) $)
+    ringexp0nn $p |- ( ph -> ( N .^ ( 0g ` R ) ) = ( 0g ` R ) ) $=
+      ( vx vy cn wcel wa cfv co wceq c1 oveq1 eqeq1d syl eqid c0g ancli cv cmgp
+      caddc cbs cmnd crg ringmnd mndidcl mgpbas a1i eleqtrd mulg1 cplusg simplr
+      ad2antrr mulgnnp1 syl2anc simpr oveq1d cmulr mgpplusg eqcomi ringrz eqtrd
+      adantr nnindd ) AADJKZLDBUAMZCNZVJOZAVIFUBAHUCZVJCNZVJOPVJCNZVJOZIUCZVJCN
+      ZVJOZVQPUENZVJCNZVJOVLHIDVMPOVNVOVJVMPVJCQRVMVQOVNVRVJVMVQVJCQRVMVTOVNWAV
+      JVMVTVJCQRVMDOVNVKVJVMDVJCQRAVJBUDMZUFMZKZVPAVJBUFMZWCABUGKZVJWEKZABUHKZW
+      FEBUISWEBVJWETZVJTZUJSZWEWCOAWEBWBWBTZWIUKULUMZWCCWBVJWCTZGUNSAVQJKZLZVSL
+      ZWAVRVJWBUOMZNZVJWQWOWDWAWSOAWOVSUPAWDWOVSWMUQWCWRCWBVQVJWNGWRTURUSWQWSVJ
+      VJWRNZVJWQVRVJVJWRWPVSUTVAWPWTVJOZVSAXAWOAWHWGXAEWKWEBWRVJVJWIBVBMZWRBXBW
+      BWLXBTVCVDWJVEUSVGVGVFVFVHS $.
+
+  $}
+
+  ${
+    aks6d1p5.1 $e |- ( ph -> K e. Field ) $.
+    aks6d1p5.2 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c5.3 $e |- P = ( chr ` K ) $.
+    aks6d1c5.4 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c5.5 $e |- ( ph -> A < P ) $.
+    aks6d1c5.6 $e |- X = ( var1 ` K ) $.
+    aks6d1c5.7 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c5.8 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    .^ ( X ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+
+    ${
+      $d A g i $.  $d A x y $.  $d G x y $.  $d K g i $.  $d g i ph $.
+      $( Lemma for Claim 5 of Theorem 6.1, G defines a map into the
+         polynomials.  (Contributed by metakunt, 5-May-2025.) $)
+      aks6d1c5lem0 $p |- ( ph -> G : ( NN0 ^m ( 0 ... A ) ) -->
+     ( Base ` ( Poly1 ` K ) ) ) $=
+        ( cfv wcel eqid cn0 cc0 cfz co cmap cpl1 cmgp cv czrh cascl cplusg cmpt
+        cgsu cbs wa ccmn ccrg fldcrngd ply1crng syl crngmgp adantr cmnd cmnmndd
+        fzfid cvv nn0ex a1i ovexd elmapd biimpd imp ffvelcdmda crngringd cmnmnd
+        wf ringcmnd crg vr1cl simpl elfzelz adantl jca czring crh zringbas rhmf
+        cz zrhrhm ply1sclcl syl2anc mndcl syl3anc wceq mgpbas eleqtrd ralrimiva
+        mulgnn0cld gsummptcl eqcomi fmptd ) ADUAUBBUCUDZUEUDZHUFRZUGRZEXBEUHZDU
+        HZRZIXFHUIRZRZXDUJRZRZXDUKRZUDZFUDZULUMUDZXDUNRZGAXGXCSZUOZXPXEUNRZXQXS
+        XTEXEXBXOXTTZAXEUPSZXRAXDUQSZYBAHUQSYCAHJURZXDHXDTZUSUTZXDXEXETZVAUTVBZ
+        XSUBBVEXSXOXTSEXBXSXFXBSZUOZXTFXEXHXNYAPXSXEVCSYIXSXEYHVDVBXSXBUAXFXGAX
+        RXBUAXGVPZAXRYKAUAXBXGVFVFUAVFSAVGVHAUBBUCVIVJVKVLVMYJXNXQXTYJXDVCSZIXQ
+        SZXLXQSZXNXQSXSYLYIAYLXRAXDUPSYLAXDAXDYFVNVQXDVOUTVBVBYJHVRSZYMXSYOYIAY
+        OXRAHYDVNVBZVBZXQXDHIOYEXQTZVSUTYJYOXJHUNRZSZYNYQYJXSXFWHSZUOYTYJXSUUAX
+        SYIVTYIUUAXSXFUBBWAWBWCXSWHYSXFXIXSYOWHYSXIVPZYPYOXIWDHWEUDSUUBHXIXITWI
+        WHYSWDHXIWFYSTZWGUTUTVMUTXKXQXDHXJYSYEXKTUUCYRWJWKXQXMXDIXLYRXMTWLWMXQX
+        TWNYJXQXDXEYGYRWOZVHWPWRWQWSXTXQWNXSXQXTUUDWTVHWPQXA $.
+    $}
+
+    ${
+      aks6d1c5p1.1 $e |- ( ph -> B e. ( 0 ... A ) ) $.
+      aks6d1c5p1.2 $e |- ( ph -> C e. ( 0 ... A ) ) $.
+      $( Lemma for claim 5, evaluate the linear factor at -c to get a root.
+         (Contributed by metakunt, 5-May-2025.) $)
+      aks6d1c5lem1 $p |- ( ph -> ( B = C <-> (
+       ( ( eval1 ` K ) `
+        ( X ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` B ) ) )
+       )
+       ` ( ( ZRHom ` K ) ` ( 0 - C ) ) ) = ( 0g ` K ) ) ) $=
+        ( wceq cc0 cmin co czrh cfv cplusg c0g cpl1 cascl ce1 czring zringplusg
+        caddc eqcomi a1i oveqd 0cnd elfzelzd zcnd subadd23d subcld eqtrd fveq2d
+        addlidd eqeq1d cdvds wbr wa cz wcel cprime cn adantr prmnn syl dvds0 cc
+        nnzd subidd eqcomd simpr oveq2d breqtrd ex wn clt 1zzd zsubcld ad2antrr
+        c1 cfz cle 1e0p1 zred posdifd mpbid 0zd zltp1led eqbrtrd nnred subge02d
+        cr elfzle1 nn0red elfzle2 lelttrd zltlem1d elfzd fzm1ndvds simpll wo wb
+        syl2anc axlttri ioran bitr2d biimpd imp anassrs jca dvdsnegb negsubdi2d
+        cneg breq2d bitrd mtbird pm2.61dan con4d crg ccrg fldcrngd crngring cbs
+        impbid eqid zringbas eleqtrdi ffvelcdmd bicomd crh zrhrhm rhmghm ghmlin
+        chrdvds cghm syl3anc wf ghmf evl1vard evl1scad evl1addd simprd ) ACDUBZ
+        UCDUDUEZJUFUGZUGZCUUQUGZJUHUGZUEZJUIUGZUBZUURKUUSJUJUGZUKUGZUGZUVDUHUGZ
+        UEZJULUGZUGUGZUVBUBAUUOUUPCUMUHUGZUEZUUQUGZUVBUBZUVCAUVNUUOAUVNCDUDUEZU
+        UQUGZUVBUBZUUOAUVMUVPUVBAUVLUVOUUQAUVLUUPCUOUEZUVOAUVKUOUUPCUVKUOUBAUOU
+        VKUNUPUQURAUVRUCUVOUOUEUVOAUCDCAUSADADUCBUAUTZVAZACACUCBTUTZVAZVBAUVOAC
+        DUWBUVTVCVFVDVDVEVGAUUOEUVOVHVIZUVQAUUOUWCAUUOUWCAUUOVJZEUCUVOVHUWDEVKV
+        LZEUCVHVIUWDEUWDEVMVLZEVNVLZAUWFUUOMVOEVPZVQVTEVRVQUWDUCCCUDUEZUVOUWDUW
+        IUCUWDCACVSVLUUOUWBVOWAWBUWDCDCUDAUUOWCWDVDWEWFAUUOUWCAUUOWGZUWCWGZAUWJ
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+        WRXAUYCUYFEWHVIUYFUWOWNVIUYCUYFDEUYCUYFUYKWPAUXGUYBUXHVOZAEXDVLUYBUXRVO
+        ZUYCUCCWNVIZUYFDWNVIUYCUXSUYOAUXSUYBTVOCUCBXEVQUYCDCUYMAUXJUYBUXKVOXCWR
+        UYCDBEUYMABXDVLUYBUXQVOUYNADBWNVIZUYBAUXOUYPUADUCBXGVQVOABEWHVIUYBPVOXH
+        XHUYCUYFEUYKUYJXIWRXJEUYFXKXOAUWCUYGXNUYBAUWCEUVOYEZVHVIZUYGAUWEUXAUWCU
+        YRXNUYIUXBEUVOYCXOAUYQUYFEVHACDUWBUVTYDYFYGVOYHVQYIWFYJYPAJYKVLZUXAUWCU
+        VQXNAJYLVLUYSAJLYMZJYNVQZUXBEJUUQUVOUVBNUUQYQZUVBYQUUFXOXRYGUUAAUVMUVAU
+        VBAUUQUMJUUGUEVLZUUPUMYOUGZVLCVUDVLUVMUVAUBAUYSVUCVUAUYSUUQUMJUUBUEVLVU
+        CJUUQVUBUUCUMJUUQUUDVQVQZAUUPVKVUDAUCDAWSUVSWJZYRYSACVKVUDUWAYRYSUVKUUT
+        UMJUUPUUQCVUDVUDYQUVKYQUUTYQZUUEUUHVGYGAUVAUVJUVBAUVJUVAAUVHUVDYOUGZVLU
+        VJUVAUBAJYOUGZUVDUUTUVGJVUHKUVFUVIUURUUSUURUVIYQZUVDYQZVUIYQZVUHYQZUYTA
+        VKVUIUUPUUQAVUCVKVUIUUQUUIVUEUMJUUQVKVUIYRVULUUJVQZVUFYTZAVUIUVDJVUHUVI
+        KUURVUJQVULVUKVUMUYTVUOUUKAUVEVUIUVDJVUHUVIUUSUURVUJVUKVULUVEYQVUMUYTAV
+        KVUICUUQVUNUWAYTVUOUULUVGYQVUGUUMUUNWBVGYG $.
+    $}
+
+    ${
+      $d .^ g i $.  $d A g i $.  $d K g i $.  $d M g i $.  $d S g i $.
+      $d W i $.  $d X g i $.  $d Y g i $.  $d g i ph $.
+      aks6d1c5p3.1 $e |- ( ph -> Y e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c5p3.2 $e |- ( ph -> W e. ( 0 ... A ) ) $.
+      aks6d1c5p3.3 $e |- ( ph -> C e. NN0 ) $.
+      aks6d1c5p3.4 $e |- ( ph -> C <_ ( Y ` W ) ) $.
+      aks6d1c5p3.5 $e |- Q = ( quot1p ` K ) $.
+      aks6d1c5p3.6 $e |- S = ( algSc ` ( Poly1 ` K ) ) $.
+      aks6d1c5p3.7 $e |- M = ( mulGrp ` ( Poly1 ` K ) ) $.
+      $( Lemma for Claim 5, polynomial division with a linear power.
+         (Contributed by metakunt, 5-May-2025.) $)
+      aks6d1c5lem3 $p |- ( ph -> ( ( G ` Y ) Q
+      ( C .^ ( X ( +g ` ( Poly1 ` K ) )
+       ( S ` ( ( ZRHom ` K ) ` W ) ) ) ) ) = ( ( ( ( Y ` W ) - C )
+        .^ ( X ( +g ` ( Poly1 ` K ) ) ( S ` ( ( ZRHom ` K ) ` W ) ) ) )
+       ( +g ` M ) ( M gsum ( i e. ( ( 0 ... A ) \ { W } ) |->
+       ( ( Y ` i ) .^ ( X ( +g ` ( Poly1 ` K ) )
+       ( S ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) ) $=
+        ( cfv cmin co czrh cpl1 cplusg cc0 cfz csn cdif cv cmpt cgsu wcel cmulr
+        cbs csg cdg1 clt wbr wceq cmnd cmgp crg ccrg fldcrngd eqid ply1crng syl
+        wa crngring ringmgp eqeltrid fveq2i cz cle cn0 cmap wf cvv wb nn0ex a1i
+        ovexd elmapg syl2anc mpbid ffvelcdmd nn0zd zsubcld nn0red mpbird elnn0z
+        subge0d jca sylibr ringcmnd cmnmnd vr1cl czring ply1sclcl mndcl syl3anc
+        ccmn crh eqcomi eleqtrrdi mulgnn0cld cfn adantr eleqtrdi mgpplusg eqtrd
+        oveq1d oveq2d recnd eqcomd eqtr2d fveq1d fveq2d cmnf wne cdomn deg1xrcl
+        oveq12d cxr c1 eqeltrd deg1nn0clb eqbrtrd zrhrhm zringbas rhmf elfzelzd
+        mgpbas eqtri crngmgp fzfid diffi eldifi adantl 3syl mulgnn0cl ralrimiva
+        elfzelz gsummptcl c0g cmncom oveqd ringassd caddc npcand w3a mulgnn0dir
+        eleqtrd 3jca cun cascl simplr mpteq2dva fvmptd wss snssd undifr mpteq1d
+        sylib neldifsnd fveq2 2fveq3 gsumunsn ringgrpd aks6d1c5lem0 deg1z cidom
+        cgrp grpsubid cdr flddrngd drngdomn ply1domn isidom 0xr deg1sclle mulg1
+        0lt1 cnzr cfield isfld biimpi simpld drngnzr 1nn0 deg1pw eqtr3d breqtrd
+        xrlelttrd deg1add idomnnzpownz mnfltd cuc1p drnguc1p q1peqb ) AMOUKZCUL
+        UMZNMKUNUKZUKZFUKZKUOUKZUPUKZUMZIUMZLHUQBURUMZMUSZUTZHVAZOUKZNUYEUXOUKZ
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+        YMUHVUQVWBWUMWULVYAWVQUXLXMWQ $.
+    $}
+
+    ${
+      $d .^ g i $.  $d A g i $.  $d K g i $.  $d W i $.  $d X g i $.
+      $d Y g i $.  $d Z g i $.  $d g i ph $.
+      aks6d1c5p2.1 $e |- ( ph -> Y e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c5p2.2 $e |- ( ph -> Z e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c5p2.3 $e |- ( ph -> ( G ` Y ) = ( G ` Z ) ) $.
+      aks6d1c5p2.4 $e |- ( ph -> W e. ( 0 ... A ) ) $.
+      aks6d1c5p2.5 $e |- ( ph -> ( Y ` W ) < ( Z ` W ) ) $.
+      $( Lemma for Claim 5, contradiction of different evaluations that map to
+         the same.  (Contributed by metakunt, 5-May-2025.) $)
+      aks6d1c5lem2 $p |- ( ph -> ( 0g ` K ) =/= ( 0g ` K ) ) $=
+        ( c0g cfv cc0 cmin co czrh cpl1 cascl cplusg cmgp cfz cdif cv cmpt cgsu
+        csn ce1 cur cmulr cbs wcel wceq eqid cfield ccrg cdr simprbi syl czring
+        isfld cz crh wf crg crngringd zrhrhm zringbas rhmf 0zd elfzelzd zsubcld
+        ffvelcdmd mgpbas ccmn ply1crng crngmgp cmnmndd cle wbr cn0 cmap cvv a1i
+        wa wb elmapg syl2anc mpbid nn0zd nn0red eqcomd jca elnn0z sylibr simpld
+        leidd mulgnn0cld oveq1d fveq2d fveq1d ringidval eqcomi eqeltrd evl1expd
+        mulg0 simprd eleqtrd cfn cmnd adantr adantl syl3anc ralrimiva gsummptcl
+        eqtrd evl1muld wne fveval1fvcl eqidd clt eqnetrd aks6d1c5lem3 eleqtrdi
+        ltled caddc nn0ex ovexd recnd subidd breqtrd evl1vard evl1scad evl1addd
+        0red ply1idvr1 cmg fzfid diffi eldifi ringcmn cmnmnd ply1sclcl r19.21bi
+        3syl mndcl evl1gprodd mgpplusg cdomn cidom fldidom isidom sylib ringmgp
+        mndidcl flddrngd drngunz cprime aks6d1c5lem1 idomnnzpownz idomnnzgmulnz
+        eldifsni necon3bid domnmuln0 necomd cq1p 3eqtrd resubcld posdifd rhmghm
+        cghm ghmlin zringplusg oveqd 0cnd zcnd npcand zrh0 eqtr3d cn ringexp0nn
+        oveq2d elnnz ringlzd neeqtrd ) AHUFUGZUHIUIUJZHUKUGZUGZIKUGZUXDUIUJZJIU
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+        UXPYCVFAUHBUULUXNUXOUUMVMZAUYBUYOVFZEUXPAUXQUXPVFZWSZUYOFUXMUXRUYAVUTSA
+        UXMYDVFVXMVVDYEZVXNUXNWOUXQKAVVIVXMVVNYEVXMUXQUXNVFAUXQUXNUXOUUNYFZWGZV
+        XNUXGYDVFZJUYOVFZUXTUYOVFZUYAUYOVFZAVXRVXMAUXGWIVFZVXRAUXGVSVFVYBAUXGVV
+        BVTUXGUUOVMUXGUUPVMYEAVXSVXMAVXSUXCJUYFUGUGUXCVGVWEXJYEVXNVUGUXSUYPVFVX
+        TAVUGVXMVUHYEZVXNVPUYPUXQUXBVXNVUGVUEVUFVYCVUJVULUUSVXNUXQUHBVXPWEWGUXH
+        UYOUXGHUXSUYPUYRVWFUYSUYTUUQXBUYOUXJUXGJUXTUYTVWGUUTYGZXLZYHZYIAEUYPUXG
+        UXMHUYJUYOUYBUXPUYFUXCUYQUYRVUSUYSUYTVXFVUDVUPAVXLEUXPAVXLEUXPVYFUURYHV
+        XKUVAXGUXGVDUGZUYDUXGVYGUXMVUSVYGVHUVBXQZUYMVHZYKYAAHUVCVFZUYIUYPVFZUYI
+        UWTYLZWSUYLUYPVFZUYLUWTYLZWSUYNUWTYLAVUBVYJAHUVDVFZVUBVYJWSAVUAVYOMHUVE
+        VMZHUVFUVGYAAVYKVYLAUYIVXDUYPUYIVXDVGAVXJWRAUYJYDVFZVXDUYPVFAVUGVYQVUHH
+        UYJVXFUVHVMUYPUYJVXDVXGVXHUVIVMXRAVUCVYLAHMUVJHUYIUWTUWTVHZVXIUVKVMXGAV
+        YMVYNAUYPEUYJUXPUYKVXGAVUBUYJWIVFVUDHUYJVXFWKVMVXKAUYKUYPVFEUXPVXNUYPUX
+        GHUYOUYBUYFUXCUYQUYRUYSUYTAVUBVXMVUDYEZAUXCUYPVFVXMVUPYEZVYEYMZYHYIAUYK
+        HEUYJUXPVXFVYPVXKWUAVXNUYKUXRUXCUYAUYFUGUGZVXAUJZUWTVXNVXLUYKWUCVGVXNUY
+        PUXGHFUYOVXAUYAUXRUYFWUBUXCUYQUYRUYSUYTVYSVYTVXNVYAWUBWUBVGVXNUYAUXMVEU
+        GZUYOVXNUYAUYOWUDVYDUYOWUDVGVXNVUTWRYBZWUDUYOVGVXNUYOWUDVUTXQWRZYBZVXNW
+        UBYNXGSVXCVXQXSYAVXNWUBHVXAUXRAVYOVXMVYPYEVXNUYPUXGHUYOUYAUYFUXCUYQUYRU
+        YSUYTVYSVYTWUGYMVXNUXQIYLZWUBUWTYLVXMWUHAUXQUXNIUVPYFVXNUXQIWUBUWTVXNBU
+        XQICDEFGHJAVUAVXMMYEACUVLVFVXMNYEOABWOVFVXMPYEABCYOWNVXMQYERSTVXPAIUXNV
+        FVXMUDYEUVMUVQXCVXQVXCUVNYPUVOXGUYPHUYMUYIUYLUWTUYSVYIVYRUVRYGYPUVSAUYH
+        UXCILUGZUXDUIUJZUXKFUJZUXMEUXPUXQLUGZUYAFUJZUSUTUJZUYDUJZUYFUGZUGZUWTAU
+        XCUYGWUPAUYEWUOUYFAUYEKGUGZUXDUXKFUJZHUVTUGZUJZLGUGZWUSWUTUJWUOAWVAUYEA
+        BUXDCWUTUXHDEFGHUXMIJKMNOPQRSTUAUDVVOAUXDVVRXKWUTVHZVWFVUSYQXFAWURWVBWU
+        SWUTUCXMABUXDCWUTUXHDEFGHUXMIJLMNOPQRSTUBUDVVOAUXDWUIVVRAWUIAUXNWOILALV
+        VGVFZUXNWOLVRZUBAVVJVVKWVDWVEWTVVLVVMWOUXNLWQWQXAXBXCZUDWGZXEZUEYSWVCVW
+        FVUSYQUWAXNXOAWUQUWTUXCWUNUYFUGUGZUYMUJZUWTAWUOUYOVFWUQWVJVGAUYPUXGHUYD
+        UYMUYOWUKWUNUYFUWTWVIUXCUYQUYRUYSUYTVUDVUPAWUKUYOVFZUXCWUKUYFUGUGZUWTVG
+        AWVKWVLWUJVWDVXAUJZVGZAUYPUXGHFUYOVXAUXKWUJUYFVWDUXCUYQUYRUYSUYTVUDVUPV
+        WISVXCAWUJVPVFZUHWUJWMWNZWSWUJWOVFAWVOWVPAWUIUXDAWUIWVGXDVVPWFAUHWUJVVQ
+        AWUIUXDWVHVVRUWBAUXDWUIYOWNUHWUJYOWNZUEAUXDWUIVVRWVHUWCXCZYSXGWUJXHXIZX
+        SZXJAWVLWVMUWTAWVKWVNWVTYAAWVMWUJUWTVXAUJUWTAVWDUWTWUJVXAAUXAIVNUNUGZUJ
+        ZUXBUGZVWDUWTAUXBVNHUWEUJVFZUXAVNVEUGZVFIWWEVFWWCVWDVGAVUEWWDVUKVNHUXBU
+        WDVMAUXAVPWWEVUOWBYRAIVPWWEVUNWBYRWWAVWCVNHUXAUXBIWWEWWEVHWWAVHVWHUWFYG
+        AWWCUHUXBUGZUWTAWWCUXAIYTUJZUXBUGWWFAWWBWWGUXBAWWAYTUXAIWWAYTVGAYTWWAUW
+        GXQWRUWHXNAWWGUHUXBAUHIAUWIAIVUNUWJUWKXNYJAVUGWWFUWTVGVUHHUXBUWTVUIVYRU
+        WLVMYJUWMUWPAHVXAWUJVUHAWVOWVQWSWUJUWNVFAWVOWVQAWUJWVSXDWVRXGWUJUWQXIVX
+        CUWOYJYJXGAWUNUYOVFWVIWVIVGAUYOEUXMUXPWUMVUTVVCVXKAWUMUYOVFEUXPVXNWUMWU
+        DUYOVXNWUDFUXMWULUYAWUDVHSVXOVXNUXNWOUXQLAWVEVXMWVFYEVXPWGWUEXLWUFYBYHY
+        IZAWVIYNXGVYHVYIYKYAAUYPHUYMWVIUWTUYSVYIVYRVUHAUYPUXGHUYOWUNUYFUXCUYQUY
+        RUYSUYTVUDVUPWWHYMUWRYJYJUWS $.
+    $}
+
+    ${
+      $d A g i $.  $d A x y z $.  $d G x y z $.  $d K g i $.  $d K z $.
+      $d g i ph $.  $d ph x y z $.  $d .^ g i $.  $d A g i x y z $.
+      $d G g i x y z $.  $d K g i z $.  $d X g i $.  $d g i ph x y z $.
+      $( Claim 5 of Theorem 6.1 ~ https://www3.nd.edu/%7eandyp/notes/AKS.pdf .
+         The mapping defined by ` G ` is injective.  (Contributed by metakunt,
+         5-May-2025.) $)
+      aks6d1c5 $p |- ( ph -> G : ( NN0 ^m ( 0 ... A ) ) -1-1->
+     ( Base ` ( Poly1 ` K ) ) ) $=
+        ( cfv wa wcel vx vy vz cn0 cc0 cfz co cmap cpl1 cbs wf cv wceq wral wf1
+        wi cmgp czrh cascl cplusg cmpt cgsu eqid ccmn fldcrngd ply1crng crngmgp
+        ccrg syl adantr fzfid cmnd cmnmndd nn0ex ovexd elmapd biimpd ffvelcdmda
+        cvv a1i imp crngringd ringcmnd cmnmnd crg vr1cl cz simpl elfzelz adantl
+        jca czring zrhrhm zringbas rhmf ply1sclcl syl2anc mndcl syl3anc eleqtrd
+        crh mgpbas mulgnn0cld ralrimiva gsummptcl eqcomi fmptd wn wne c0g eqidd
+        wrex wo simpr neneqd wfn simp-4r mpbid ffn simpllr eqfnfv2 notbid ianor
+        wb mpd sylib notnotd orcnd rexnal sylibr rexbii clt wbr nn0red ad2antrr
+        df-ne ad6antr simplr aks6d1c5lem2 ex simprl simprr jca31 lttri2d cfield
+        cprime eqcomd jaodan sylbid rexlimddv pm2.65da notbii notnotb dff13 ) A
+        UDUEBUFUGZUHUGZHUIRZUJRZGUKZUAULZGRZUBULZGRZUMZUUTUVBUMZUPZUBUUPUNZUAUU
+        PUNZSUUPUURGUOAUUSUVHADUUPUUQUQRZEUUOEULZDULZRZIUVJHURRZRZUUQUSRZRZUUQU
+        TRZUGZFUGZVAVBUGZUURGAUVKUUPTZSZUVTUVIUJRZUURUWBUWCEUVIUUOUVSUWCVCZAUVI
+        VDTZUWAAUUQVHTZUWEAHVHTUWFAHJVEZUUQHUUQVCZVFVIZUUQUVIUVIVCZVGVIVJZUWBUE
+        BVKUWBUVSUWCTEUUOUWBUVJUUOTZSZUWCFUVIUVLUVRUWDPUWBUVIVLTUWLUWBUVIUWKVMV
+        JUWBUUOUDUVJUVKAUWAUUOUDUVKUKZAUWAUWNAUDUUOUVKVSVSUDVSTZAVNVTAUEBUFVOVP
+        VQWAVRUWMUVRUURUWCUWMUUQVLTZIUURTZUVPUURTZUVRUURTUWBUWPUWLAUWPUWAAUUQVD
+        TUWPAUUQAUUQUWIWBWCUUQWDVIVJVJUWMHWETZUWQUWBUWSUWLAUWSUWAAHUWGWBVJZVJZU
+        URUUQHIOUWHUURVCZWFVIUWMUWSUVNHUJRZTZUWRUXAUWMUWBUVJWGTZSUXDUWMUWBUXEUW
+        BUWLWHUWLUXEUWBUVJUEBWIWJWKUWBWGUXCUVJUVMUWBUWSWGUXCUVMUKZUWTUWSUVMWLHX
+        AUGTUXFHUVMUVMVCWMWGUXCWLHUVMWNUXCVCZWOVIVIVRVIUVOUURUUQHUVNUXCUWHUVOVC
+        UXGUXBWPWQUURUVQUUQIUVPUXBUVQVCWRWSUURUWCUMUWMUURUUQUVIUWJUXBXBZVTWTXCX
+        DXEUWCUURUMUWBUURUWCUXHXFVTWTQXGAUVGUAUUPAUUTUUPTZSZUVFUBUUPUXJUVBUUPTZ
+        SZUVDUVEUXLUVDSZUVEXHZXHZUVEUXMUUTUVBXIZXHUXOUXMUXPHXJRZUXQUMUXMUXPSZUX
+        QXKUXRUXQUXQUXRUCULZUUTRZUXSUVBRZXIZUXQUXQXIZUCUUOUXRUXTUYAUMZXHZUCUUOX
+        LZUYBUCUUOXLUXRUYDUCUUOUNZXHZUYFUXRUUOUUOUMZXHZUYHUXRUYIUYGSZXHZUYJUYHX
+        MUXRUXNUYLUXRUUTUVBUXMUXPXNXOUXRUXNUYLUXRUVEUYKUXRUUTUUOXPZUVBUUOXPZUVE
+        UYKYDUXRUUOUDUUTUKZUYMUXRUXIUYOAUXIUXKUVDUXPXQZUXRUDUUOUUTVSVSUWOUXRVNV
+        TZUXRUEBUFVOZVPZXRUUOUDUUTXSVIUXRUUOUDUVBUKZUYNUXRUXKUYTUXJUXKUVDUXPXTZ
+        UXRUDUUOUVBVSVSUYQUYRVPZXRUUOUDUVBXSVIUCUUOUUOUUTUVBYAWQYBVQYEUYIUYGYCY
+        FUXRUYIUXRUUOXKYGYHUYDUCUUOYIYJUYBUYEUCUUOUXTUYAYPYKYJUXRUXSUUOTZUYBSZS
+        ZUXRVUCSZUYBSUYCVUEUXRVUCUYBUXRVUDWHUXRVUCUYBUUAUXRVUCUYBUUBUUCVUFUYBUY
+        CVUFUYBUXTUYAYLYMZUYAUXTYLYMZXMZUYCVUFUXTUYAVUFUXTUXRUUOUDUXSUUTUXRUXIU
+        YOUYPUXRUXIUYOUYSVQYEVRYNVUFUYAUXRUUOUDUXSUVBUXRUXKUYTVUAUXRUXKUYTVUBVQ
+        YEVRYNUUDVUFVUIUYCVUFVUGUYCVUHVUFVUGSBCDEFGHUXSIUUTUVBAHUUETZUXIUXKUVDU
+        XPVUCVUGJYQACUUFTZUXIUXKUVDUXPVUCVUGKYQLABUDTZUXIUXKUVDUXPVUCVUGMYQABCY
+        LYMZUXIUXKUVDUXPVUCVUGNYQOPQUXRUXIVUCVUGUYPYOUXRUXKVUCVUGVUAYOUXLUVDUXP
+        VUCVUGXQUXRVUCVUGYRVUFVUGXNYSVUFVUHSZBCDEFGHUXSIUVBUUTAVUJUXIUXKUVDUXPV
+        UCVUHJYQAVUKUXIUXKUVDUXPVUCVUHKYQLAVULUXIUXKUVDUXPVUCVUHMYQAVUMUXIUXKUV
+        DUXPVUCVUHNYQOPQUXRUXKVUCVUHVUAYOUXRUXIVUCVUHUYPYOVUNUVAUVCUXLUVDUXPVUC
+        VUHXQUUGUXRVUCVUHYRVUFVUHXNYSUUHYTUUIWAVIUUJXOUUKUXPUXNUUTUVBYPUULYFUVE
+        UUMYJYTXDXDWKUAUBUUPUURGUUNYJ $.
+    $}
+  $}
+
   ${
     $( The value of 5 choose 2.  (Contributed by metakunt, 8-Jun-2024.) $)
     5bc2eq10 $p |- ( 5 _C 2 ) = ; 1 0 $=


Reviews:
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-02T07:32:23Z
user: login: tirix
body: There are 3 usages of `ad10antr` so far, and I'm afraid, I'm guilty for all of them!

Anyway, very good progress here. I believe this shows that your definitions are well chosen.
state: APPROVED
author_association: CONTRIBUTOR
submitted_at: 2025-05-02T07:35:58Z
user: login: avekens
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-02T16:56:17Z
user: login: metakunt
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-02T17:11:15Z
user: login: metakunt
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-02T17:12:58Z
user: login: metakunt
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-02T17:15:45Z
user: login: avekens
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-03T10:47:30Z
user: login: metakunt
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-05T18:35:14Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-06T07:58:41Z
user: login: tirix
body: Nice! I think we should not wait for too long before this is merged.
state: APPROVED
author_association: CONTRIBUTOR
submitted_at: 2025-05-06T08:00:40Z
user: login: metakunt
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-06T08:46:55Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-06T10:04:21Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-06T10:09:00Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-06T10:10:04Z
user: login: metakunt
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-06T10:49:05Z
user: login: metakunt
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-06T12:09:23Z
user: login: avekens
state: APPROVED
author_association: CONTRIBUTOR
submitted_at: 2025-05-06T13:49:02Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-05-06T13:56:14Z
user: login: icecream17
state: APPROVED
author_association: CONTRIBUTOR
submitted_at: 2025-05-06T14:05:20Z

Review comments:
diff_hunk: @@ -677132,6 +677132,253 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
path: set.mm
user: login: tirix
body: Usually when using this method, I avoid set variables for your `s`, `r`, `o`, etc., and I use class variables instead, normally I try to use the corresponding capital letter.
 ...which often results in conflicts: here, `S`, `R`, `P` are already used.

That would be a bit more elegant but completely unnecessary, since this is a single-use lemma.
What's the opinion of others?
created_at: 2025-05-02T07:32:22Z
author_association: CONTRIBUTOR
diff_hunk: @@ -677132,6 +677132,253 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
path: set.mm
user: login: avekens
body: I am not familiar with the naming pattern used here (aks...). If ~aks6d1c2p3 is actually a lemma, it should get the corresponding suffix (aks6d1c2p4lem?). Then I would not mind that there are setvar variable in the hypotheses.
created_at: 2025-05-02T16:56:17Z
author_association: CONTRIBUTOR
diff_hunk: @@ -677132,6 +677132,253 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
path: set.mm
user: login: metakunt
body: @tirix yeah you are right. This was done just to break up the huge antecedent.

@avekens that is the suffix. aks for the name of the theorem, 6 is theorem 6, d1 stands for dot 1. So theorem 6.1,
c2 stands for Claim 2. p3 stands for part 3. I couldn't find a better naming scheme.
created_at: 2025-05-02T17:11:15Z
author_association: CONTRIBUTOR
diff_hunk: @@ -677132,6 +677132,253 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
path: set.mm
user: login: metakunt
body: This isn't even a lemma, it's a "simple" calculation. I just couldn't justify it for the first time to not break up the proof since it got very gnarly. Carrying that many hyps in the antecedent makes natural deduction not autocomplete.
created_at: 2025-05-02T17:12:58Z
author_association: CONTRIBUTOR
diff_hunk: @@ -677132,6 +677132,253 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
path: set.mm
user: login: metakunt
body: Also this was the first time I could justify cutting several hypotheses and inline the definitions.
created_at: 2025-05-02T17:15:44Z
author_association: CONTRIBUTOR
diff_hunk: @@ -677132,6 +677132,253 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
path: set.mm
user: login: avekens
body: > This isn't even a lemma, it's a "simple" calculation. I just couldn't justify it for the first time to not break up the proof since it got very gnarly. Carrying that many hyps in the antecedent makes natural deduction not autocomplete.

We use "lemma" as described in Wikipedia: 

> a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem".

In this meaning, a simple calculation extracted from a proof can also be called a lemma. The comment should start with "Lemma for ~ <theorem for which the lemma is used> .", and its label should be prefixed with "lem" (or "lem<No>" if there are more than one lemma). Then such theorems are suppresses in theorem lists.

See our conventions, item "Organizing proofs."


created_at: 2025-05-03T10:47:29Z
author_association: CONTRIBUTOR
diff_hunk: @@ -677132,6 +677132,253 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
path: set.mm
user: login: metakunt
body: I have applied the suggestions. Could you please check?
created_at: 2025-05-05T18:35:14Z
author_association: CONTRIBUTOR
diff_hunk: @@ -677236,6 +677236,863 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
+      aks6d1c2p3.7 $e |- ( ph -> I = ( p E q ) ) $.
+      aks6d1c2p3.8 $e |- ( ph -> I e. NN0 ) $.
+      $( Lemma for ~ aks6d1c2 to simplify context.  (Contributed by metakunt,
+         1-May-2025.) $)
+      aks6d1c2p3lem $p |- ( ph -> ( J ( .g ` ( mulGrp ` K ) ) ( (
+      ( eval1 ` K ) ` ( G ` s ) ) ` M ) ) = ( I ( .g ` ( mulGrp ` K ) ) ( (
+        ( eval1 ` K ) ` ( G ` s ) ) ` M ) ) ) $=
+        ( vv cv cfv ce1 cmgp cmg co cexp cdiv cmul cn0 cvv cmpo wceq a1i simprl
+        wa oveq2d simprr oveq12d cc0 cfz wcel elfznn0 ovexd ovmpod eqtrd oveq1d
+        syl oveq2 fveq2d eqeq12d wbr wral syl2anc eqid aks6d1c1rh aks6d1c1p1rcl
+        mpbid cbs cn simpld aks6d1c1p1 rspcdva eqcomd w3a nnnn0d 3jca eqtr2d wf
+        cprimroots fveq2 cmap wb nn0ex elmapg cpl1 simprd caddc cplusg cmnd crg
+        ccrg fldcrngd crngring ringmgp nn0mulcld c0g wi ccmn crngmgp isprimroot
+        cdvds biimpd mpd simp1d mulgnn0dir mulgnn0ass simp2d mulgnn0cld mndrid
+        mulgnn0z ) AUEUHUKWMZUBWNZUFWOWNZWNZWNZUFWPWNZWQWNZWRGULWMZWSWRZUIGWTWR
+        ZRWMZWSWRZXAWRZUVRUVTWRZUDUVRUVTWRZAUEUWFUVRUVTAUEUWAUWDSWRUWFWGAQUPUWA
+        UWDXBXBGQWMZWSWRZUWCUPWMZWSWRZXAWRZUWFSXCSQUPXBXBUWMXDXEAVHXFZAUWIUWAXE
+        ZUWKUWDXEZXHXHZUWJUWBUWLUWEXAUWQUWIUWAGWSAUWOUWPXGXIUWQUWKUWDUWCWSAUWOU
+        WPXJXIXKAUWAXLEXMWRZXNUWAXBXNWEUWAEXOXTZAUWDUWRXNUWDXBXNWFUWDEXOXTZAUWB
+        UWEXAXPXQXRZXSAUWHGUNWMZWSWRZUWCUMWMZWSWRZXAWRZUVRUVTWRZUWGAUDUXFUVRUVT
+        AUDUXBUXDSWRUXFWJAQUPUXBUXDXBXBUWMUXFSXCUWNAUWIUXBXEZUWKUXDXEZXHXHZUWJU
+        XCUWLUXEXAUXJUWIUXBGWSAUXHUXIXGXIUXJUWKUXDUWCWSAUXHUXIXJXIXKAUXBUWRXNUX
+        BXBXNWHUXBEXOXTZAUXDUWRXNUXDXBXNWIUXDEXOXTZAUXCUXEXAXPXQXRZXSAUXGUXFUHU
+        VTWRZUVQWNZUWGAUXFCWMZUVQWNZUVTWRZUXFUXPUVTWRZUVQWNZXEZUXGUXOXECUVSIUUB
+        WRZUHUXPUHXEZUXRUXGUXTUXOUYCUXQUVRUXFUVTUXPUHUVQUUCZXIUYCUXSUXNUVQUXPUH
+        UXFUVTYAYBYCAUXFUVOHYDUYACUYBYEABCDGHIUXBLMNPUXFUVNUBUFUXDUIUOUQURUSUTV
+        AVBVCVDAUVNXBXLDXMWRZUUDWRXNZUYEXBUVNUUAZWDAXBXCXNZUYEXCXNUYFUYGUUEUYHA
+        UUFXFAXLDXMXPXBUYEUVNXCXCUUGYFYJZVFVGUXKUXLUXFYGVJVKYHZACUFUUHWNYKWNZUV
+        THILMUXFUVTUVOUVSUVPUQAUXFYLXNZUVOUYKXNZACUYKUVTHILMUXFUVTUVOUVSUVPUQUY
+        JYIZUUIZAUYLUYMUYNYMYNYJVLYOAUXOUWFUHUVTWRZUVQWNZUWGAUXNUYPUVQAUXNUDUHU
+        VTWRZUYPAUXFUDUHUVTAUDUXFUXMYPXSAUYPUEUHUVTWRZUYRAUWFUEUHUVTAUEUWFUXAYP
+        XSAUYSUDKIXAWRZUUJWRZUHUVTWRZUYRAUEVUAUHUVTWCXSAVUBUYRUYTUHUVTWRZUVSUUK
+        WNZWRZUYRAUVSUULXNZUDXBXNZUYTXBXNZUHUVSYKWNZXNZYQVUBVUEXEAUFUUMXNZVUFAU
+        FUUNXNZVUKAUFUSUUOZUFUUPXTUFUVSUVSYGZUUQXTZAVUGVUHVUJWKAKIAKWBYRZAIVAYR
+        ZUURAVUJIUHUVTWRZUVSUUSWNZXEZWLWMZUHUVTWRVUSXEIVVAUVDYDUUTWLXBYEZAUHUYB
+        XNZVUJVUTVVBYQZVLAVVCVVDAUVSUVTIUHWLAVULUVSUVAXNVUMUFUVSVUNUVBXTVUQUVTY
+        GZUVCUVEUVFZUVGZYSVUIVUDUVTUVSUDUYTUHVUIYGZVVEVUDYGZUVHYFAVUEUYRVUSVUDW
+        RZUYRAVUCVUSUYRVUDAVUCKVURUVTWRZVUSAVUFKXBXNZIXBXNZVUJYQVUCVVKXEVUOAVVL
+        VVMVUJVUPVUQVVGYSVUIUVTUVSKIUHVVHVVEUVIYFAVVKKVUSUVTWRZVUSAVURVUSKUVTAV
+        UJVUTVVBVVFUVJXIAVUFVVLVVNVUSXEVUOVUPVUIUVTUVSKVUSVVHVVEVUSYGZUVMYFXRXR
+        XIAVUFUYRVUIXNVVJUYRXEVUOAVUIUVTUVSUDUHVVHVVEVUOWKVVGUVKVUIVUDUVSUYRVUS
+        VVHVVIVVOUVLYFXRXRXRYTXRYBAUWGUYQAUWFUXQUVTWRZUWFUXPUVTWRZUVQWNZXEZUWGU
+        YQXECUYBUHUYCVVPUWGVVRUYQUYCUXQUVRUWFUVTUYDXIUYCVVQUYPUVQUXPUHUWFUVTYAY
+        BYCAUWFUVOHYDVVSCUYBYEABCDGHIUWALMNPUWFUVNUBUFUWDUIUOUQURUSUTVAVBVCVDUY
+        IVFVGUWSUWTUWFYGVJVKYHZACUYKUVTHILMUWFUVTUVOUVSUVPUQUYOAUWFYLXNUYMACUYK
+        UVTHILMUWFUVTUVOUVSUVPUQVVTYIYMYNYJVLYOYPXRXRYTXR $.
+    $}
+
+    ${
+      $d .~ a $.  $d A a o p q r s $.  $d A g i o p q r s $.  $d A h s $.
+      $d A k l o p q r s $.  $d A o p q r s x $.  $d B a o p q r $.
+      $d B g i o p q r $.  $d B k l o p q r $.  $d B o p q r x $.
+      $d E a o p q r $.  $d E g i o p q r $.  $d E k l o p q r $.
+      $d E o p q r x $.  $d G e f o p q r y $.  $d G h $.  $d H s $.
+      $d I a o p q r $.  $d I g i o p q r $.  $d I k l o p q r $.
+      $d I o p q r x $.  $d J a o p q r $.  $d J g i o p q r $.
+      $d J k l o p q r $.  $d J o p q r x $.  $d K a o p q r s $.
+      $d K e f o p q r s y $.  $d K g i o p q r s $.  $d K h s $.  $d K v $.
+      $d K o p q r s x $.  $d M h $.  $d M o p q r y $.  $d M v $.
+      $d N a o r $.  $d N e f o r y $.  $d N k l o r $.  $d N o r x $.
+      $d P e f y $.  $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.
+      $d R x $.  $d S s $.  $d a o p ph q r s $.  $d g i o p ph q r s $.
+      $d h ph s $.  $d k l o p ph q r s $.  $d ph v $.  $d ph q r s x $.
+      $( Claim 2 of Theorem 6.1 AKS, Preparation for injectivity proof.
+         (Contributed by metakunt, 1-May-2025.) $)
+      aks6d1c2p4 $p |- ( ph -> ( # ` ( H " ( NN0 ^m ( 0 ... A ) ) ) ) <_
+       ( N ^ B ) ) $=
+        ( vs vv vr vo vp vq cn0 cc0 cfz co cima chash cfv c0g cexp wcel cvv cle
+        wbr fvexd syl cv wa wceq a1i fveq2d cbs eqid adantr simpr nnnn0d mpd cn
+        cmul c1 wf adantl simprd eqeltrd oveq2d wss nn0expcld cz wne wb syl3anc
+        mpbid jca elnn0z sylibr wrex nn0red cr syl2anc r19.29vva ad7antr eqcomd
+        eqtrd reexpcld letrd recnd eqbrtrd cmap ce1 csn cfn cnvexg imaexd fmptd
+        ccnv nfv ffnd fnfund cpl1 csg fveq1d ccrg fldcrngd cmpt fvmptd cmgp cmg
+        cdvds wi wral cprimroots w3a crngmgp isprimroot biimpd simp1d eleqtrrdi
+        ccmn mgpbas cdiv cfield cprime cgcd elmapi 0nn0 cascl cplusg aks6d1c1rh
+        cv1 czrh aks6d1c1p1rcl fveval1fvcl ply1crng cmnmndd cmpo simprl oveq12d
+        crs simprr fz0ssnn0 sselda sseli ovexd prmnn nn0zd dvdsval2 nnred nnrpd
+        ovmpod nnne0d nn0ge0d divge0d nn0mulcld cxp eleqtrd aks6d1c2p1 c0 csqrt
+        wfn cuz cfl czn hashscontpowcl resqrtcld flcld sqrtge0d 0zd sylib eluz1
+        flge ax-mp fzn0 ssxpb mpbird ovelimab crg crngringd vr1cl eqcomi eleq2i
+        xpnz mulgnn0cld evl1vard evl1expd caddc ad6antr simp-6r simp-5r simp-4r
+        0z simpllr simplr ad3antrrr ad2antrr aks6d1c2p3lem ad4antr 3eqtrd eqidd
+        clt evl1subd cgrp crnggrpd grpsubid wfun cdm cpws grpsubcl eqeltrid crh
+        elsng evl1rhm rhmf ffvelcdmda pwsbas eleqtrrd elmapd ffn fnfun fvimacnv
+        fndm funimassd ssexd cdg1 cnzr cdr biimpi simpld drngnzr deg1pw 3brtr3d
+        ex isfld deg1sub cidom fldidom deg1nn0clb fta1g hashbnd hashcl redivcld
+        hashss remulcld 0red 1red prmgt1 ltled expge0d elfzuz3 leexp2ad mullidd
+        0le1 nnge1d nnzd dvdsle lemuldivd lemul12ad oveq1d mulexpd eqtr2d leidd
+        divcan2d ) AUBWEWFDWGWHZUUAWHZWIZWJWKZJUEUUBWKZWKZUUHZUEWLWKZUUCZWIZWJW
+        KZUHEWMWHZAVXRAVXQUUDWNZVXRWEWNAVXQWOWNVYEWEWNZVXRVYEWPWQZVYGAVXQVYDWOA
+        VYAVYCWOAVXTWOWNVYAWOWNAJVXSWRVXTWOUUEWSUUFZAVSVXPVYDUBAVSUUIAVXPUBAVXP
+        WOUBAOVXPUGOWTZUAWKZVXSWKZWKZWOUBAVYKVXPWNXAUGVYMWRVHUUGUUJUUKAVSWTZVXP
+        WNZXAZVYOUBWKZVXTWKZVYCWNZVYRVYDWNZVYQVYTVYSVYBXBZVYQVYSVYRUDUISWHZUCUI
+        SWHZUEUULWKZUUMWKZWHZVXSWKZWKZVYBVYQVYRVXTWUHVYQJWUGVXSJWUGXBZVYQVPXCXD
+        UUNVYQWUIVYRWUDVXSWKWKZWUKUEUUMWKZWHZVYBVYQWUGWUEXEWKZWNZWUIWUMXBVYQUEX
+        EWKZWULWUEUEWUNWUCWUFWUDVXSWUKWUKVYRVXSXFZWUEXFZWUPXFZWUNXFZAUEUUOWNZVY
+        PAUEUNUUPZXGZVYQVYRUGVYOUAWKZVXSWKZWKZWUPVYQOVYOVYNWVFVXPUBWOUBOVXPVYNU
+        UQXBVYQVHXCVYQVYKVYOXBZXAZUGVYMWVEWVHVYLWVDVXSWVHVYKVYOUAVYQWVGXHXDXDUU
+        NAVYPXHZVYQUGWVEWRUURZVYQWUPWUEUEWUNWVDVXSUGWUQWURWUSWUTWVCAUGWUPWNVYPA
+        UGUEUUSWKZXEWKZWUPAUGWVLWNZIUGWVKUUTWKZWHWVKWLWKZXBZVTWTZUGWVNWHWVOXBIW
+        VQUVAWQUVBVTWEUVCZAUGWVKIUVDWHWNZWVMWVPWVRUVEZVGAWVSWVTAWVKWVNIUGVTAWVA
+        WVKUVKWNWVBUEWVKWVKXFZUVFWSAIUPXIWVNXFZUVGUVHXJUVIWUPUEWVKWWAWUSUVLUVJX
+        GVYQGWFWMWHUHGUVMWHZWFWMWHXLWHZXKWNWVDWUNWNVYQCWUNWVNHILMWWDWVNWVDWVKVX
+        SULVYQBCDGHIWFLMNPWWDVYOUAUEWFUHUJULUMAUEUVNWNZVYPUNXGAGUVOWNZVYPUOXGAI
+        XKWNZVYPUPXGAUHXKWNZVYPUQXGAGUHUVAWQZVYPURXGAUHIUVPWHXMXBZVYPUSXGVYPVXO
+        WEVYOXNAVYOWEVXOUVQXOVAADWEWNZVYPVBXGWFWEWNVYQUVRXCZWWLWWDXFAUHUEUWBWKU
+        JWTUEUWCWKWKWUEUVSWKWKWUEUVTWKWHHWQUJXMDWGWHUVCZVYPVEXGABWUPGBWTWVNWHUU
+        QUEUEUWKWHWNZVYPVFXGUWAUWDXPUWEXQZVYQWUCWUNWNZVYRWUCVXSWKWKZWUKXBAWWPVY
+        PAWUCWUEUUSWKZXEWKZWUNAWWSSWWRUDUIWWSXFZVNAWWRAWUEUUOWNZWWRUVKWNAWVAWXA
+        WVBWUEUEWURUWFWSZWUEWWRWWRXFZUVFWSUWGZAUDWAWTZWBWTZRWHZXBZUDWEWNZWAWBWF
+        EWGWHZWXJAWXEWXJWNZXAZWXFWXJWNZXAZWXHXAZUDWXGWEWXNWXHXHZWXNWXGWEWNWXHWX
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+        AXVCWXKWXMXVPVUGVXKVXLWXNVYFYWGVXMYTYRYTXGYTXVSYMYTYTYTYRYR $.
+    $}
+  $}
+
+  ${
+    $d A f $.  $d A x y $.  $d B f $.  $d F f $.  $d F x y $.
+    hashnexinj.1 $e |- ( ph -> A e. Fin ) $.
+    hashnexinj.2 $e |- ( ph -> B e. Fin ) $.
+    hashnexinj.3 $e |- ( ph -> ( # ` B ) < ( # ` A ) ) $.
+    hashnexinj.4 $e |- ( ph -> F : A --> B ) $.
+    $( If the number of elements of the domain are greater than the number of
+       elements in a codomain, then there are two different values that map to
+       the same.  (Contributed by metakunt, 2-May-2025.) $)
+    hashnexinj $p |- ( ph -> E. x e. A E. y e. A
+     ( ( F ` x ) = ( F ` y ) /\ x =/= y ) ) $=
+      ( vf cfv wa wn cfn wcel syl notbid mpd wral cv wceq wne wrex wf1 wal cdom
+      wf wex wbr chash cle clt cn0 hashcl nn0red ltnled mpbid wb hashdom biimpd
+      syl2anc wi brdomg alnex sylibr cmap elmapdd f1eq1 spcgv dff13 iman anbi2i
+      co df-ne xchbinxr 2ralbii ralnex2 bitri a1i mpnanrd notnotrd ) ABUAZFLCUA
+      ZFLUBZWCWDUCZMZCDUDBDUDZADEFUHZWHNZJADEFUEZNZWIWJMZNZADEKUAZUEZNZKUFZWLAW
+      PKUIZNZWRADEUGUJZNZWTADUKLZEUKLZULUJZNZXBAXDXCUMUJXFIAXDXCAXDAEOPZXDUNPHE
+      UOQUPAXCADOPZXCUNPGDUOQUPUQURAXFXBAXEXAAXHXGXEXAUSGHDEOUTVBRVASAXGXBWTVCH
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+      KDEWOFVIRVJQSAWLWNAWKWMWKWMUSAWKWIWEWCWDUBZVCZCDTBDTZMWMBCDEFVKXLWJWIXLWG
+      NZCDTBDTWJXKXMBCDDXKWEXJNZMWGWEXJVLWFXNWEWCWDVOVMVPVQWGBCDDVRVSVMVSVTRVAS
+      WAWB $.
+  $}
+
+  ${
+    $d A w x y z $.  $d F w x y z $.  $d ph x y $.
+    hashnexinjle.1 $e |- ( ph -> A e. Fin ) $.
+    hashnexinjle.2 $e |- ( ph -> B e. Fin ) $.
+    hashnexinjle.3 $e |- ( ph -> ( # ` B ) < ( # ` A ) ) $.
+    hashnexinjle.4 $e |- ( ph -> F : A --> B ) $.
+    hashnexinjle.5 $e |- ( ph -> A C_ RR ) $.
+    $( If the number of elements of the domain are greater than the number of
+       elements in a codomain, then there are two different values that map to
+       the same.  Also we introduce a one sided inequality to simplify a
+       duplicateable proof.  (Contributed by metakunt, 2-May-2025.) $)
+    hashnexinjle $p |- ( ph -> E. x e. A E. y e. A
+     ( ( F ` x ) = ( F ` y ) /\ x < y ) ) $=
+      ( vw vz cv cfv wceq clt wbr wa wrex simpr wb fveq2 eqeq2d anbi12d fveqeq2
+      breq2 breq1 cbvrex2vw a1i biimpd imp sylib rexcom hashnexinj wcel simplrl
+      wne wo jca orcd eqcomd olcd simprr simpl simprl sselda syl adantr lttri2d
+      cr mpbid mpjaodan ex reximdvva r19.43 rexbii mpd ) ABNZFOZCNZFOZPZVSWAQRZ
+      SZCDTZBDTZWGWBVTPZWAVSQRZSZCDTZBDTZAWGUAAWLSZWEBDTCDTZWGWMLNZFOZMNZFOZPZW
+      OWQQRZSZLDTMDTZWNAWLXBAWLXBWLXBUBAWJXAWBWRPZWAWQQRZSBCMLDDVSWQPZWHXCWIXDX
+      EVTWRWBVSWQFUCUDVSWQWAQUGUEWAWOPXCWSXDWTWAWOWRFUFWAWOWQQUHUEUIUJUKULXAWEW
+      PWBPZWOWAQRZSMLCBDDWQWAPZWSXFWTXGXHWRWBWPWQWAFUCUDWQWAWOQUGUEWOVSPXFWCXGW
+      DWOVSWBFUFWOVSWAQUHUEUIUMWECBDDUNUMAWCVSWAURZSZCDTBDTZWGWLUSZABCDEFGHIJUO
+      AXKXLAXKSZWFWKUSZBDTZXLXMWEWJUSZCDTZBDTZXOAXKXRAXJXPBCDDAVSDUPZWADUPZSZSZ
+      XJXPYBXJSZWDXPWIYCWDSZWEWJYDWCWDYBWCXIWDUQYCWDUAUTVAYCWISZWJWEYEWHWIYEVTW
+      BYBWCXIWIUQVBYCWIUAUTVCYCXIWDWIUSYBWCXIVDYCVSWAYBVSVKUPZXJYBAXSSYFYBAXSAY
+      AVEZAXSXTVFUTADVKVSKVGVHVIYBWAVKUPZXJYBAXTSYHYBAXTYGAXSXTVDUTADVKWAKVGVHV
+      IVJVLVMVNVOULXQXNBDWEWJCDVPVQUMWFWKBDVPUMVNVRVM $.
+  $}
+
+  ${
+    $d .~ a $.  $d A a b c d $.  $d A b c d g i $.  $d A b c d h $.
+    $d A b c d k l $.  $d A b c d x $.  $d B a b c d $.  $d B b c d g i $.
+    $d B b c d k l $.  $d B u w $.  $d B b c d x $.  $d C a b c d $.
+    $d C b c d g i $.  $d C b c d h $.  $d C b c d k l $.  $d C b c s t $.
+    $d C b c d x $.  $d E a $.  $d E g i $.  $d E k l $.  $d E o $.  $d E t $.
+    $d E u w $.  $d E x $.  $d G e f y $.  $d G h $.  $d H b c d $.  $d K a $.
+    $d K e f y $.  $d K g i $.  $d K h $.  $d K x $.  $d L b c d t $.
+    $d L b c s t $.  $d M h $.  $d M y $.  $d N a b c d $.  $d N e f y $.
+    $d N b c d k l $.  $d N b c d x $.  $d P e f y $.  $d P k l $.  $d P x $.
+    $d R a d $.  $d R e f y $.  $d R d g i $.  $d R d h $.  $d R d k l $.
+    $d R d x $.  $d a b c d ph $.  $d g i ph $.  $d h ph $.  $d k l ph $.
+    $d o ph $.  $d ph s t $.  $d ph w $.  $d ph x $.
+    aks6d1c2a.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2a.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2a.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2a.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2a.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2a.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2a.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2a.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2a.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2a.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2a.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2a.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2a.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2a.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2a.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2a.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2a.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2a.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2a.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2a.20 $e |- ( ph -> ( Q e. Prime /\ Q || N /\ P =/= Q ) ) $.
+    $( Claim 2 of Theorem 6.1 of ~ https://www3.nd.edu/andyp/notes/AKS.pdf
+       (Contributed by metakunt, 2-May-2025.) $)
+    aks6d1c2 $p |- ( ph -> ( # ` ( H " ( NN0 ^m ( 0 ... A ) ) ) ) <_
+       ( N ^ B ) ) $=
+      ( vb vc vd vt vs vo vu vw cv clt wbr cmul co caddc wceq cn wa cn0 cc0 cfz
+      wrex cima chash cfv cle wcel simprl jca simprr cmgp cmg ad5antr cprime c1
+      cdvds wf cbs cmpt simpllr eqid simplr simpr imp syl cz cvv nfcv a1i fvexd
+      fveq2d fvmptd eqcomd eqtrd wb adantr wss cxp nn0red nn0ge0d syl2anc mpbid
+      wne cr sylibr mpbird wf1 sstrd sseq1d ad2antrr syl3anc recnd cfn breqtrrd
+      crn cmap cexp simpl cv1 cpl1 csg cfield cgcd czrh cascl cplusg cprimroots
+      wral crs simp-5r simp-4r aks6d1c2p4 ex rexlimdva cmin fveq2 cbvmpt nnnn0d
+      fz0ssnn0 c0 cuz csqrt cfl czn hashscontpowcl resqrtcld flcld sqrtge0d 0zd
+      flge elnn0z eleq1d nn0zd eluz fzn0 xpnz biimpi ssxpb imass2 simp1d simp2d
+      nfv simp3d aks6d1c2p2 ffnd fnfund ffvelcdmda funimassd sseldd nnzd zndvds
+      f1f zsubcld divides biimpd mpd cdiv nnred resubcld crp nnrpd rpred rpgt0d
+      posdifd divgt0d zred mulcomd gt0ne0d divmuld breqtrd cc eqeltrd reximssdv
+      elnnz subaddd wfun fzfid xpfi imafi czring crh crg zncrng crngring zrhrhm
+      ccrg imaexd hashclb hashcl sqrtmsq sqrtmul 1red readdcld flltp1 ltmul12ad
+      oveq1d eqbrtrd hashfz0 oveq12d hashxp wf1o ovexd xpexg mptexd cres sselda
+      wex wfn fnfvimad fmptd feqresmpt feq1d eqidd f1eq123d df-ima rneqd eqtr2d
+      f1resf1 dff1o5 f1oeq1 spcedv wi hasheqf1oi zringbas rhmf resss rnss nnssz
+      frn sseld fnfvima nnssre hashnexinjle reximddv2 r19.29vva ) AVGVOZVHVOZVP
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+      WBYIWWEWBYIYBAVVQXNYMVVRVVSVVT $.
+  $}
+
+  ${
+    $d B x $.  $d D x $.
+    $( Special case related to ~ rspsbc .  (Contributed by metakunt,
+       5-May-2025.) $)
+    rspcsbnea $p |- ( ( A e. B /\ A. x e. B C =/= D ) ->
+     [_ A / x ]_ C =/= D ) $=
+      ( wcel wne wral csb wsbc rspsbc wceq wn wb df-ne sbcbii a1i sbcng sbceq1g
+      bitrd notbid biidd necon3bbid sylibd imp ) BCFZDEGZACHZABDIZEGZUFUHUGABJZ
+      UJUGABCKUFUKUIELZMZUJUFUKDELZMZABJZUMUKUPNUFUGUOABDEOPQUFUPUNABJZMUMUNABC
+      RUFUQULABDECSUATTUFULUIEUFULUBUCTUDUE $.
+  $}
+
+  ${
+    $d .^ x y $.  $d A x y $.  $d N x $.  $d R x y $.  $d ph x y $.
+    idomnnzpownz.1 $e |- ( ph -> R e. IDomn ) $.
+    idomnnzpownz.2 $e |- ( ph -> A e. ( Base ` R ) ) $.
+    idomnnzpownz.3 $e |- ( ph -> A =/= ( 0g ` R ) ) $.
+    idomnnzpownz.4 $e |- ( ph -> N e. NN0 ) $.
+    idomnnzpownz.5 $e |- .^ = ( .g ` ( mulGrp ` R ) ) $.
+    $( A non-zero power in an integral domain is non-zero.  (Contributed by
+       metakunt, 5-May-2025.) $)
+    idomnnzpownz $p |- ( ph -> ( N .^ A ) =/= ( 0g ` R ) ) $=
+      ( wcel wa co cfv wne wceq oveq1 eqid syl adantr vx vy cn0 c0g ancli cv c1
+      cc0 caddc neeq1d cur cmgp cbs eleqtrdi mulg0 ringidval eqtr4di cnzr cidom
+      mgpbas cdomn ccrg isidom simprbi domnnzr nzrnz eqnetrd cmnd crg idomringd
+      cplusg ringmgp simplr ad2antrr mulgnn0p1 syl3anc cmulr mgpplusg a1i oveqd
+      eqcomd mulgnn0cl eqcomi eleqtrd simpr jca domnmuln0 nn0indd ) AAEUCKZLEBD
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+      WGVPVGVGWHS $.
+  $}
+
+  ${
+    $d A m y z $.  $d A x y z $.  $d G m y z $.  $d G x y z $.  $d N m n y z $.
+    $d N n x y z $.  $d R m n y z $.  $d R n x y z $.  $d m n ph y z $.
+    $d ph x y z $.
+    idomnnzgmulnz.1 $e |- G = ( mulGrp ` R ) $.
+    idomnnzgmulnz.2 $e |- ( ph -> R e. IDomn ) $.
+    idomnnzgmulnz.3 $e |- ( ph -> N e. Fin ) $.
+    idomnnzgmulnz.4 $e |- ( ( ph /\ n e. N ) -> A e. ( Base ` R ) ) $.
+    idomnnzgmulnz.5 $e |- ( ( ph /\ n e. N ) -> A =/= ( 0g ` R ) ) $.
+    $( A finite product of non-zero elements in an integral domain is non-zero.
+       (Contributed by metakunt, 5-May-2025.) $)
+    idomnnzgmulnz $p |- ( ph -> ( G gsum ( n e. N |-> A ) )
+    =/= ( 0g ` R ) ) $=
+      ( vm cmpt cgsu co cfv wne wceq wcel adantr vx vy vz cv c0g csn cun mpteq1
+      c0 oveq2d neeq1d mpt0 a1i eqid gsum0 eqtrd cur ringidval cidom cnzr cdomn
+      eqcomi ccrg isidom simprbi domnnzr syl nzrnz 3syl eqnetrd wss cdif wa csb
+      cplusg nfcv nfcsb1v csbeq1a cbvmpt oveq2i cbs ccmn simplbi crngmgp simprl
+      cfn ssfid wral ad2antrr simpr sseldd ralrimiva ad3antrrr rspcsbela mgpbas
+      syl2anc eleqtrd eldifi adantl wn eldifn csbeq1 gsumunsn gsummptcl equcoms
+      eqcomd jca rspcsbnea cmulr mgpplusg domnmuln0 syl3anc ex findcard2d ) AED
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+      QUUKUVSUMWQDUULYEBXBXCUPUUKUUBUUOUVKSZUUOXRQZVMUWDUUPXRQZVMUURXRQUUJUUBYD
+      AUUBUUIAYTUUBHUUEVGTTUUKUWFUWGUUKUVKLEYAUUMUVSUVFUVHUUKUVNLYAUVPWLXDUUKUU
+      OYCXRUUKUUNYBENUUNYBRUUKLDYAUUMBUVBUVAUULDUDRBUUMBUUMRDLUVCXEXFVSUMUJUUJY
+      DWJVJXGUUKUWDUWHUWEUUJUWHYDUUJUVTBXRQZDFWHZUWHUWAAUWJUUIAUWIDFKWLTDYEFBXR
+      XHWPTXGUVKCUUQUUOUUPXRUVRCXIPZUUQCUWKEGUWKUNXJVBUUFXKXLVJXMIXN $.
+  $}
+
+  ${
+    $d .^ x y $.  $d N x $.  $d R x y $.  $d ph x y $.
+    ringexp0nn.1 $e |- ( ph -> R e. Ring ) $.
+    ringexp0nn.2 $e |- ( ph -> N e. NN ) $.
+    ringexp0nn.3 $e |- .^ = ( .g ` ( mulGrp ` R ) ) $.
+    $( Zero to the power of a positive integer is zero.  (Contributed by
+       metakunt, 5-May-2025.) $)
+    ringexp0nn $p |- ( ph -> ( N .^ ( 0g ` R ) ) = ( 0g ` R ) ) $=
+      ( vx vy cn wcel wa cfv co wceq c1 oveq1 eqeq1d syl eqid c0g ancli cv cmgp
+      caddc cbs cmnd crg ringmnd mndidcl mgpbas a1i eleqtrd mulg1 cplusg simplr
+      ad2antrr mulgnnp1 syl2anc simpr oveq1d cmulr mgpplusg eqcomi ringrz eqtrd
+      adantr nnindd ) AADJKZLDBUAMZCNZVJOZAVIFUBAHUCZVJCNZVJOPVJCNZVJOZIUCZVJCN
+      ZVJOZVQPUENZVJCNZVJOVLHIDVMPOVNVOVJVMPVJCQRVMVQOVNVRVJVMVQVJCQRVMVTOVNWAV
+      JVMVTVJCQRVMDOVNVKVJVMDVJCQRAVJBUDMZUFMZKZVPAVJBUFMZWCABUGKZVJWEKZABUHKZW
+      FEBUISWEBVJWETZVJTZUJSZWEWCOAWEBWBWBTZWIUKULUMZWCCWBVJWCTZGUNSAVQJKZLZVSL
+      ZWAVRVJWBUOMZNZVJWQWOWDWAWSOAWOVSUPAWDWOVSWMUQWCWRCWBVQVJWNGWRTURUSWQWSVJ
+      VJWRNZVJWQVRVJVJWRWPVSUTVAWPWTVJOZVSAXAWOAWHWGXAEWKWEBWRVJVJWIBVBMZWRBXBW
+      BWLXBTVCVDWJVEUSVGVGVFVFVHS $.
+
+  $}
+
+  ${
+    aks6d1p5.1 $e |- ( ph -> K e. Field ) $.
+    aks6d1p5.2 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c5.3 $e |- P = ( chr ` K ) $.
+    aks6d1c5.4 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c5.5 $e |- ( ph -> A < P ) $.
+    aks6d1c5.6 $e |- X = ( var1 ` K ) $.
+    aks6d1c5.7 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c5.8 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    .^ ( X ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+
+    ${
+      $d A g i $.  $d A x y $.  $d G x y $.  $d K g i $.  $d g i ph $.
+      $( Lemma for Claim 5 of Theorem 6.1, G defines a map into the
+         polynomials.  (Contributed by metakunt, 5-May-2025.) $)
+      aks6d1c5p0 $p |- ( ph -> G : ( NN0 ^m ( 0 ... A ) ) -->
+     ( Base ` ( Poly1 ` K ) ) ) $=
+        ( cfv wcel eqid cn0 cc0 cfz co cmap cpl1 cmgp cv czrh cascl cplusg cmpt
+        cgsu cbs wa ccmn ccrg fldcrngd ply1crng syl crngmgp adantr cmnd cmnmndd
+        fzfid cvv nn0ex a1i ovexd elmapd biimpd imp ffvelcdmda crngringd cmnmnd
+        wf ringcmnd crg vr1cl simpl elfzelz adantl jca czring crh zringbas rhmf
+        cz zrhrhm ply1sclcl syl2anc mndcl syl3anc wceq mgpbas eleqtrd ralrimiva
+        mulgnn0cld gsummptcl eqcomi fmptd ) ADUAUBBUCUDZUEUDZHUFRZUGRZEXBEUHZDU
+        HZRZIXFHUIRZRZXDUJRZRZXDUKRZUDZFUDZULUMUDZXDUNRZGAXGXCSZUOZXPXEUNRZXQXS
+        XTEXEXBXOXTTZAXEUPSZXRAXDUQSZYBAHUQSYCAHJURZXDHXDTZUSUTZXDXEXETZVAUTVBZ
+        XSUBBVEXSXOXTSEXBXSXFXBSZUOZXTFXEXHXNYAPXSXEVCSYIXSXEYHVDVBXSXBUAXFXGAX
+        RXBUAXGVPZAXRYKAUAXBXGVFVFUAVFSAVGVHAUBBUCVIVJVKVLVMYJXNXQXTYJXDVCSZIXQ
+        SZXLXQSZXNXQSXSYLYIAYLXRAXDUPSYLAXDAXDYFVNVQXDVOUTVBVBYJHVRSZYMXSYOYIAY
+        OXRAHYDVNVBZVBZXQXDHIOYEXQTZVSUTYJYOXJHUNRZSZYNYQYJXSXFWHSZUOYTYJXSUUAX
+        SYIVTYIUUAXSXFUBBWAWBWCXSWHYSXFXIXSYOWHYSXIVPZYPYOXIWDHWEUDSUUBHXIXITWI
+        WHYSWDHXIWFYSTZWGUTUTVMUTXKXQXDHXJYSYEXKTUUCYRWJWKXQXMXDIXLYRXMTWLWMXQX
+        TWNYJXQXDXEYGYRWOZVHWPWRWQWSXTXQWNXSXQXTUUDWTVHWPQXA $.
+    $}
+
+    ${
+      aks6d1c5p1.1 $e |- ( ph -> B e. ( 0 ... A ) ) $.
+      aks6d1c5p1.2 $e |- ( ph -> C e. ( 0 ... A ) ) $.
+      $( Lemma for claim 5, evaluate the linear factor at -c to get a root.
+         (Contributed by metakunt, 5-May-2025.) $)
+      aks6d1c5p1 $p |- ( ph -> ( B = C <-> (
+       ( ( eval1 ` K ) `
+        ( X ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` B ) ) )
+       )
+       ` ( ( ZRHom ` K ) ` ( 0 - C ) ) ) = ( 0g ` K ) ) ) $=
path: set.mm
user: login: tirix
body: We have ` -u C ` to express "negative C"
created_at: 2025-05-06T07:58:41Z
author_association: CONTRIBUTOR
diff_hunk: @@ -677236,6 +677236,863 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
+      aks6d1c2p3.7 $e |- ( ph -> I = ( p E q ) ) $.
+      aks6d1c2p3.8 $e |- ( ph -> I e. NN0 ) $.
+      $( Lemma for ~ aks6d1c2 to simplify context.  (Contributed by metakunt,
+         1-May-2025.) $)
+      aks6d1c2p3lem $p |- ( ph -> ( J ( .g ` ( mulGrp ` K ) ) ( (
+      ( eval1 ` K ) ` ( G ` s ) ) ` M ) ) = ( I ( .g ` ( mulGrp ` K ) ) ( (
+        ( eval1 ` K ) ` ( G ` s ) ) ` M ) ) ) $=
+        ( vv cv cfv ce1 cmgp cmg co cexp cdiv cmul cn0 cvv cmpo wceq a1i simprl
+        wa oveq2d simprr oveq12d cc0 cfz wcel elfznn0 ovexd ovmpod eqtrd oveq1d
+        syl oveq2 fveq2d eqeq12d wbr wral syl2anc eqid aks6d1c1rh aks6d1c1p1rcl
+        mpbid cbs cn simpld aks6d1c1p1 rspcdva eqcomd w3a nnnn0d 3jca eqtr2d wf
+        cprimroots fveq2 cmap wb nn0ex elmapg cpl1 simprd caddc cplusg cmnd crg
+        ccrg fldcrngd crngring ringmgp nn0mulcld c0g wi ccmn crngmgp isprimroot
+        cdvds biimpd mpd simp1d mulgnn0dir mulgnn0ass simp2d mulgnn0cld mndrid
+        mulgnn0z ) AUEUHUKWMZUBWNZUFWOWNZWNZWNZUFWPWNZWQWNZWRGULWMZWSWRZUIGWTWR
+        ZRWMZWSWRZXAWRZUVRUVTWRZUDUVRUVTWRZAUEUWFUVRUVTAUEUWAUWDSWRUWFWGAQUPUWA
+        UWDXBXBGQWMZWSWRZUWCUPWMZWSWRZXAWRZUWFSXCSQUPXBXBUWMXDXEAVHXFZAUWIUWAXE
+        ZUWKUWDXEZXHXHZUWJUWBUWLUWEXAUWQUWIUWAGWSAUWOUWPXGXIUWQUWKUWDUWCWSAUWOU
+        WPXJXIXKAUWAXLEXMWRZXNUWAXBXNWEUWAEXOXTZAUWDUWRXNUWDXBXNWFUWDEXOXTZAUWB
+        UWEXAXPXQXRZXSAUWHGUNWMZWSWRZUWCUMWMZWSWRZXAWRZUVRUVTWRZUWGAUDUXFUVRUVT
+        AUDUXBUXDSWRUXFWJAQUPUXBUXDXBXBUWMUXFSXCUWNAUWIUXBXEZUWKUXDXEZXHXHZUWJU
+        XCUWLUXEXAUXJUWIUXBGWSAUXHUXIXGXIUXJUWKUXDUWCWSAUXHUXIXJXIXKAUXBUWRXNUX
+        BXBXNWHUXBEXOXTZAUXDUWRXNUXDXBXNWIUXDEXOXTZAUXCUXEXAXPXQXRZXSAUXGUXFUHU
+        VTWRZUVQWNZUWGAUXFCWMZUVQWNZUVTWRZUXFUXPUVTWRZUVQWNZXEZUXGUXOXECUVSIUUB
+        WRZUHUXPUHXEZUXRUXGUXTUXOUYCUXQUVRUXFUVTUXPUHUVQUUCZXIUYCUXSUXNUVQUXPUH
+        UXFUVTYAYBYCAUXFUVOHYDUYACUYBYEABCDGHIUXBLMNPUXFUVNUBUFUXDUIUOUQURUSUTV
+        AVBVCVDAUVNXBXLDXMWRZUUDWRXNZUYEXBUVNUUAZWDAXBXCXNZUYEXCXNUYFUYGUUEUYHA
+        UUFXFAXLDXMXPXBUYEUVNXCXCUUGYFYJZVFVGUXKUXLUXFYGVJVKYHZACUFUUHWNYKWNZUV
+        THILMUXFUVTUVOUVSUVPUQAUXFYLXNZUVOUYKXNZACUYKUVTHILMUXFUVTUVOUVSUVPUQUY
+        JYIZUUIZAUYLUYMUYNYMYNYJVLYOAUXOUWFUHUVTWRZUVQWNZUWGAUXNUYPUVQAUXNUDUHU
+        VTWRZUYPAUXFUDUHUVTAUDUXFUXMYPXSAUYPUEUHUVTWRZUYRAUWFUEUHUVTAUEUWFUXAYP
+        XSAUYSUDKIXAWRZUUJWRZUHUVTWRZUYRAUEVUAUHUVTWCXSAVUBUYRUYTUHUVTWRZUVSUUK
+        WNZWRZUYRAUVSUULXNZUDXBXNZUYTXBXNZUHUVSYKWNZXNZYQVUBVUEXEAUFUUMXNZVUFAU
+        FUUNXNZVUKAUFUSUUOZUFUUPXTUFUVSUVSYGZUUQXTZAVUGVUHVUJWKAKIAKWBYRZAIVAYR
+        ZUURAVUJIUHUVTWRZUVSUUSWNZXEZWLWMZUHUVTWRVUSXEIVVAUVDYDUUTWLXBYEZAUHUYB
+        XNZVUJVUTVVBYQZVLAVVCVVDAUVSUVTIUHWLAVULUVSUVAXNVUMUFUVSVUNUVBXTVUQUVTY
+        GZUVCUVEUVFZUVGZYSVUIVUDUVTUVSUDUYTUHVUIYGZVVEVUDYGZUVHYFAVUEUYRVUSVUDW
+        RZUYRAVUCVUSUYRVUDAVUCKVURUVTWRZVUSAVUFKXBXNZIXBXNZVUJYQVUCVVKXEVUOAVVL
+        VVMVUJVUPVUQVVGYSVUIUVTUVSKIUHVVHVVEUVIYFAVVKKVUSUVTWRZVUSAVURVUSKUVTAV
+        UJVUTVVBVVFUVJXIAVUFVVLVVNVUSXEVUOVUPVUIUVTUVSKVUSVVHVVEVUSYGZUVMYFXRXR
+        XIAVUFUYRVUIXNVVJUYRXEVUOAVUIUVTUVSUDUHVVHVVEVUOWKVVGUVKVUIVUDUVSUYRVUS
+        VVHVVIVVOUVLYFXRXRXRYTXRYBAUWGUYQAUWFUXQUVTWRZUWFUXPUVTWRZUVQWNZXEZUWGU
+        YQXECUYBUHUYCVVPUWGVVRUYQUYCUXQUVRUWFUVTUYDXIUYCVVQUYPUVQUXPUHUWFUVTYAY
+        BYCAUWFUVOHYDVVSCUYBYEABCDGHIUWALMNPUWFUVNUBUFUWDUIUOUQURUSUTVAVBVCVDUY
+        IVFVGUWSUWTUWFYGVJVKYHZACUYKUVTHILMUWFUVTUVOUVSUVPUQUYOAUWFYLXNUYMACUYK
+        UVTHILMUWFUVTUVOUVSUVPUQVVTYIYMYNYJVLYOYPXRXRYTXR $.
+    $}
+
+    ${
+      $d .~ a $.  $d A a o p q r s $.  $d A g i o p q r s $.  $d A h s $.
+      $d A k l o p q r s $.  $d A o p q r s x $.  $d B a o p q r $.
+      $d B g i o p q r $.  $d B k l o p q r $.  $d B o p q r x $.
+      $d E a o p q r $.  $d E g i o p q r $.  $d E k l o p q r $.
+      $d E o p q r x $.  $d G e f o p q r y $.  $d G h $.  $d H s $.
+      $d I a o p q r $.  $d I g i o p q r $.  $d I k l o p q r $.
+      $d I o p q r x $.  $d J a o p q r $.  $d J g i o p q r $.
+      $d J k l o p q r $.  $d J o p q r x $.  $d K a o p q r s $.
+      $d K e f o p q r s y $.  $d K g i o p q r s $.  $d K h s $.  $d K v $.
+      $d K o p q r s x $.  $d M h $.  $d M o p q r y $.  $d M v $.
+      $d N a o r $.  $d N e f o r y $.  $d N k l o r $.  $d N o r x $.
+      $d P e f y $.  $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.
+      $d R x $.  $d S s $.  $d a o p ph q r s $.  $d g i o p ph q r s $.
+      $d h ph s $.  $d k l o p ph q r s $.  $d ph v $.  $d ph q r s x $.
+      $( Claim 2 of Theorem 6.1 AKS, Preparation for injectivity proof.
+         (Contributed by metakunt, 1-May-2025.) $)
+      aks6d1c2p4 $p |- ( ph -> ( # ` ( H " ( NN0 ^m ( 0 ... A ) ) ) ) <_
+       ( N ^ B ) ) $=
+        ( vs vv vr vo vp vq cn0 cc0 cfz co cima chash cfv c0g cexp wcel cvv cle
+        wbr fvexd syl cv wa wceq a1i fveq2d cbs eqid adantr simpr nnnn0d mpd cn
+        cmul c1 wf adantl simprd eqeltrd oveq2d wss nn0expcld cz wne wb syl3anc
+        mpbid jca elnn0z sylibr wrex nn0red cr syl2anc r19.29vva ad7antr eqcomd
+        eqtrd reexpcld letrd recnd eqbrtrd cmap ce1 csn cfn cnvexg imaexd fmptd
+        ccnv nfv ffnd fnfund cpl1 csg fveq1d ccrg fldcrngd cmpt fvmptd cmgp cmg
+        cdvds wi wral cprimroots w3a crngmgp isprimroot biimpd simp1d eleqtrrdi
+        ccmn mgpbas cdiv cfield cprime cgcd elmapi 0nn0 cascl cplusg aks6d1c1rh
+        cv1 czrh aks6d1c1p1rcl fveval1fvcl ply1crng cmnmndd cmpo simprl oveq12d
+        crs simprr fz0ssnn0 sselda sseli ovexd prmnn nn0zd dvdsval2 nnred nnrpd
+        ovmpod nnne0d nn0ge0d divge0d nn0mulcld cxp eleqtrd aks6d1c2p1 c0 csqrt
+        wfn cuz cfl czn hashscontpowcl resqrtcld flcld sqrtge0d 0zd sylib eluz1
+        flge ax-mp fzn0 ssxpb mpbird ovelimab crg crngringd vr1cl eqcomi eleq2i
+        xpnz mulgnn0cld evl1vard evl1expd caddc ad6antr simp-6r simp-5r simp-4r
+        0z simpllr simplr ad3antrrr ad2antrr aks6d1c2p3lem ad4antr 3eqtrd eqidd
+        clt evl1subd cgrp crnggrpd grpsubid wfun cdm cpws grpsubcl eqeltrid crh
+        elsng evl1rhm rhmf ffvelcdmda pwsbas eleqtrrd elmapd ffn fnfun fvimacnv
+        fndm funimassd ssexd cdg1 cnzr cdr biimpi simpld drngnzr deg1pw 3brtr3d
+        ex isfld deg1sub cidom fldidom deg1nn0clb fta1g hashbnd hashcl redivcld
+        hashss remulcld 0red 1red prmgt1 ltled expge0d elfzuz3 leexp2ad mullidd
+        0le1 nnge1d nnzd dvdsle lemuldivd lemul12ad oveq1d mulexpd eqtr2d leidd
+        divcan2d ) AUBWEWFDWGWHZUUAWHZWIZWJWKZJUEUUBWKZWKZUUHZUEWLWKZUUCZWIZWJW
+        KZUHEWMWHZAVXRAVXQUUDWNZVXRWEWNAVXQWOWNVYEWEWNZVXRVYEWPWQZVYGAVXQVYDWOA
+        VYAVYCWOAVXTWOWNVYAWOWNAJVXSWRVXTWOUUEWSUUFZAVSVXPVYDUBAVSUUIAVXPUBAVXP
+        WOUBAOVXPUGOWTZUAWKZVXSWKZWKZWOUBAVYKVXPWNXAUGVYMWRVHUUGUUJUUKAVSWTZVXP
+        WNZXAZVYOUBWKZVXTWKZVYCWNZVYRVYDWNZVYQVYTVYSVYBXBZVYQVYSVYRUDUISWHZUCUI
+        SWHZUEUULWKZUUMWKZWHZVXSWKZWKZVYBVYQVYRVXTWUHVYQJWUGVXSJWUGXBZVYQVPXCXD
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+        EWKZWULWUEUEWUNWUCWUFWUDVXSWUKWUKVYRVXSXFZWUEXFZWUPXFZWUNXFZAUEUUOWNZVY
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+        NAVYPXHZVYQUGWVEWRUURZVYQWUPWUEUEWUNWVDVXSUGWUQWURWUSWUTWVCAUGWUPWNVYPA
+        UGUEUUSWKZXEWKZWUPAUGWVLWNZIUGWVKUUTWKZWHWVKWLWKZXBZVTWTZUGWVNWHWVOXBIW
+        VQUVAWQUVBVTWEUVCZAUGWVKIUVDWHWNZWVMWVPWVRUVEZVGAWVSWVTAWVKWVNIUGVTAWVA
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+        JWTUEUWCWKWKWUEUVSWKWKWUEUVTWKWHHWQUJXMDWGWHUVCZVYPVEXGABWUPGBWTWVNWHUU
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+        EWGWHZWXJAWXEWXJWNZXAZWXFWXJWNZXAZWXHXAZUDWXGWEWXNWXHXHZWXNWXGWEWNWXHWX
+        NWXGGWXEWMWHZWWCWXFWMWHZXLWHZWEWXNQUKWXEWXFWEWEGQWTZWMWHZWWCUKWTZWMWHZX
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+        AXVCWXKWXMXVPVUGVXKVXLWXNVYFYWGVXMYTYRYTXGYTXVSYMYTYTYTYRYR $.
+    $}
+  $}
+
+  ${
+    $d A f $.  $d A x y $.  $d B f $.  $d F f $.  $d F x y $.
+    hashnexinj.1 $e |- ( ph -> A e. Fin ) $.
+    hashnexinj.2 $e |- ( ph -> B e. Fin ) $.
+    hashnexinj.3 $e |- ( ph -> ( # ` B ) < ( # ` A ) ) $.
+    hashnexinj.4 $e |- ( ph -> F : A --> B ) $.
+    $( If the number of elements of the domain are greater than the number of
+       elements in a codomain, then there are two different values that map to
+       the same.  (Contributed by metakunt, 2-May-2025.) $)
+    hashnexinj $p |- ( ph -> E. x e. A E. y e. A
+     ( ( F ` x ) = ( F ` y ) /\ x =/= y ) ) $=
+      ( vf cfv wa wn cfn wcel syl notbid mpd wral cv wceq wne wrex wf1 wal cdom
+      wf wex wbr chash cle clt cn0 hashcl nn0red ltnled mpbid wb hashdom biimpd
+      syl2anc wi brdomg alnex sylibr cmap elmapdd f1eq1 spcgv dff13 iman anbi2i
+      co df-ne xchbinxr 2ralbii ralnex2 bitri a1i mpnanrd notnotrd ) ABUAZFLCUA
+      ZFLUBZWCWDUCZMZCDUDBDUDZADEFUHZWHNZJADEFUEZNZWIWJMZNZADEKUAZUEZNZKUFZWLAW
+      PKUIZNZWRADEUGUJZNZWTADUKLZEUKLZULUJZNZXBAXDXCUMUJXFIAXDXCAXDAEOPZXDUNPHE
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+      KDEWOFVIRVJQSAWLWNAWKWMWKWMUSAWKWIWEWCWDUBZVCZCDTBDTZMWMBCDEFVKXLWJWIXLWG
+      NZCDTBDTWJXKXMBCDDXKWEXJNZMWGWEXJVLWFXNWEWCWDVOVMVPVQWGBCDDVRVSVMVSVTRVAS
+      WAWB $.
+  $}
+
+  ${
+    $d A w x y z $.  $d F w x y z $.  $d ph x y $.
+    hashnexinjle.1 $e |- ( ph -> A e. Fin ) $.
+    hashnexinjle.2 $e |- ( ph -> B e. Fin ) $.
+    hashnexinjle.3 $e |- ( ph -> ( # ` B ) < ( # ` A ) ) $.
+    hashnexinjle.4 $e |- ( ph -> F : A --> B ) $.
+    hashnexinjle.5 $e |- ( ph -> A C_ RR ) $.
+    $( If the number of elements of the domain are greater than the number of
+       elements in a codomain, then there are two different values that map to
+       the same.  Also we introduce a one sided inequality to simplify a
+       duplicateable proof.  (Contributed by metakunt, 2-May-2025.) $)
+    hashnexinjle $p |- ( ph -> E. x e. A E. y e. A
+     ( ( F ` x ) = ( F ` y ) /\ x < y ) ) $=
+      ( vw vz cv cfv wceq clt wbr wa wrex simpr wb fveq2 eqeq2d anbi12d fveqeq2
+      breq2 breq1 cbvrex2vw a1i biimpd imp sylib rexcom hashnexinj wcel simplrl
+      wne wo jca orcd eqcomd olcd simprr simpl simprl sselda syl adantr lttri2d
+      cr mpbid mpjaodan ex reximdvva r19.43 rexbii mpd ) ABNZFOZCNZFOZPZVSWAQRZ
+      SZCDTZBDTZWGWBVTPZWAVSQRZSZCDTZBDTZAWGUAAWLSZWEBDTCDTZWGWMLNZFOZMNZFOZPZW
+      OWQQRZSZLDTMDTZWNAWLXBAWLXBWLXBUBAWJXAWBWRPZWAWQQRZSBCMLDDVSWQPZWHXCWIXDX
+      EVTWRWBVSWQFUCUDVSWQWAQUGUEWAWOPXCWSXDWTWAWOWRFUFWAWOWQQUHUEUIUJUKULXAWEW
+      PWBPZWOWAQRZSMLCBDDWQWAPZWSXFWTXGXHWRWBWPWQWAFUCUDWQWAWOQUGUEWOVSPXFWCXGW
+      DWOVSWBFUFWOVSWAQUHUEUIUMWECBDDUNUMAWCVSWAURZSZCDTBDTZWGWLUSZABCDEFGHIJUO
+      AXKXLAXKSZWFWKUSZBDTZXLXMWEWJUSZCDTZBDTZXOAXKXRAXJXPBCDDAVSDUPZWADUPZSZSZ
+      XJXPYBXJSZWDXPWIYCWDSZWEWJYDWCWDYBWCXIWDUQYCWDUAUTVAYCWISZWJWEYEWHWIYEVTW
+      BYBWCXIWIUQVBYCWIUAUTVCYCXIWDWIUSYBWCXIVDYCVSWAYBVSVKUPZXJYBAXSSYFYBAXSAY
+      AVEZAXSXTVFUTADVKVSKVGVHVIYBWAVKUPZXJYBAXTSYHYBAXTYGAXSXTVDUTADVKWAKVGVHV
+      IVJVLVMVNVOULXQXNBDWEWJCDVPVQUMWFWKBDVPUMVNVRVM $.
+  $}
+
+  ${
+    $d .~ a $.  $d A a b c d $.  $d A b c d g i $.  $d A b c d h $.
+    $d A b c d k l $.  $d A b c d x $.  $d B a b c d $.  $d B b c d g i $.
+    $d B b c d k l $.  $d B u w $.  $d B b c d x $.  $d C a b c d $.
+    $d C b c d g i $.  $d C b c d h $.  $d C b c d k l $.  $d C b c s t $.
+    $d C b c d x $.  $d E a $.  $d E g i $.  $d E k l $.  $d E o $.  $d E t $.
+    $d E u w $.  $d E x $.  $d G e f y $.  $d G h $.  $d H b c d $.  $d K a $.
+    $d K e f y $.  $d K g i $.  $d K h $.  $d K x $.  $d L b c d t $.
+    $d L b c s t $.  $d M h $.  $d M y $.  $d N a b c d $.  $d N e f y $.
+    $d N b c d k l $.  $d N b c d x $.  $d P e f y $.  $d P k l $.  $d P x $.
+    $d R a d $.  $d R e f y $.  $d R d g i $.  $d R d h $.  $d R d k l $.
+    $d R d x $.  $d a b c d ph $.  $d g i ph $.  $d h ph $.  $d k l ph $.
+    $d o ph $.  $d ph s t $.  $d ph w $.  $d ph x $.
+    aks6d1c2a.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2a.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2a.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2a.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2a.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2a.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2a.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2a.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2a.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2a.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2a.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2a.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2a.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2a.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2a.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2a.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2a.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2a.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2a.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2a.20 $e |- ( ph -> ( Q e. Prime /\ Q || N /\ P =/= Q ) ) $.
+    $( Claim 2 of Theorem 6.1 of ~ https://www3.nd.edu/andyp/notes/AKS.pdf
+       (Contributed by metakunt, 2-May-2025.) $)
+    aks6d1c2 $p |- ( ph -> ( # ` ( H " ( NN0 ^m ( 0 ... A ) ) ) ) <_
+       ( N ^ B ) ) $=
+      ( vb vc vd vt vs vo vu vw cv clt wbr cmul co caddc wceq cn wa cn0 cc0 cfz
+      wrex cima chash cfv cle wcel simprl jca simprr cmgp cmg ad5antr cprime c1
+      cdvds wf cbs cmpt simpllr eqid simplr simpr imp syl cz cvv nfcv a1i fvexd
+      fveq2d fvmptd eqcomd eqtrd wb adantr wss cxp nn0red nn0ge0d syl2anc mpbid
+      wne cr sylibr mpbird wf1 sstrd sseq1d ad2antrr syl3anc recnd cfn breqtrrd
+      crn cmap cexp simpl cv1 cpl1 csg cfield cgcd czrh cascl cplusg cprimroots
+      wral crs simp-5r simp-4r aks6d1c2p4 ex rexlimdva cmin fveq2 cbvmpt nnnn0d
+      fz0ssnn0 c0 cuz csqrt cfl czn hashscontpowcl resqrtcld flcld sqrtge0d 0zd
+      flge elnn0z eleq1d nn0zd eluz fzn0 xpnz biimpi ssxpb imass2 simp1d simp2d
+      nfv simp3d aks6d1c2p2 ffnd fnfund ffvelcdmda funimassd sseldd nnzd zndvds
+      f1f zsubcld divides biimpd mpd cdiv nnred resubcld crp nnrpd rpred rpgt0d
+      posdifd divgt0d zred mulcomd gt0ne0d divmuld breqtrd cc eqeltrd reximssdv
+      elnnz subaddd wfun fzfid xpfi imafi czring crh crg zncrng crngring zrhrhm
+      ccrg imaexd hashclb hashcl sqrtmsq sqrtmul 1red readdcld flltp1 ltmul12ad
+      oveq1d eqbrtrd hashfz0 oveq12d hashxp wf1o ovexd xpexg mptexd cres sselda
+      wex wfn fnfvimad fmptd feqresmpt feq1d eqidd f1eq123d df-ima rneqd eqtr2d
+      f1resf1 dff1o5 f1oeq1 spcedv wi hasheqf1oi zringbas rhmf resss rnss nnssz
+      frn sseld fnfvima nnssre hashnexinjle reximddv2 r19.29vva ) AVGVOZVHVOZVP
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+      VXNWAWWMWXDVWEVXNWWMJVWDWWMJWXBYQZWWMVWDWWMVWDWWOUXKYQZUXLVXLWWKVXOWOZXSW
+      WMVXNJVWDWWMVXNWWSYQZWXEWXFWWMJWXCUXMUXNYKUXOWNVWDUXSYJWWMVWFVWBWWMVXNVWE
+      WAVWFVWBWAWWMVWEVXNWXGXRWWMVWBVWAVWEWWMVWBWWQYQWWMVWAWWRYQWWMVWEVXNUXPWXG
+      WXHUXQUXTYGXRUXRWNAVGVHFWUHVXGAFYRWLVYNYRWLZAQUYAVYMYRWLZWXIWWCAVYLYRWLZW
+      XKWXJAWEEUYBZWXLVYLVYLUYCYFQVYMUYDYFAFVYNYRWWDUVQYKAWUHYRWLZWUIWDWLZWURAW
+      UHXLWLWXMWXNXTAUBVYPUYEWUPUYFVSZAWUPUYGWLZUBWXOWLZAWUPUYKWLZWXPAVYFWXRVYI
+      JWUPWUQUYHXJWUPUYIXJWUPUBUSUYJXJZUYLWUHXLUYMXJYKZAWUIVYNWIWJZFWIWJVPAWUIV
+      YMWIWJZWYAVPAWUIVYLWIWJZWYCVRVSZWYBVPAWUIEWTVTVSZWYEVRVSZWYDVPAWUIWUIWUIV
+      RVSUVGWJZWYFVPAWYGWUIAWUIYIWLZWEWUIWKVQZWYGWUIWAAWUIAWXMWXNWXTWUHUYNXJZYD
+      ZAWUIWYJYEZWUIUYOYFXRAWYGWUJWUJVRVSZWYFVPAWYHWYIWCZWYNWYGWYMWAAWYHWYIWYKW
+      YLWNZWYOWUIWUIUYPYFAWUJWYEWUJWYEAWUIWYKWYLUVKZAEWTAEAWUGWULAWUOWULAWUMWUN
+      AWUJWYPUVLAWVBWUNAWUIWYKWYLUVMZAWVCWVDWVEWYPWVFWVGYFYGWNWVHYJWVJYKZYDAUYQ
+      UYRZWYPWYSWYQAWUJWUKWTVTVSZWYEVPAWVCWUJWYTVPVQWYPWUJUYSXJAEWUKWTVTWVIVUAY
+      SZWYQXUAUYTVUBVUBAWYCWYEWYCWYEVRAWUGWYCWYEWAWYREVUCXJZXUBVUDYSAWXKWXKWCWY
+      BWYDWAAWXKWXKWXLWXLWNVYLVYLVUEXJYSAVYMVYNVMVOZVUFZVMVULZWYBWYAWAZAXUDVYMV
+      YNVNVYMVNVOZQWJZXDZVUFZVMXLXUIAVNVYMXUHXLAVYLXLWLZXUKWCVYMXLWLZAXUKXUKAWE
+      EWFVUGZXUMWNVYLVYLXLXLVUHXJZVUIAVYMVYNXUIYLZXUIYTZVYNWAZWCXUJAXUOXUQAVYMV
+      YNQVYMVUJZYLZXUOAWVOVYQVYMVYNXURXBZXUSWVTWVMAXUTVYMVYNXUIXBAVNVYMXUHVYNXU
+      IAXUGVYMWLZWCVYOXUGVYMQAQVYOVUMXVAWWBYAAVYMVYOXUGWVMVUKAXVAXHVUNXUIXFVUOA
+      VYMVYNXURXUIAVNVYOWBVYMQWWAWVMVUPZVUQYKVYOWBVYMVYNQVVCYPAVYMVYMVYNVYNXURX
+      UIXVBAVYMVURAVYNVURVUSYGAVYNXURYTZXUPVYNXVCWAAQVYMVUTXNAXURXUIXVBVVAVVBWN
+      VYMVYNXUIVVDYJVYMVYNXUCXUIVVEVVFAXULXUEXUFVVGXUNVYMVYNVMXLVVHXJUXAUXOAFVY
+      NWIWWDXPYSAVJFVXFWUHVXGAVXEFWLZWCUBXKVUMZVYPXKYBZVXEVYPWLZVXFWUHWLAXVEXVD
+      AXKWUPXCWJZUBAWXQXKXVHUBXBWXSXKXVHUYEWUPUBVVIXVHXFVVJXJUWJYAAXVFXVDAVYPWB
+      XKAVYPQYTZWBAVYPXVIYBQVYOVUJZYTZXVIYBZAXVJQYBZXVLXVMAQVYOVVKXNXVJQVVLXJAV
+      YPXVKXVIVYPXVKWAAQVYOVUTXNYNYKAWVPXVIWBYBWWAVYOWBQVVNXJYMWBXKYBAVVMXNYMYA
+      AXVDXVGAFVYPVXEAFVYPYBVYRWVNAFVYNVYPWWDYNYKVVOXIXKVYPUBVXEVVPYPVXGXFVUOAF
+      WBYIWWEWBYIYBAVVQXNYMVVRVVSVVT $.
+  $}
+
+  ${
+    $d B x $.  $d D x $.
+    $( Special case related to ~ rspsbc .  (Contributed by metakunt,
+       5-May-2025.) $)
+    rspcsbnea $p |- ( ( A e. B /\ A. x e. B C =/= D ) ->
+     [_ A / x ]_ C =/= D ) $=
+      ( wcel wne wral csb wsbc rspsbc wceq wn wb df-ne sbcbii a1i sbcng sbceq1g
+      bitrd notbid biidd necon3bbid sylibd imp ) BCFZDEGZACHZABDIZEGZUFUHUGABJZ
+      UJUGABCKUFUKUIELZMZUJUFUKDELZMZABJZUMUKUPNUFUGUOABDEOPQUFUPUNABJZMUMUNABC
+      RUFUQULABDECSUATTUFULUIEUFULUBUCTUDUE $.
+  $}
+
+  ${
+    $d .^ x y $.  $d A x y $.  $d N x $.  $d R x y $.  $d ph x y $.
+    idomnnzpownz.1 $e |- ( ph -> R e. IDomn ) $.
+    idomnnzpownz.2 $e |- ( ph -> A e. ( Base ` R ) ) $.
+    idomnnzpownz.3 $e |- ( ph -> A =/= ( 0g ` R ) ) $.
+    idomnnzpownz.4 $e |- ( ph -> N e. NN0 ) $.
+    idomnnzpownz.5 $e |- .^ = ( .g ` ( mulGrp ` R ) ) $.
+    $( A non-zero power in an integral domain is non-zero.  (Contributed by
+       metakunt, 5-May-2025.) $)
+    idomnnzpownz $p |- ( ph -> ( N .^ A ) =/= ( 0g ` R ) ) $=
+      ( wcel wa co cfv wne wceq oveq1 eqid syl adantr vx vy cn0 c0g ancli cv c1
+      cc0 caddc neeq1d cur cmgp cbs eleqtrdi mulg0 ringidval eqtr4di cnzr cidom
+      mgpbas cdomn ccrg isidom simprbi domnnzr nzrnz eqnetrd cmnd crg idomringd
+      cplusg ringmgp simplr ad2antrr mulgnn0p1 syl3anc cmulr mgpplusg a1i oveqd
+      eqcomd mulgnn0cl eqcomi eleqtrd simpr jca domnmuln0 nn0indd ) AAEUCKZLEBD
+      MZCUDNZOZAWIIUEAUAUFZBDMZWKOUHBDMZWKOUBUFZBDMZWKOZWPUGUIMZBDMZWKOWLUAUBEW
+      MUHPWNWOWKWMUHBDQUJWMWPPWNWQWKWMWPBDQUJWMWSPWNWTWKWMWSBDQUJWMEPWNWJWKWMEB
+      DQUJAWOCUKNZWKAWOCULNZUDNZXAABXBUMNZKZWOXCPABCUMNZXDGXFCXBXBRZXFRZUTZUNZX
+      DDXBBXCXDRZXCRJUOSCXAXBXGXARZUPUQACURKZXAWKOACUSKZXMFXNCVAKZXMXNCVBKXOCVC
+      VDZCVESSCXAWKXLWKRZVFSVGAWPUCKZLZWRLZWTWQBXBVKNZMZWKXTXBVHKZXRXEWTYBPXSYC
+      WRAYCXRACVIKYCACFVJCXBXGVLSTTZAXRWRVMZAXEXRWRXJVNZXDYADXBWPBXKJYARVOVPXTY
+      BWQBCVQNZMZWKXTYAYGWQBXTYGYAYGYAPXTCYGXBXGYGRZVRVSWAVTXTXOWQXFKZWRLBXFKZB
+      WKOZLZYHWKOXSXOWRAXOXRAXNXOFXPSTTXTYJWRXTWQXDXFXTYCXRXEWQXDKYDYEYFXDDXBWP
+      BXKJWBVPXDXFPXTXFXDXIWCVSWDXSWRWEWFXSYMWRAYMXRAYKYLGHWFTTXFCYGWQBWKXHYIXQ
+      WGVPVGVGWHS $.
+  $}
+
+  ${
+    $d A m y z $.  $d A x y z $.  $d G m y z $.  $d G x y z $.  $d N m n y z $.
+    $d N n x y z $.  $d R m n y z $.  $d R n x y z $.  $d m n ph y z $.
+    $d ph x y z $.
+    idomnnzgmulnz.1 $e |- G = ( mulGrp ` R ) $.
+    idomnnzgmulnz.2 $e |- ( ph -> R e. IDomn ) $.
+    idomnnzgmulnz.3 $e |- ( ph -> N e. Fin ) $.
+    idomnnzgmulnz.4 $e |- ( ( ph /\ n e. N ) -> A e. ( Base ` R ) ) $.
+    idomnnzgmulnz.5 $e |- ( ( ph /\ n e. N ) -> A =/= ( 0g ` R ) ) $.
+    $( A finite product of non-zero elements in an integral domain is non-zero.
+       (Contributed by metakunt, 5-May-2025.) $)
+    idomnnzgmulnz $p |- ( ph -> ( G gsum ( n e. N |-> A ) )
+    =/= ( 0g ` R ) ) $=
+      ( vm cmpt cgsu co cfv wne wceq wcel adantr vx vy vz cv c0g csn cun mpteq1
+      c0 oveq2d neeq1d mpt0 a1i eqid gsum0 eqtrd cur ringidval cidom cnzr cdomn
+      eqcomi ccrg isidom simprbi domnnzr syl nzrnz 3syl eqnetrd wss cdif wa csb
+      cplusg nfcv nfcsb1v csbeq1a cbvmpt oveq2i cbs ccmn simplbi crngmgp simprl
+      cfn ssfid wral ad2antrr simpr sseldd ralrimiva ad3antrrr rspcsbela mgpbas
+      syl2anc eleqtrd eldifi adantl wn eldifn csbeq1 gsumunsn gsummptcl equcoms
+      eqcomd jca rspcsbnea cmulr mgpplusg domnmuln0 syl3anc ex findcard2d ) AED
+      UAUDZBMZNOZCUEPZQEDUIBMZNOZXRQEDUBUDZBMZNOZXRQZEDYAUCUDZUFUGZBMZNOZXRQZED
+      FBMZNOZXRQUAUBUCFXOUIRZXQXTXRYLXPXSENDXOUIBUHUJUKXOYARZXQYCXRYMXPYBENDXOY
+      ABUHUJUKXOYFRZXQYHXRYNXPYGENDXOYFBUHUJUKXOFRZXQYKXRYOXPYJENDXOFBUHUJUKAXT
+      EUEPZXRAXTEUINOZYPAXSUIENXSUIRADBULUMUJYQYPRAEYPYPUNUOUMUPAYPCUQPZXRYPYRR
+      AYRYPCYREGYRUNZURVBUMACUSSZCUTSZYRXRQHYTCVASZUUAYTCVCSZUUBCVDZVEZCVFVGCYR
+      XRYSXRUNZVHVIVJVJAYAFVKZYEFYAVLSZVMZVMZYDYIUUJYDVMZYHELYADLUDZBVNZMZNOZDY
+      EBVNZEVOPZOZXRUUKYHELYFUUMMZNOZUURYHUUTRUUKYGUUSENDLYFBUUMLBVPZDUULBVQZDU
+      ULBVRZVSVTUMUUKYAEWAPZUUQLEYEFUUMUUPUVDUNUUQUNUUJEWBSZYDAUVEUUIAUUCUVEAYT
+      UUCHYTUUCUUBUUDWCVGCEGWDVGTTZUUJYAWFSYDUUJFYAAFWFSUUIITAUUGUUHWEZWGTZUUKU
+      ULYASZVMZUUMCWAPZUVDUVJUULFSBUVKSZDFWHZUUMUVKSZUVJYAFUULUUJUUGYDUVIUVGWIU
+      UKUVIWJWKAUVMUUIYDUVIAUVLDFJWLZWMDUULFBUVKWNWPZUVKUVDRZUVJUVKCEGUVKUNZWOZ
+      UMWQUUJYEFSZYDUUIUVTAUUHUVTUUGYEFYAWRWSWSZTZUUJYEYASWTZYDUUIUWCAUUHUWCUUG
+      YEFYAXAWSWSTUUKUUPUVKUVDUUKUVTUVMUUPUVKSZUWBAUVMUUIYDUVOWIDYEFBUVKWNWPZUV
+      QUUKUVSUMWQDUULYEBXBXCUPUUKUUBUUOUVKSZUUOXRQZVMUWDUUPXRQZVMUURXRQUUJUUBYD
+      AUUBUUIAYTUUBHUUEVGTTUUKUWFUWGUUKUVKLEYAUUMUVSUVFUVHUUKUVNLYAUVPWLXDUUKUU
+      OYCXRUUKUUNYBENUUNYBRUUKLDYAUUMBUVBUVAUULDUDRBUUMBUUMRDLUVCXEXFVSUMUJUUJY
+      DWJVJXGUUKUWDUWHUWEUUJUWHYDUUJUVTBXRQZDFWHZUWHUWAAUWJUUIAUWIDFKWLTDYEFBXR
+      XHWPTXGUVKCUUQUUOUUPXRUVRCXIPZUUQCUWKEGUWKUNXJVBUUFXKXLVJXMIXN $.
+  $}
+
+  ${
+    $d .^ x y $.  $d N x $.  $d R x y $.  $d ph x y $.
+    ringexp0nn.1 $e |- ( ph -> R e. Ring ) $.
+    ringexp0nn.2 $e |- ( ph -> N e. NN ) $.
+    ringexp0nn.3 $e |- .^ = ( .g ` ( mulGrp ` R ) ) $.
+    $( Zero to the power of a positive integer is zero.  (Contributed by
+       metakunt, 5-May-2025.) $)
+    ringexp0nn $p |- ( ph -> ( N .^ ( 0g ` R ) ) = ( 0g ` R ) ) $=
+      ( vx vy cn wcel wa cfv co wceq c1 oveq1 eqeq1d syl eqid c0g ancli cv cmgp
+      caddc cbs cmnd crg ringmnd mndidcl mgpbas a1i eleqtrd mulg1 cplusg simplr
+      ad2antrr mulgnnp1 syl2anc simpr oveq1d cmulr mgpplusg eqcomi ringrz eqtrd
+      adantr nnindd ) AADJKZLDBUAMZCNZVJOZAVIFUBAHUCZVJCNZVJOPVJCNZVJOZIUCZVJCN
+      ZVJOZVQPUENZVJCNZVJOVLHIDVMPOVNVOVJVMPVJCQRVMVQOVNVRVJVMVQVJCQRVMVTOVNWAV
+      JVMVTVJCQRVMDOVNVKVJVMDVJCQRAVJBUDMZUFMZKZVPAVJBUFMZWCABUGKZVJWEKZABUHKZW
+      FEBUISWEBVJWETZVJTZUJSZWEWCOAWEBWBWBTZWIUKULUMZWCCWBVJWCTZGUNSAVQJKZLZVSL
+      ZWAVRVJWBUOMZNZVJWQWOWDWAWSOAWOVSUPAWDWOVSWMUQWCWRCWBVQVJWNGWRTURUSWQWSVJ
+      VJWRNZVJWQVRVJVJWRWPVSUTVAWPWTVJOZVSAXAWOAWHWGXAEWKWEBWRVJVJWIBVBMZWRBXBW
+      BWLXBTVCVDWJVEUSVGVGVFVFVHS $.
+
+  $}
+
+  ${
+    aks6d1p5.1 $e |- ( ph -> K e. Field ) $.
+    aks6d1p5.2 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c5.3 $e |- P = ( chr ` K ) $.
+    aks6d1c5.4 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c5.5 $e |- ( ph -> A < P ) $.
+    aks6d1c5.6 $e |- X = ( var1 ` K ) $.
+    aks6d1c5.7 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c5.8 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    .^ ( X ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+
+    ${
+      $d A g i $.  $d A x y $.  $d G x y $.  $d K g i $.  $d g i ph $.
+      $( Lemma for Claim 5 of Theorem 6.1, G defines a map into the
+         polynomials.  (Contributed by metakunt, 5-May-2025.) $)
+      aks6d1c5p0 $p |- ( ph -> G : ( NN0 ^m ( 0 ... A ) ) -->
+     ( Base ` ( Poly1 ` K ) ) ) $=
+        ( cfv wcel eqid cn0 cc0 cfz co cmap cpl1 cmgp cv czrh cascl cplusg cmpt
+        cgsu cbs wa ccmn ccrg fldcrngd ply1crng syl crngmgp adantr cmnd cmnmndd
+        fzfid cvv nn0ex a1i ovexd elmapd biimpd imp ffvelcdmda crngringd cmnmnd
+        wf ringcmnd crg vr1cl simpl elfzelz adantl jca czring crh zringbas rhmf
+        cz zrhrhm ply1sclcl syl2anc mndcl syl3anc wceq mgpbas eleqtrd ralrimiva
+        mulgnn0cld gsummptcl eqcomi fmptd ) ADUAUBBUCUDZUEUDZHUFRZUGRZEXBEUHZDU
+        HZRZIXFHUIRZRZXDUJRZRZXDUKRZUDZFUDZULUMUDZXDUNRZGAXGXCSZUOZXPXEUNRZXQXS
+        XTEXEXBXOXTTZAXEUPSZXRAXDUQSZYBAHUQSYCAHJURZXDHXDTZUSUTZXDXEXETZVAUTVBZ
+        XSUBBVEXSXOXTSEXBXSXFXBSZUOZXTFXEXHXNYAPXSXEVCSYIXSXEYHVDVBXSXBUAXFXGAX
+        RXBUAXGVPZAXRYKAUAXBXGVFVFUAVFSAVGVHAUBBUCVIVJVKVLVMYJXNXQXTYJXDVCSZIXQ
+        SZXLXQSZXNXQSXSYLYIAYLXRAXDUPSYLAXDAXDYFVNVQXDVOUTVBVBYJHVRSZYMXSYOYIAY
+        OXRAHYDVNVBZVBZXQXDHIOYEXQTZVSUTYJYOXJHUNRZSZYNYQYJXSXFWHSZUOYTYJXSUUAX
+        SYIVTYIUUAXSXFUBBWAWBWCXSWHYSXFXIXSYOWHYSXIVPZYPYOXIWDHWEUDSUUBHXIXITWI
+        WHYSWDHXIWFYSTZWGUTUTVMUTXKXQXDHXJYSYEXKTUUCYRWJWKXQXMXDIXLYRXMTWLWMXQX
+        TWNYJXQXDXEYGYRWOZVHWPWRWQWSXTXQWNXSXQXTUUDWTVHWPQXA $.
+    $}
+
+    ${
+      aks6d1c5p1.1 $e |- ( ph -> B e. ( 0 ... A ) ) $.
+      aks6d1c5p1.2 $e |- ( ph -> C e. ( 0 ... A ) ) $.
+      $( Lemma for claim 5, evaluate the linear factor at -c to get a root.
+         (Contributed by metakunt, 5-May-2025.) $)
+      aks6d1c5p1 $p |- ( ph -> ( B = C <-> (
+       ( ( eval1 ` K ) `
+        ( X ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` B ) ) )
+       )
+       ` ( ( ZRHom ` K ) ` ( 0 - C ) ) ) = ( 0g ` K ) ) ) $=
path: set.mm
user: login: metakunt
body: That's true but I always find the current API for -u awkward to use. Maybe there are lemmas missing in main that would make it more ergonomic. It mostly just comes to just unfolding the definition which gives not much benefit.
created_at: 2025-05-06T08:46:55Z
author_association: CONTRIBUTOR
diff_hunk: @@ -84,7 +84,11 @@ make a github issue.)
 
 DONE:
 Date      Old       New         Notes
- 2-May-25 bj-sbnf   sbnf        Moved from BJ's mathbox to main set.mm
+05-May-25 idomd     [same]      Moved from TA's mathbox to main set.mm
path: changes-set.txt
user: login: tirix
body: I think this should be `idomdomd`
created_at: 2025-05-06T10:04:21Z
author_association: CONTRIBUTOR
diff_hunk: @@ -677236,6 +677236,863 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
+      aks6d1c2p3.7 $e |- ( ph -> I = ( p E q ) ) $.
+      aks6d1c2p3.8 $e |- ( ph -> I e. NN0 ) $.
+      $( Lemma for ~ aks6d1c2 to simplify context.  (Contributed by metakunt,
+         1-May-2025.) $)
+      aks6d1c2p3lem $p |- ( ph -> ( J ( .g ` ( mulGrp ` K ) ) ( (
+      ( eval1 ` K ) ` ( G ` s ) ) ` M ) ) = ( I ( .g ` ( mulGrp ` K ) ) ( (
+        ( eval1 ` K ) ` ( G ` s ) ) ` M ) ) ) $=
+        ( vv cv cfv ce1 cmgp cmg co cexp cdiv cmul cn0 cvv cmpo wceq a1i simprl
+        wa oveq2d simprr oveq12d cc0 cfz wcel elfznn0 ovexd ovmpod eqtrd oveq1d
+        syl oveq2 fveq2d eqeq12d wbr wral syl2anc eqid aks6d1c1rh aks6d1c1p1rcl
+        mpbid cbs cn simpld aks6d1c1p1 rspcdva eqcomd w3a nnnn0d 3jca eqtr2d wf
+        cprimroots fveq2 cmap wb nn0ex elmapg cpl1 simprd caddc cplusg cmnd crg
+        ccrg fldcrngd crngring ringmgp nn0mulcld c0g wi ccmn crngmgp isprimroot
+        cdvds biimpd mpd simp1d mulgnn0dir mulgnn0ass simp2d mulgnn0cld mndrid
+        mulgnn0z ) AUEUHUKWMZUBWNZUFWOWNZWNZWNZUFWPWNZWQWNZWRGULWMZWSWRZUIGWTWR
+        ZRWMZWSWRZXAWRZUVRUVTWRZUDUVRUVTWRZAUEUWFUVRUVTAUEUWAUWDSWRUWFWGAQUPUWA
+        UWDXBXBGQWMZWSWRZUWCUPWMZWSWRZXAWRZUWFSXCSQUPXBXBUWMXDXEAVHXFZAUWIUWAXE
+        ZUWKUWDXEZXHXHZUWJUWBUWLUWEXAUWQUWIUWAGWSAUWOUWPXGXIUWQUWKUWDUWCWSAUWOU
+        WPXJXIXKAUWAXLEXMWRZXNUWAXBXNWEUWAEXOXTZAUWDUWRXNUWDXBXNWFUWDEXOXTZAUWB
+        UWEXAXPXQXRZXSAUWHGUNWMZWSWRZUWCUMWMZWSWRZXAWRZUVRUVTWRZUWGAUDUXFUVRUVT
+        AUDUXBUXDSWRUXFWJAQUPUXBUXDXBXBUWMUXFSXCUWNAUWIUXBXEZUWKUXDXEZXHXHZUWJU
+        XCUWLUXEXAUXJUWIUXBGWSAUXHUXIXGXIUXJUWKUXDUWCWSAUXHUXIXJXIXKAUXBUWRXNUX
+        BXBXNWHUXBEXOXTZAUXDUWRXNUXDXBXNWIUXDEXOXTZAUXCUXEXAXPXQXRZXSAUXGUXFUHU
+        VTWRZUVQWNZUWGAUXFCWMZUVQWNZUVTWRZUXFUXPUVTWRZUVQWNZXEZUXGUXOXECUVSIUUB
+        WRZUHUXPUHXEZUXRUXGUXTUXOUYCUXQUVRUXFUVTUXPUHUVQUUCZXIUYCUXSUXNUVQUXPUH
+        UXFUVTYAYBYCAUXFUVOHYDUYACUYBYEABCDGHIUXBLMNPUXFUVNUBUFUXDUIUOUQURUSUTV
+        AVBVCVDAUVNXBXLDXMWRZUUDWRXNZUYEXBUVNUUAZWDAXBXCXNZUYEXCXNUYFUYGUUEUYHA
+        UUFXFAXLDXMXPXBUYEUVNXCXCUUGYFYJZVFVGUXKUXLUXFYGVJVKYHZACUFUUHWNYKWNZUV
+        THILMUXFUVTUVOUVSUVPUQAUXFYLXNZUVOUYKXNZACUYKUVTHILMUXFUVTUVOUVSUVPUQUY
+        JYIZUUIZAUYLUYMUYNYMYNYJVLYOAUXOUWFUHUVTWRZUVQWNZUWGAUXNUYPUVQAUXNUDUHU
+        VTWRZUYPAUXFUDUHUVTAUDUXFUXMYPXSAUYPUEUHUVTWRZUYRAUWFUEUHUVTAUEUWFUXAYP
+        XSAUYSUDKIXAWRZUUJWRZUHUVTWRZUYRAUEVUAUHUVTWCXSAVUBUYRUYTUHUVTWRZUVSUUK
+        WNZWRZUYRAUVSUULXNZUDXBXNZUYTXBXNZUHUVSYKWNZXNZYQVUBVUEXEAUFUUMXNZVUFAU
+        FUUNXNZVUKAUFUSUUOZUFUUPXTUFUVSUVSYGZUUQXTZAVUGVUHVUJWKAKIAKWBYRZAIVAYR
+        ZUURAVUJIUHUVTWRZUVSUUSWNZXEZWLWMZUHUVTWRVUSXEIVVAUVDYDUUTWLXBYEZAUHUYB
+        XNZVUJVUTVVBYQZVLAVVCVVDAUVSUVTIUHWLAVULUVSUVAXNVUMUFUVSVUNUVBXTVUQUVTY
+        GZUVCUVEUVFZUVGZYSVUIVUDUVTUVSUDUYTUHVUIYGZVVEVUDYGZUVHYFAVUEUYRVUSVUDW
+        RZUYRAVUCVUSUYRVUDAVUCKVURUVTWRZVUSAVUFKXBXNZIXBXNZVUJYQVUCVVKXEVUOAVVL
+        VVMVUJVUPVUQVVGYSVUIUVTUVSKIUHVVHVVEUVIYFAVVKKVUSUVTWRZVUSAVURVUSKUVTAV
+        UJVUTVVBVVFUVJXIAVUFVVLVVNVUSXEVUOVUPVUIUVTUVSKVUSVVHVVEVUSYGZUVMYFXRXR
+        XIAVUFUYRVUIXNVVJUYRXEVUOAVUIUVTUVSUDUHVVHVVEVUOWKVVGUVKVUIVUDUVSUYRVUS
+        VVHVVIVVOUVLYFXRXRXRYTXRYBAUWGUYQAUWFUXQUVTWRZUWFUXPUVTWRZUVQWNZXEZUWGU
+        YQXECUYBUHUYCVVPUWGVVRUYQUYCUXQUVRUWFUVTUYDXIUYCVVQUYPUVQUXPUHUWFUVTYAY
+        BYCAUWFUVOHYDVVSCUYBYEABCDGHIUWALMNPUWFUVNUBUFUWDUIUOUQURUSUTVAVBVCVDUY
+        IVFVGUWSUWTUWFYGVJVKYHZACUYKUVTHILMUWFUVTUVOUVSUVPUQUYOAUWFYLXNUYMACUYK
+        UVTHILMUWFUVTUVOUVSUVPUQVVTYIYMYNYJVLYOYPXRXRYTXR $.
+    $}
+
+    ${
+      $d .~ a $.  $d A a o p q r s $.  $d A g i o p q r s $.  $d A h s $.
+      $d A k l o p q r s $.  $d A o p q r s x $.  $d B a o p q r $.
+      $d B g i o p q r $.  $d B k l o p q r $.  $d B o p q r x $.
+      $d E a o p q r $.  $d E g i o p q r $.  $d E k l o p q r $.
+      $d E o p q r x $.  $d G e f o p q r y $.  $d G h $.  $d H s $.
+      $d I a o p q r $.  $d I g i o p q r $.  $d I k l o p q r $.
+      $d I o p q r x $.  $d J a o p q r $.  $d J g i o p q r $.
+      $d J k l o p q r $.  $d J o p q r x $.  $d K a o p q r s $.
+      $d K e f o p q r s y $.  $d K g i o p q r s $.  $d K h s $.  $d K v $.
+      $d K o p q r s x $.  $d M h $.  $d M o p q r y $.  $d M v $.
+      $d N a o r $.  $d N e f o r y $.  $d N k l o r $.  $d N o r x $.
+      $d P e f y $.  $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.
+      $d R x $.  $d S s $.  $d a o p ph q r s $.  $d g i o p ph q r s $.
+      $d h ph s $.  $d k l o p ph q r s $.  $d ph v $.  $d ph q r s x $.
+      $( Claim 2 of Theorem 6.1 AKS, Preparation for injectivity proof.
+         (Contributed by metakunt, 1-May-2025.) $)
+      aks6d1c2p4 $p |- ( ph -> ( # ` ( H " ( NN0 ^m ( 0 ... A ) ) ) ) <_
+       ( N ^ B ) ) $=
+        ( vs vv vr vo vp vq cn0 cc0 cfz co cima chash cfv c0g cexp wcel cvv cle
+        wbr fvexd syl cv wa wceq a1i fveq2d cbs eqid adantr simpr nnnn0d mpd cn
+        cmul c1 wf adantl simprd eqeltrd oveq2d wss nn0expcld cz wne wb syl3anc
+        mpbid jca elnn0z sylibr wrex nn0red cr syl2anc r19.29vva ad7antr eqcomd
+        eqtrd reexpcld letrd recnd eqbrtrd cmap ce1 csn cfn cnvexg imaexd fmptd
+        ccnv nfv ffnd fnfund cpl1 csg fveq1d ccrg fldcrngd cmpt fvmptd cmgp cmg
+        cdvds wi wral cprimroots w3a crngmgp isprimroot biimpd simp1d eleqtrrdi
+        ccmn mgpbas cdiv cfield cprime cgcd elmapi 0nn0 cascl cplusg aks6d1c1rh
+        cv1 czrh aks6d1c1p1rcl fveval1fvcl ply1crng cmnmndd cmpo simprl oveq12d
+        crs simprr fz0ssnn0 sselda sseli ovexd prmnn nn0zd dvdsval2 nnred nnrpd
+        ovmpod nnne0d nn0ge0d divge0d nn0mulcld cxp eleqtrd aks6d1c2p1 c0 csqrt
+        wfn cuz cfl czn hashscontpowcl resqrtcld flcld sqrtge0d 0zd sylib eluz1
+        flge ax-mp fzn0 ssxpb mpbird ovelimab crg crngringd vr1cl eqcomi eleq2i
+        xpnz mulgnn0cld evl1vard evl1expd caddc ad6antr simp-6r simp-5r simp-4r
+        0z simpllr simplr ad3antrrr ad2antrr aks6d1c2p3lem ad4antr 3eqtrd eqidd
+        clt evl1subd cgrp crnggrpd grpsubid wfun cdm cpws grpsubcl eqeltrid crh
+        elsng evl1rhm rhmf ffvelcdmda pwsbas eleqtrrd elmapd ffn fnfun fvimacnv
+        fndm funimassd ssexd cdg1 cnzr cdr biimpi simpld drngnzr deg1pw 3brtr3d
+        ex isfld deg1sub cidom fldidom deg1nn0clb fta1g hashbnd hashcl redivcld
+        hashss remulcld 0red 1red prmgt1 ltled expge0d elfzuz3 leexp2ad mullidd
+        0le1 nnge1d nnzd dvdsle lemuldivd lemul12ad oveq1d mulexpd eqtr2d leidd
+        divcan2d ) AUBWEWFDWGWHZUUAWHZWIZWJWKZJUEUUBWKZWKZUUHZUEWLWKZUUCZWIZWJW
+        KZUHEWMWHZAVXRAVXQUUDWNZVXRWEWNAVXQWOWNVYEWEWNZVXRVYEWPWQZVYGAVXQVYDWOA
+        VYAVYCWOAVXTWOWNVYAWOWNAJVXSWRVXTWOUUEWSUUFZAVSVXPVYDUBAVSUUIAVXPUBAVXP
+        WOUBAOVXPUGOWTZUAWKZVXSWKZWKZWOUBAVYKVXPWNXAUGVYMWRVHUUGUUJUUKAVSWTZVXP
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+        AXVCWXKWXMXVPVUGVXKVXLWXNVYFYWGVXMYTYRYTXGYTXVSYMYTYTYTYRYR $.
+    $}
+  $}
+
+  ${
+    $d A f $.  $d A x y $.  $d B f $.  $d F f $.  $d F x y $.
+    hashnexinj.1 $e |- ( ph -> A e. Fin ) $.
+    hashnexinj.2 $e |- ( ph -> B e. Fin ) $.
+    hashnexinj.3 $e |- ( ph -> ( # ` B ) < ( # ` A ) ) $.
+    hashnexinj.4 $e |- ( ph -> F : A --> B ) $.
+    $( If the number of elements of the domain are greater than the number of
+       elements in a codomain, then there are two different values that map to
+       the same.  (Contributed by metakunt, 2-May-2025.) $)
+    hashnexinj $p |- ( ph -> E. x e. A E. y e. A
+     ( ( F ` x ) = ( F ` y ) /\ x =/= y ) ) $=
+      ( vf cfv wa wn cfn wcel syl notbid mpd wral cv wceq wne wrex wf1 wal cdom
+      wf wex wbr chash cle clt cn0 hashcl nn0red ltnled mpbid wb hashdom biimpd
+      syl2anc wi brdomg alnex sylibr cmap elmapdd f1eq1 spcgv dff13 iman anbi2i
+      co df-ne xchbinxr 2ralbii ralnex2 bitri a1i mpnanrd notnotrd ) ABUAZFLCUA
+      ZFLUBZWCWDUCZMZCDUDBDUDZADEFUHZWHNZJADEFUEZNZWIWJMZNZADEKUAZUEZNZKUFZWLAW
+      PKUIZNZWRADEUGUJZNZWTADUKLZEUKLZULUJZNZXBAXDXCUMUJXFIAXDXCAXDAEOPZXDUNPHE
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+      KDEWOFVIRVJQSAWLWNAWKWMWKWMUSAWKWIWEWCWDUBZVCZCDTBDTZMWMBCDEFVKXLWJWIXLWG
+      NZCDTBDTWJXKXMBCDDXKWEXJNZMWGWEXJVLWFXNWEWCWDVOVMVPVQWGBCDDVRVSVMVSVTRVAS
+      WAWB $.
+  $}
+
+  ${
+    $d A w x y z $.  $d F w x y z $.  $d ph x y $.
+    hashnexinjle.1 $e |- ( ph -> A e. Fin ) $.
+    hashnexinjle.2 $e |- ( ph -> B e. Fin ) $.
+    hashnexinjle.3 $e |- ( ph -> ( # ` B ) < ( # ` A ) ) $.
+    hashnexinjle.4 $e |- ( ph -> F : A --> B ) $.
+    hashnexinjle.5 $e |- ( ph -> A C_ RR ) $.
+    $( If the number of elements of the domain are greater than the number of
+       elements in a codomain, then there are two different values that map to
+       the same.  Also we introduce a one sided inequality to simplify a
+       duplicateable proof.  (Contributed by metakunt, 2-May-2025.) $)
+    hashnexinjle $p |- ( ph -> E. x e. A E. y e. A
+     ( ( F ` x ) = ( F ` y ) /\ x < y ) ) $=
+      ( vw vz cv cfv wceq clt wbr wa wrex simpr wb fveq2 eqeq2d anbi12d fveqeq2
+      breq2 breq1 cbvrex2vw a1i biimpd imp sylib rexcom hashnexinj wcel simplrl
+      wne wo jca orcd eqcomd olcd simprr simpl simprl sselda syl adantr lttri2d
+      cr mpbid mpjaodan ex reximdvva r19.43 rexbii mpd ) ABNZFOZCNZFOZPZVSWAQRZ
+      SZCDTZBDTZWGWBVTPZWAVSQRZSZCDTZBDTZAWGUAAWLSZWEBDTCDTZWGWMLNZFOZMNZFOZPZW
+      OWQQRZSZLDTMDTZWNAWLXBAWLXBWLXBUBAWJXAWBWRPZWAWQQRZSBCMLDDVSWQPZWHXCWIXDX
+      EVTWRWBVSWQFUCUDVSWQWAQUGUEWAWOPXCWSXDWTWAWOWRFUFWAWOWQQUHUEUIUJUKULXAWEW
+      PWBPZWOWAQRZSMLCBDDWQWAPZWSXFWTXGXHWRWBWPWQWAFUCUDWQWAWOQUGUEWOVSPXFWCXGW
+      DWOVSWBFUFWOVSWAQUHUEUIUMWECBDDUNUMAWCVSWAURZSZCDTBDTZWGWLUSZABCDEFGHIJUO
+      AXKXLAXKSZWFWKUSZBDTZXLXMWEWJUSZCDTZBDTZXOAXKXRAXJXPBCDDAVSDUPZWADUPZSZSZ
+      XJXPYBXJSZWDXPWIYCWDSZWEWJYDWCWDYBWCXIWDUQYCWDUAUTVAYCWISZWJWEYEWHWIYEVTW
+      BYBWCXIWIUQVBYCWIUAUTVCYCXIWDWIUSYBWCXIVDYCVSWAYBVSVKUPZXJYBAXSSYFYBAXSAY
+      AVEZAXSXTVFUTADVKVSKVGVHVIYBWAVKUPZXJYBAXTSYHYBAXTYGAXSXTVDUTADVKWAKVGVHV
+      IVJVLVMVNVOULXQXNBDWEWJCDVPVQUMWFWKBDVPUMVNVRVM $.
+  $}
+
+  ${
+    $d .~ a $.  $d A a b c d $.  $d A b c d g i $.  $d A b c d h $.
+    $d A b c d k l $.  $d A b c d x $.  $d B a b c d $.  $d B b c d g i $.
+    $d B b c d k l $.  $d B u w $.  $d B b c d x $.  $d C a b c d $.
+    $d C b c d g i $.  $d C b c d h $.  $d C b c d k l $.  $d C b c s t $.
+    $d C b c d x $.  $d E a $.  $d E g i $.  $d E k l $.  $d E o $.  $d E t $.
+    $d E u w $.  $d E x $.  $d G e f y $.  $d G h $.  $d H b c d $.  $d K a $.
+    $d K e f y $.  $d K g i $.  $d K h $.  $d K x $.  $d L b c d t $.
+    $d L b c s t $.  $d M h $.  $d M y $.  $d N a b c d $.  $d N e f y $.
+    $d N b c d k l $.  $d N b c d x $.  $d P e f y $.  $d P k l $.  $d P x $.
+    $d R a d $.  $d R e f y $.  $d R d g i $.  $d R d h $.  $d R d k l $.
+    $d R d x $.  $d a b c d ph $.  $d g i ph $.  $d h ph $.  $d k l ph $.
+    $d o ph $.  $d ph s t $.  $d ph w $.  $d ph x $.
+    aks6d1c2a.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2a.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2a.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2a.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2a.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2a.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2a.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2a.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2a.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2a.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2a.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2a.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2a.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2a.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2a.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2a.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2a.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2a.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2a.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2a.20 $e |- ( ph -> ( Q e. Prime /\ Q || N /\ P =/= Q ) ) $.
+    $( Claim 2 of Theorem 6.1 of ~ https://www3.nd.edu/andyp/notes/AKS.pdf
+       (Contributed by metakunt, 2-May-2025.) $)
+    aks6d1c2 $p |- ( ph -> ( # ` ( H " ( NN0 ^m ( 0 ... A ) ) ) ) <_
+       ( N ^ B ) ) $=
+      ( vb vc vd vt vs vo vu vw cv clt wbr cmul co caddc wceq cn wa cn0 cc0 cfz
+      wrex cima chash cfv cle wcel simprl jca simprr cmgp cmg ad5antr cprime c1
+      cdvds wf cbs cmpt simpllr eqid simplr simpr imp syl cz cvv nfcv a1i fvexd
+      fveq2d fvmptd eqcomd eqtrd wb adantr wss cxp nn0red nn0ge0d syl2anc mpbid
+      wne cr sylibr mpbird wf1 sstrd sseq1d ad2antrr syl3anc recnd cfn breqtrrd
+      crn cmap cexp simpl cv1 cpl1 csg cfield cgcd czrh cascl cplusg cprimroots
+      wral crs simp-5r simp-4r aks6d1c2p4 ex rexlimdva cmin fveq2 cbvmpt nnnn0d
+      fz0ssnn0 c0 cuz csqrt cfl czn hashscontpowcl resqrtcld flcld sqrtge0d 0zd
+      flge elnn0z eleq1d nn0zd eluz fzn0 xpnz biimpi ssxpb imass2 simp1d simp2d
+      nfv simp3d aks6d1c2p2 ffnd fnfund ffvelcdmda funimassd sseldd nnzd zndvds
+      f1f zsubcld divides biimpd mpd cdiv nnred resubcld crp nnrpd rpred rpgt0d
+      posdifd divgt0d zred mulcomd gt0ne0d divmuld breqtrd cc eqeltrd reximssdv
+      elnnz subaddd wfun fzfid xpfi imafi czring crh crg zncrng crngring zrhrhm
+      ccrg imaexd hashclb hashcl sqrtmsq sqrtmul 1red readdcld flltp1 ltmul12ad
+      oveq1d eqbrtrd hashfz0 oveq12d hashxp wf1o ovexd xpexg mptexd cres sselda
+      wex wfn fnfvimad fmptd feqresmpt feq1d eqidd f1eq123d df-ima rneqd eqtr2d
+      f1resf1 dff1o5 f1oeq1 spcedv wi hasheqf1oi zringbas rhmf resss rnss nnssz
+      frn sseld fnfvima nnssre hashnexinjle reximddv2 r19.29vva ) AVGVOZVHVOZVP
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+      AXKWUPXCWJZUBAWXQXKXVHUBXBWXSXKXVHUYEWUPUBVVIXVHXFVVJXJUWJYAAXVFXVDAVYPWB
+      XKAVYPQYTZWBAVYPXVIYBQVYOVUJZYTZXVIYBZAXVJQYBZXVLXVMAQVYOVVKXNXVJQVVLXJAV
+      YPXVKXVIVYPXVKWAAQVYOVUTXNYNYKAWVPXVIWBYBWWAVYOWBQVVNXJYMWBXKYBAVVMXNYMYA
+      AXVDXVGAFVYPVXEAFVYPYBVYRWVNAFVYNVYPWWDYNYKVVOXIXKVYPUBVXEVVPYPVXGXFVUOAF
+      WBYIWWEWBYIYBAVVQXNYMVVRVVSVVT $.
+  $}
+
+  ${
+    $d B x $.  $d D x $.
+    $( Special case related to ~ rspsbc .  (Contributed by metakunt,
+       5-May-2025.) $)
+    rspcsbnea $p |- ( ( A e. B /\ A. x e. B C =/= D ) ->
+     [_ A / x ]_ C =/= D ) $=
+      ( wcel wne wral csb wsbc rspsbc wceq wn wb df-ne sbcbii a1i sbcng sbceq1g
+      bitrd notbid biidd necon3bbid sylibd imp ) BCFZDEGZACHZABDIZEGZUFUHUGABJZ
+      UJUGABCKUFUKUIELZMZUJUFUKDELZMZABJZUMUKUPNUFUGUOABDEOPQUFUPUNABJZMUMUNABC
+      RUFUQULABDECSUATTUFULUIEUFULUBUCTUDUE $.
+  $}
+
+  ${
+    $d .^ x y $.  $d A x y $.  $d N x $.  $d R x y $.  $d ph x y $.
+    idomnnzpownz.1 $e |- ( ph -> R e. IDomn ) $.
+    idomnnzpownz.2 $e |- ( ph -> A e. ( Base ` R ) ) $.
+    idomnnzpownz.3 $e |- ( ph -> A =/= ( 0g ` R ) ) $.
+    idomnnzpownz.4 $e |- ( ph -> N e. NN0 ) $.
+    idomnnzpownz.5 $e |- .^ = ( .g ` ( mulGrp ` R ) ) $.
+    $( A non-zero power in an integral domain is non-zero.  (Contributed by
+       metakunt, 5-May-2025.) $)
+    idomnnzpownz $p |- ( ph -> ( N .^ A ) =/= ( 0g ` R ) ) $=
+      ( wcel wa co cfv wne wceq oveq1 eqid syl adantr vx vy cn0 c0g ancli cv c1
+      cc0 caddc neeq1d cur cmgp cbs eleqtrdi mulg0 ringidval eqtr4di cnzr cidom
+      mgpbas cdomn ccrg isidom simprbi domnnzr nzrnz eqnetrd cmnd crg idomringd
+      cplusg ringmgp simplr ad2antrr mulgnn0p1 syl3anc cmulr mgpplusg a1i oveqd
+      eqcomd mulgnn0cl eqcomi eleqtrd simpr jca domnmuln0 nn0indd ) AAEUCKZLEBD
+      MZCUDNZOZAWIIUEAUAUFZBDMZWKOUHBDMZWKOUBUFZBDMZWKOZWPUGUIMZBDMZWKOWLUAUBEW
+      MUHPWNWOWKWMUHBDQUJWMWPPWNWQWKWMWPBDQUJWMWSPWNWTWKWMWSBDQUJWMEPWNWJWKWMEB
+      DQUJAWOCUKNZWKAWOCULNZUDNZXAABXBUMNZKZWOXCPABCUMNZXDGXFCXBXBRZXFRZUTZUNZX
+      DDXBBXCXDRZXCRJUOSCXAXBXGXARZUPUQACURKZXAWKOACUSKZXMFXNCVAKZXMXNCVBKXOCVC
+      VDZCVESSCXAWKXLWKRZVFSVGAWPUCKZLZWRLZWTWQBXBVKNZMZWKXTXBVHKZXRXEWTYBPXSYC
+      WRAYCXRACVIKYCACFVJCXBXGVLSTTZAXRWRVMZAXEXRWRXJVNZXDYADXBWPBXKJYARVOVPXTY
+      BWQBCVQNZMZWKXTYAYGWQBXTYGYAYGYAPXTCYGXBXGYGRZVRVSWAVTXTXOWQXFKZWRLBXFKZB
+      WKOZLZYHWKOXSXOWRAXOXRAXNXOFXPSTTXTYJWRXTWQXDXFXTYCXRXEWQXDKYDYEYFXDDXBWP
+      BXKJWBVPXDXFPXTXFXDXIWCVSWDXSWRWEWFXSYMWRAYMXRAYKYLGHWFTTXFCYGWQBWKXHYIXQ
+      WGVPVGVGWHS $.
+  $}
+
+  ${
+    $d A m y z $.  $d A x y z $.  $d G m y z $.  $d G x y z $.  $d N m n y z $.
+    $d N n x y z $.  $d R m n y z $.  $d R n x y z $.  $d m n ph y z $.
+    $d ph x y z $.
+    idomnnzgmulnz.1 $e |- G = ( mulGrp ` R ) $.
+    idomnnzgmulnz.2 $e |- ( ph -> R e. IDomn ) $.
+    idomnnzgmulnz.3 $e |- ( ph -> N e. Fin ) $.
+    idomnnzgmulnz.4 $e |- ( ( ph /\ n e. N ) -> A e. ( Base ` R ) ) $.
+    idomnnzgmulnz.5 $e |- ( ( ph /\ n e. N ) -> A =/= ( 0g ` R ) ) $.
+    $( A finite product of non-zero elements in an integral domain is non-zero.
+       (Contributed by metakunt, 5-May-2025.) $)
+    idomnnzgmulnz $p |- ( ph -> ( G gsum ( n e. N |-> A ) )
+    =/= ( 0g ` R ) ) $=
+      ( vm cmpt cgsu co cfv wne wceq wcel adantr vx vy vz cv c0g csn cun mpteq1
+      c0 oveq2d neeq1d mpt0 a1i eqid gsum0 eqtrd cur ringidval cidom cnzr cdomn
+      eqcomi ccrg isidom simprbi domnnzr syl nzrnz 3syl eqnetrd wss cdif wa csb
+      cplusg nfcv nfcsb1v csbeq1a cbvmpt oveq2i cbs ccmn simplbi crngmgp simprl
+      cfn ssfid wral ad2antrr simpr sseldd ralrimiva ad3antrrr rspcsbela mgpbas
+      syl2anc eleqtrd eldifi adantl wn eldifn csbeq1 gsumunsn gsummptcl equcoms
+      eqcomd jca rspcsbnea cmulr mgpplusg domnmuln0 syl3anc ex findcard2d ) AED
+      UAUDZBMZNOZCUEPZQEDUIBMZNOZXRQEDUBUDZBMZNOZXRQZEDYAUCUDZUFUGZBMZNOZXRQZED
+      FBMZNOZXRQUAUBUCFXOUIRZXQXTXRYLXPXSENDXOUIBUHUJUKXOYARZXQYCXRYMXPYBENDXOY
+      ABUHUJUKXOYFRZXQYHXRYNXPYGENDXOYFBUHUJUKXOFRZXQYKXRYOXPYJENDXOFBUHUJUKAXT
+      EUEPZXRAXTEUINOZYPAXSUIENXSUIRADBULUMUJYQYPRAEYPYPUNUOUMUPAYPCUQPZXRYPYRR
+      AYRYPCYREGYRUNZURVBUMACUSSZCUTSZYRXRQHYTCVASZUUAYTCVCSZUUBCVDZVEZCVFVGCYR
+      XRYSXRUNZVHVIVJVJAYAFVKZYEFYAVLSZVMZVMZYDYIUUJYDVMZYHELYADLUDZBVNZMZNOZDY
+      EBVNZEVOPZOZXRUUKYHELYFUUMMZNOZUURYHUUTRUUKYGUUSENDLYFBUUMLBVPZDUULBVQZDU
+      ULBVRZVSVTUMUUKYAEWAPZUUQLEYEFUUMUUPUVDUNUUQUNUUJEWBSZYDAUVEUUIAUUCUVEAYT
+      UUCHYTUUCUUBUUDWCVGCEGWDVGTTZUUJYAWFSYDUUJFYAAFWFSUUIITAUUGUUHWEZWGTZUUKU
+      ULYASZVMZUUMCWAPZUVDUVJUULFSBUVKSZDFWHZUUMUVKSZUVJYAFUULUUJUUGYDUVIUVGWIU
+      UKUVIWJWKAUVMUUIYDUVIAUVLDFJWLZWMDUULFBUVKWNWPZUVKUVDRZUVJUVKCEGUVKUNZWOZ
+      UMWQUUJYEFSZYDUUIUVTAUUHUVTUUGYEFYAWRWSWSZTZUUJYEYASWTZYDUUIUWCAUUHUWCUUG
+      YEFYAXAWSWSTUUKUUPUVKUVDUUKUVTUVMUUPUVKSZUWBAUVMUUIYDUVOWIDYEFBUVKWNWPZUV
+      QUUKUVSUMWQDUULYEBXBXCUPUUKUUBUUOUVKSZUUOXRQZVMUWDUUPXRQZVMUURXRQUUJUUBYD
+      AUUBUUIAYTUUBHUUEVGTTUUKUWFUWGUUKUVKLEYAUUMUVSUVFUVHUUKUVNLYAUVPWLXDUUKUU
+      OYCXRUUKUUNYBENUUNYBRUUKLDYAUUMBUVBUVAUULDUDRBUUMBUUMRDLUVCXEXFVSUMUJUUJY
+      DWJVJXGUUKUWDUWHUWEUUJUWHYDUUJUVTBXRQZDFWHZUWHUWAAUWJUUIAUWIDFKWLTDYEFBXR
+      XHWPTXGUVKCUUQUUOUUPXRUVRCXIPZUUQCUWKEGUWKUNXJVBUUFXKXLVJXMIXN $.
+  $}
+
+  ${
+    $d .^ x y $.  $d N x $.  $d R x y $.  $d ph x y $.
+    ringexp0nn.1 $e |- ( ph -> R e. Ring ) $.
+    ringexp0nn.2 $e |- ( ph -> N e. NN ) $.
+    ringexp0nn.3 $e |- .^ = ( .g ` ( mulGrp ` R ) ) $.
+    $( Zero to the power of a positive integer is zero.  (Contributed by
+       metakunt, 5-May-2025.) $)
+    ringexp0nn $p |- ( ph -> ( N .^ ( 0g ` R ) ) = ( 0g ` R ) ) $=
+      ( vx vy cn wcel wa cfv co wceq c1 oveq1 eqeq1d syl eqid c0g ancli cv cmgp
+      caddc cbs cmnd crg ringmnd mndidcl mgpbas a1i eleqtrd mulg1 cplusg simplr
+      ad2antrr mulgnnp1 syl2anc simpr oveq1d cmulr mgpplusg eqcomi ringrz eqtrd
+      adantr nnindd ) AADJKZLDBUAMZCNZVJOZAVIFUBAHUCZVJCNZVJOPVJCNZVJOZIUCZVJCN
+      ZVJOZVQPUENZVJCNZVJOVLHIDVMPOVNVOVJVMPVJCQRVMVQOVNVRVJVMVQVJCQRVMVTOVNWAV
+      JVMVTVJCQRVMDOVNVKVJVMDVJCQRAVJBUDMZUFMZKZVPAVJBUFMZWCABUGKZVJWEKZABUHKZW
+      FEBUISWEBVJWETZVJTZUJSZWEWCOAWEBWBWBTZWIUKULUMZWCCWBVJWCTZGUNSAVQJKZLZVSL
+      ZWAVRVJWBUOMZNZVJWQWOWDWAWSOAWOVSUPAWDWOVSWMUQWCWRCWBVQVJWNGWRTURUSWQWSVJ
+      VJWRNZVJWQVRVJVJWRWPVSUTVAWPWTVJOZVSAXAWOAWHWGXAEWKWEBWRVJVJWIBVBMZWRBXBW
+      BWLXBTVCVDWJVEUSVGVGVFVFVHS $.
+
+  $}
+
+  ${
+    aks6d1p5.1 $e |- ( ph -> K e. Field ) $.
+    aks6d1p5.2 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c5.3 $e |- P = ( chr ` K ) $.
+    aks6d1c5.4 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c5.5 $e |- ( ph -> A < P ) $.
+    aks6d1c5.6 $e |- X = ( var1 ` K ) $.
+    aks6d1c5.7 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c5.8 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    .^ ( X ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+
+    ${
+      $d A g i $.  $d A x y $.  $d G x y $.  $d K g i $.  $d g i ph $.
+      $( Lemma for Claim 5 of Theorem 6.1, G defines a map into the
+         polynomials.  (Contributed by metakunt, 5-May-2025.) $)
+      aks6d1c5p0 $p |- ( ph -> G : ( NN0 ^m ( 0 ... A ) ) -->
+     ( Base ` ( Poly1 ` K ) ) ) $=
+        ( cfv wcel eqid cn0 cc0 cfz co cmap cpl1 cmgp cv czrh cascl cplusg cmpt
+        cgsu cbs wa ccmn ccrg fldcrngd ply1crng syl crngmgp adantr cmnd cmnmndd
+        fzfid cvv nn0ex a1i ovexd elmapd biimpd imp ffvelcdmda crngringd cmnmnd
+        wf ringcmnd crg vr1cl simpl elfzelz adantl jca czring crh zringbas rhmf
+        cz zrhrhm ply1sclcl syl2anc mndcl syl3anc wceq mgpbas eleqtrd ralrimiva
+        mulgnn0cld gsummptcl eqcomi fmptd ) ADUAUBBUCUDZUEUDZHUFRZUGRZEXBEUHZDU
+        HZRZIXFHUIRZRZXDUJRZRZXDUKRZUDZFUDZULUMUDZXDUNRZGAXGXCSZUOZXPXEUNRZXQXS
+        XTEXEXBXOXTTZAXEUPSZXRAXDUQSZYBAHUQSYCAHJURZXDHXDTZUSUTZXDXEXETZVAUTVBZ
+        XSUBBVEXSXOXTSEXBXSXFXBSZUOZXTFXEXHXNYAPXSXEVCSYIXSXEYHVDVBXSXBUAXFXGAX
+        RXBUAXGVPZAXRYKAUAXBXGVFVFUAVFSAVGVHAUBBUCVIVJVKVLVMYJXNXQXTYJXDVCSZIXQ
+        SZXLXQSZXNXQSXSYLYIAYLXRAXDUPSYLAXDAXDYFVNVQXDVOUTVBVBYJHVRSZYMXSYOYIAY
+        OXRAHYDVNVBZVBZXQXDHIOYEXQTZVSUTYJYOXJHUNRZSZYNYQYJXSXFWHSZUOYTYJXSUUAX
+        SYIVTYIUUAXSXFUBBWAWBWCXSWHYSXFXIXSYOWHYSXIVPZYPYOXIWDHWEUDSUUBHXIXITWI
+        WHYSWDHXIWFYSTZWGUTUTVMUTXKXQXDHXJYSYEXKTUUCYRWJWKXQXMXDIXLYRXMTWLWMXQX
+        TWNYJXQXDXEYGYRWOZVHWPWRWQWSXTXQWNXSXQXTUUDWTVHWPQXA $.
+    $}
+
+    ${
+      aks6d1c5p1.1 $e |- ( ph -> B e. ( 0 ... A ) ) $.
+      aks6d1c5p1.2 $e |- ( ph -> C e. ( 0 ... A ) ) $.
+      $( Lemma for claim 5, evaluate the linear factor at -c to get a root.
+         (Contributed by metakunt, 5-May-2025.) $)
+      aks6d1c5p1 $p |- ( ph -> ( B = C <-> (
+       ( ( eval1 ` K ) `
+        ( X ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` B ) ) )
+       )
+       ` ( ( ZRHom ` K ) ` ( 0 - C ) ) ) = ( 0g ` K ) ) ) $=
path: set.mm
user: login: tirix
body: Ok - this is a lemmon, so I suppose its formulation is not crucial.
created_at: 2025-05-06T10:09:00Z
author_association: CONTRIBUTOR
diff_hunk: @@ -677132,6 +677132,253 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
path: set.mm
user: login: tirix
body: I think the usual naming would have been to use `...lem1`, `...lem2`, `...lem3` instead of `...p1`, `...p2`, `...p3`.
created_at: 2025-05-06T10:10:04Z
author_association: CONTRIBUTOR
diff_hunk: @@ -84,7 +84,11 @@ make a github issue.)
 
 DONE:
 Date      Old       New         Notes
- 2-May-25 bj-sbnf   sbnf        Moved from BJ's mathbox to main set.mm
+05-May-25 idomd     [same]      Moved from TA's mathbox to main set.mm
path: changes-set.txt
user: login: metakunt
body: Good point, it should be fixed now.
created_at: 2025-05-06T10:49:05Z
author_association: CONTRIBUTOR
diff_hunk: @@ -677132,6 +677132,253 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
path: set.mm
user: login: metakunt
body: All theorems in this PR should now follow this naming scheme.
created_at: 2025-05-06T12:09:23Z
author_association: CONTRIBUTOR
diff_hunk: @@ -677132,6 +677132,253 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
path: set.mm
user: login: tirix
body: Alexander approved, so I'm merging now.
created_at: 2025-05-06T13:56:14Z
author_association: CONTRIBUTOR
diff_hunk: @@ -677236,6 +677236,863 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
+      aks6d1c2p3.7 $e |- ( ph -> I = ( p E q ) ) $.
+      aks6d1c2p3.8 $e |- ( ph -> I e. NN0 ) $.
+      $( Lemma for ~ aks6d1c2 to simplify context.  (Contributed by metakunt,
+         1-May-2025.) $)
+      aks6d1c2lem3 $p |- ( ph -> ( J ( .g ` ( mulGrp ` K ) ) ( (
+      ( eval1 ` K ) ` ( G ` s ) ) ` M ) ) = ( I ( .g ` ( mulGrp ` K ) ) ( (
+        ( eval1 ` K ) ` ( G ` s ) ) ` M ) ) ) $=
+        ( vv cv cfv ce1 cmgp cmg co cexp cdiv cmul cn0 cvv cmpo wceq a1i simprl
+        wa oveq2d simprr oveq12d cc0 cfz wcel elfznn0 ovexd ovmpod eqtrd oveq1d
+        syl oveq2 fveq2d eqeq12d wbr wral syl2anc eqid aks6d1c1rh aks6d1c1p1rcl
+        mpbid cbs cn simpld aks6d1c1p1 rspcdva eqcomd w3a nnnn0d 3jca eqtr2d wf
+        cprimroots fveq2 cmap wb nn0ex elmapg cpl1 simprd caddc cplusg cmnd crg
+        ccrg fldcrngd crngring ringmgp nn0mulcld c0g wi ccmn crngmgp isprimroot
+        cdvds biimpd mpd simp1d mulgnn0dir mulgnn0ass simp2d mulgnn0cld mndrid
+        mulgnn0z ) AUEUHUKWMZUBWNZUFWOWNZWNZWNZUFWPWNZWQWNZWRGULWMZWSWRZUIGWTWR
+        ZRWMZWSWRZXAWRZUVRUVTWRZUDUVRUVTWRZAUEUWFUVRUVTAUEUWAUWDSWRUWFWGAQUPUWA
+        UWDXBXBGQWMZWSWRZUWCUPWMZWSWRZXAWRZUWFSXCSQUPXBXBUWMXDXEAVHXFZAUWIUWAXE
+        ZUWKUWDXEZXHXHZUWJUWBUWLUWEXAUWQUWIUWAGWSAUWOUWPXGXIUWQUWKUWDUWCWSAUWOU
+        WPXJXIXKAUWAXLEXMWRZXNUWAXBXNWEUWAEXOXTZAUWDUWRXNUWDXBXNWFUWDEXOXTZAUWB
+        UWEXAXPXQXRZXSAUWHGUNWMZWSWRZUWCUMWMZWSWRZXAWRZUVRUVTWRZUWGAUDUXFUVRUVT
+        AUDUXBUXDSWRUXFWJAQUPUXBUXDXBXBUWMUXFSXCUWNAUWIUXBXEZUWKUXDXEZXHXHZUWJU
+        XCUWLUXEXAUXJUWIUXBGWSAUXHUXIXGXIUXJUWKUXDUWCWSAUXHUXIXJXIXKAUXBUWRXNUX
+        BXBXNWHUXBEXOXTZAUXDUWRXNUXDXBXNWIUXDEXOXTZAUXCUXEXAXPXQXRZXSAUXGUXFUHU
+        VTWRZUVQWNZUWGAUXFCWMZUVQWNZUVTWRZUXFUXPUVTWRZUVQWNZXEZUXGUXOXECUVSIUUB
+        WRZUHUXPUHXEZUXRUXGUXTUXOUYCUXQUVRUXFUVTUXPUHUVQUUCZXIUYCUXSUXNUVQUXPUH
+        UXFUVTYAYBYCAUXFUVOHYDUYACUYBYEABCDGHIUXBLMNPUXFUVNUBUFUXDUIUOUQURUSUTV
+        AVBVCVDAUVNXBXLDXMWRZUUDWRXNZUYEXBUVNUUAZWDAXBXCXNZUYEXCXNUYFUYGUUEUYHA
+        UUFXFAXLDXMXPXBUYEUVNXCXCUUGYFYJZVFVGUXKUXLUXFYGVJVKYHZACUFUUHWNYKWNZUV
+        THILMUXFUVTUVOUVSUVPUQAUXFYLXNZUVOUYKXNZACUYKUVTHILMUXFUVTUVOUVSUVPUQUY
+        JYIZUUIZAUYLUYMUYNYMYNYJVLYOAUXOUWFUHUVTWRZUVQWNZUWGAUXNUYPUVQAUXNUDUHU
+        VTWRZUYPAUXFUDUHUVTAUDUXFUXMYPXSAUYPUEUHUVTWRZUYRAUWFUEUHUVTAUEUWFUXAYP
+        XSAUYSUDKIXAWRZUUJWRZUHUVTWRZUYRAUEVUAUHUVTWCXSAVUBUYRUYTUHUVTWRZUVSUUK
+        WNZWRZUYRAUVSUULXNZUDXBXNZUYTXBXNZUHUVSYKWNZXNZYQVUBVUEXEAUFUUMXNZVUFAU
+        FUUNXNZVUKAUFUSUUOZUFUUPXTUFUVSUVSYGZUUQXTZAVUGVUHVUJWKAKIAKWBYRZAIVAYR
+        ZUURAVUJIUHUVTWRZUVSUUSWNZXEZWLWMZUHUVTWRVUSXEIVVAUVDYDUUTWLXBYEZAUHUYB
+        XNZVUJVUTVVBYQZVLAVVCVVDAUVSUVTIUHWLAVULUVSUVAXNVUMUFUVSVUNUVBXTVUQUVTY
+        GZUVCUVEUVFZUVGZYSVUIVUDUVTUVSUDUYTUHVUIYGZVVEVUDYGZUVHYFAVUEUYRVUSVUDW
+        RZUYRAVUCVUSUYRVUDAVUCKVURUVTWRZVUSAVUFKXBXNZIXBXNZVUJYQVUCVVKXEVUOAVVL
+        VVMVUJVUPVUQVVGYSVUIUVTUVSKIUHVVHVVEUVIYFAVVKKVUSUVTWRZVUSAVURVUSKUVTAV
+        UJVUTVVBVVFUVJXIAVUFVVLVVNVUSXEVUOVUPVUIUVTUVSKVUSVVHVVEVUSYGZUVMYFXRXR
+        XIAVUFUYRVUIXNVVJUYRXEVUOAVUIUVTUVSUDUHVVHVVEVUOWKVVGUVKVUIVUDUVSUYRVUS
+        VVHVVIVVOUVLYFXRXRXRYTXRYBAUWGUYQAUWFUXQUVTWRZUWFUXPUVTWRZUVQWNZXEZUWGU
+        YQXECUYBUHUYCVVPUWGVVRUYQUYCUXQUVRUWFUVTUYDXIUYCVVQUYPUVQUXPUHUWFUVTYAY
+        BYCAUWFUVOHYDVVSCUYBYEABCDGHIUWALMNPUWFUVNUBUFUWDUIUOUQURUSUTVAVBVCVDUY
+        IVFVGUWSUWTUWFYGVJVKYHZACUYKUVTHILMUWFUVTUVOUVSUVPUQUYOAUWFYLXNUYMACUYK
+        UVTHILMUWFUVTUVOUVSUVPUQVVTYIYMYNYJVLYOYPXRXRYTXR $.
+    $}
+
+    ${
+      $d .~ a $.  $d A a o p q r s $.  $d A g i o p q r s $.  $d A h s $.
+      $d A k l o p q r s $.  $d A o p q r s x $.  $d B a o p q r $.
+      $d B g i o p q r $.  $d B k l o p q r $.  $d B o p q r x $.
+      $d E a o p q r $.  $d E g i o p q r $.  $d E k l o p q r $.
+      $d E o p q r x $.  $d G e f o p q r y $.  $d G h $.  $d H s $.
+      $d I a o p q r $.  $d I g i o p q r $.  $d I k l o p q r $.
+      $d I o p q r x $.  $d J a o p q r $.  $d J g i o p q r $.
+      $d J k l o p q r $.  $d J o p q r x $.  $d K a o p q r s $.
+      $d K e f o p q r s y $.  $d K g i o p q r s $.  $d K h s $.  $d K v $.
+      $d K o p q r s x $.  $d M h $.  $d M o p q r y $.  $d M v $.
+      $d N a o r $.  $d N e f o r y $.  $d N k l o r $.  $d N o r x $.
+      $d P e f y $.  $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.
+      $d R x $.  $d S s $.  $d a o p ph q r s $.  $d g i o p ph q r s $.
+      $d h ph s $.  $d k l o p ph q r s $.  $d ph v $.  $d ph q r s x $.
+      $( Claim 2 of Theorem 6.1 AKS, Preparation for injectivity proof.
+         (Contributed by metakunt, 1-May-2025.) $)
+      aks6d1c2lem4 $p |- ( ph -> ( # ` ( H " ( NN0 ^m ( 0 ... A ) ) ) ) <_
+       ( N ^ B ) ) $=
+        ( vs vv vr vo vp vq cn0 cc0 cfz co cima chash cfv c0g cexp wcel cvv cle
+        wbr fvexd syl cv wa wceq a1i fveq2d cbs eqid adantr simpr nnnn0d mpd cn
+        cmul c1 wf adantl simprd eqeltrd oveq2d wss nn0expcld cz wne wb syl3anc
+        mpbid jca elnn0z sylibr wrex nn0red cr syl2anc r19.29vva ad7antr eqcomd
+        eqtrd reexpcld letrd recnd eqbrtrd cmap ce1 csn cfn cnvexg imaexd fmptd
+        ccnv nfv ffnd fnfund cpl1 csg fveq1d ccrg fldcrngd cmpt fvmptd cmgp cmg
+        cdvds wi wral cprimroots w3a crngmgp isprimroot biimpd simp1d eleqtrrdi
+        ccmn mgpbas cdiv cfield cprime cgcd elmapi 0nn0 cascl cplusg aks6d1c1rh
+        cv1 czrh aks6d1c1p1rcl fveval1fvcl ply1crng cmnmndd cmpo simprl oveq12d
+        crs simprr fz0ssnn0 sselda sseli ovexd prmnn nn0zd dvdsval2 nnred nnrpd
+        ovmpod nnne0d nn0ge0d divge0d nn0mulcld cxp eleqtrd aks6d1c2p1 c0 csqrt
+        wfn cuz cfl czn hashscontpowcl resqrtcld flcld sqrtge0d 0zd sylib eluz1
+        flge ax-mp fzn0 ssxpb mpbird ovelimab crg crngringd vr1cl eqcomi eleq2i
+        xpnz mulgnn0cld evl1vard evl1expd caddc ad6antr simp-6r simp-5r simp-4r
+        clt simpllr simplr ad3antrrr ad2antrr aks6d1c2lem3 ad4antr 3eqtrd eqidd
+        0z evl1subd cgrp crnggrpd grpsubid elsng wfun cdm grpsubcl eqeltrid crh
+        cpws evl1rhm rhmf ffvelcdmda pwsbas eleqtrrd elmapd fnfun fndm fvimacnv
+        ex ffn funimassd ssexd cdg1 cnzr cdr isfld biimpi simpld drngnzr deg1pw
+        3brtr3d deg1sub cidom fldidom deg1nn0clb hashbnd hashcl hashss redivcld
+        fta1g remulcld 0red 1red prmgt1 expge0d nnge1d elfzuz3 leexp2ad mullidd
+        0le1 ltled nnzd dvdsle lemuldivd lemul12ad divcan2d oveq1d eqtr2d leidd
+        mulexpd ) AUBWEWFDWGWHZUUAWHZWIZWJWKZJUEUUBWKZWKZUUHZUEWLWKZUUCZWIZWJWK
+        ZUHEWMWHZAVXRAVXQUUDWNZVXRWEWNAVXQWOWNVYEWEWNZVXRVYEWPWQZVYGAVXQVYDWOAV
+        YAVYCWOAVXTWOWNVYAWOWNAJVXSWRVXTWOUUEWSUUFZAVSVXPVYDUBAVSUUIAVXPUBAVXPW
+        OUBAOVXPUGOWTZUAWKZVXSWKZWKZWOUBAVYKVXPWNXAUGVYMWRVHUUGUUJUUKAVSWTZVXPW
+        NZXAZVYOUBWKZVXTWKZVYCWNZVYRVYDWNZVYQVYTVYSVYBXBZVYQVYSVYRUDUISWHZUCUIS
+        WHZUEUULWKZUUMWKZWHZVXSWKZWKZVYBVYQVYRVXTWUHVYQJWUGVXSJWUGXBZVYQVPXCXDU
+        UNVYQWUIVYRWUDVXSWKWKZWUKUEUUMWKZWHZVYBVYQWUGWUEXEWKZWNZWUIWUMXBVYQUEXE
+        WKZWULWUEUEWUNWUCWUFWUDVXSWUKWUKVYRVXSXFZWUEXFZWUPXFZWUNXFZAUEUUOWNZVYP
+        AUEUNUUPZXGZVYQVYRUGVYOUAWKZVXSWKZWKZWUPVYQOVYOVYNWVFVXPUBWOUBOVXPVYNUU
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+        CXFZAWWEUEVWGWNUNUEVWHWSXYTAJYVCYBZYUKYVBAXWBXYRYVEYUKYCXWCXYTWUNYUIWUE
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+        UFVYDVXQWOVWLYLZVXQVYEWOVWJYDVXQVWKWSYJAVYEYVGYJZAVYFAUHEXUDXVPXTYJZYVH
+        AVYEYUJVYFYVIAYUJYUMYKYUNAYUMYULYKYVAAYULUDYKYUTAUDXVTYJXQXQXQYVJYVFAYU
+        JYUMVYFWPYUNAYUMYULVYFWPYVAAYULUDVYFWPYUTAWXHUDVYFWPWQWAWBWXJWXJWXOUDWX
+        GVYFWPWXPWXNWXGVYFWPWQWXHWXNWXGWXSVYFWPWYNWXNWXSGEWMWHZWWCEWMWHZXLWHZVY
+        FWXNWXQWXRWXNGWXEWXLGYKWNZWXMAYVNWXKAWWFYVNUOWWFGWYPUWTWSZXGXGZWYLYQZWX
+        NWWCWXFWXLWWCYKWNZWXMAYVRWXKAUHGXUEYVOXUCVWMZXGXGZWYMYQZVWOWXLYVMYKWNWX
+        MWXLYVKYVLAYVKYKWNZWXKAGEYVOXVPYQXGZAYVLYKWNZWXKAWWCEYVSXVPYQXGZVWOXGWX
+        LVYFYKWNZWXMAYWFWXKAUHEXUEXVPYQXGXGZWXNWXQYVKWXRYVLYVQWXLYWBWXMYWCXGYWA
+        WXLYWDWXMYWEXGWXNGWXEYVPWYLWXLWFGWPWQZWXMAYWHWXKAWFXMGAVWPAVWQZYVOWFXMW
+        PWQAVXDXCAXMGYWIYVOAWWFXMGVUCWQUOGVWRWSVXEYRXGXGVWSWXNWWCWXFYVTWYMWXLXU
+        AWXMAXUAWXKXUGXGXGVWSWXNGWXEEYVPWXLXMGWPWQZWXMAYWJWXKAGWYQVWTXGXGWXLEWX
+        EUXMWKWNZWXMWXKYWKAWXEWFEVXAXOXGVXBWXNWWCWXFEYVTWXLXMWWCWPWQZWXMAYWLWXK
+        AXMGXLWHZUHWPWQYWLAYWMGUHWPAGAGYVOYSZVXCAWWIGUHWPWQZURAXUBWWHWWIYWOUVBA
+        GWYQVXFUQGUHVXGYLXJYTAXMUHGYWIXUEXUFVXHYEXGXGWXMEWXFUXMWKWNWXLWXFWFEVXA
+        XOVXBVXIWXNYVMVYFVYFWPWXNVYFGWWCXLWHZEWMWHYVMWXNUHYWPEWMWXLUHYWPXBZWXMA
+        YWQWXKAYWPUHAUHGAUHXUEYSYWNXUCVXJYOXGXGVXKWXNGWWCEWXNGYVPYSWXNWWCYVTYSA
+        XVCWXKWXMXVPVUGVXNVXLWXNVYFYWGVXMYTYRYTXGYTXVSYMYTYTYTYRYR $.
+    $}
+  $}
+
+  ${
+    $d A f $.  $d A x y $.  $d B f $.  $d F f $.  $d F x y $.
+    hashnexinj.1 $e |- ( ph -> A e. Fin ) $.
+    hashnexinj.2 $e |- ( ph -> B e. Fin ) $.
+    hashnexinj.3 $e |- ( ph -> ( # ` B ) < ( # ` A ) ) $.
+    hashnexinj.4 $e |- ( ph -> F : A --> B ) $.
+    $( If the number of elements of the domain are greater than the number of
+       elements in a codomain, then there are two different values that map to
+       the same.  (Contributed by metakunt, 2-May-2025.) $)
+    hashnexinj $p |- ( ph -> E. x e. A E. y e. A
+     ( ( F ` x ) = ( F ` y ) /\ x =/= y ) ) $=
+      ( vf cfv wa wn cfn wcel syl notbid mpd wral cv wceq wne wrex wf1 wal cdom
+      wf wex wbr chash cle clt cn0 hashcl nn0red ltnled mpbid wb hashdom biimpd
+      syl2anc wi brdomg alnex sylibr cmap elmapdd f1eq1 spcgv dff13 iman anbi2i
+      co df-ne xchbinxr 2ralbii ralnex2 bitri a1i mpnanrd notnotrd ) ABUAZFLCUA
+      ZFLUBZWCWDUCZMZCDUDBDUDZADEFUHZWHNZJADEFUEZNZWIWJMZNZADEKUAZUEZNZKUFZWLAW
+      PKUIZNZWRADEUGUJZNZWTADUKLZEUKLZULUJZNZXBAXDXCUMUJXFIAXDXCAXDAEOPZXDUNPHE
+      UOQUPAXCADOPZXCUNPGDUOQUPUQURAXFXBAXEXAAXHXGXEXAUSGHDEOUTVBRVASAXGXBWTVCH
+      XGXBWTXGXAWSDEOKVDRVAQSWPKVEVFAFEDVGVNZPWRWLVCAEDFOOHGJVHWQWLKFXIWOFUBWPW
+      KDEWOFVIRVJQSAWLWNAWKWMWKWMUSAWKWIWEWCWDUBZVCZCDTBDTZMWMBCDEFVKXLWJWIXLWG
+      NZCDTBDTWJXKXMBCDDXKWEXJNZMWGWEXJVLWFXNWEWCWDVOVMVPVQWGBCDDVRVSVMVSVTRVAS
+      WAWB $.
+  $}
+
+  ${
+    $d A w x y z $.  $d F w x y z $.  $d ph x y $.
+    hashnexinjle.1 $e |- ( ph -> A e. Fin ) $.
+    hashnexinjle.2 $e |- ( ph -> B e. Fin ) $.
+    hashnexinjle.3 $e |- ( ph -> ( # ` B ) < ( # ` A ) ) $.
+    hashnexinjle.4 $e |- ( ph -> F : A --> B ) $.
+    hashnexinjle.5 $e |- ( ph -> A C_ RR ) $.
+    $( If the number of elements of the domain are greater than the number of
+       elements in a codomain, then there are two different values that map to
+       the same.  Also we introduce a one sided inequality to simplify a
+       duplicateable proof.  (Contributed by metakunt, 2-May-2025.) $)
+    hashnexinjle $p |- ( ph -> E. x e. A E. y e. A
path: set.mm
user: login: icecream17
body: ```diff
-hashexinjle
+hashexinjlt
```
created_at: 2025-05-06T14:01:07Z
author_association: CONTRIBUTOR
diff_hunk: @@ -677236,6 +677236,863 @@ fixed reference functional determined by this vector (corresponding to
       WEWIWEWKWEUAWOWEWLWEUBUCUDUEUFUGUHUIUJUKULUMUNUOWF $.
   $}
 
+  ${
+    aks6d1c2.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2.20 $e |- ( ph -> I e. C ) $.
+    aks6d1c2.21 $e |- ( ph -> J e. C ) $.
+    aks6d1c2.22 $e |- ( ph -> I < J ) $.
+    aks6d1c2.23 $e |- .^ = ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) $.
+    aks6d1c2.24 $e |- X = ( var1 ` K ) $.
+    aks6d1c2.25 $e |- S = ( ( J .^ X ) ( -g ` ( Poly1 ` K ) ) ( I .^ X ) ) $.
+    aks6d1c2.26 $e |- ( ph -> U e. NN ) $.
+    aks6d1c2.27 $e |- ( ph -> J = ( I + ( U x. R ) ) ) $.
+    ${
+      $d .~ a $.  $d A a $.  $d A g i $.  $d A x $.  $d G e f y $.  $d K a $.
+      $d K e f y $.  $d K g i $.  $d K v $.  $d K x $.  $d M v $.  $d M y $.
+      $d N a $.  $d N e f y $.  $d N k l $.  $d N x $.  $d P e f y $.
+      $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.  $d R x $.  $d a ph $.
+      $d e f o y $.  $d e f p y $.  $d e f q y $.  $d e f r y $.  $d e f s y $.
+      $d g i ph $.  $d g i s $.  $d k l o $.  $d k l p $.  $d k l ph $.
+      $d k l q $.  $d k l r $.  $d ph v $.  $d ph x $.
+      aks6d1c2p3.1 $e |- ( ph -> s e. ( NN0 ^m ( 0 ... A ) ) ) $.
+      aks6d1c2p3.2 $e |- ( ph -> r e. ( 0 ... B ) ) $.
+      aks6d1c2p3.3 $e |- ( ph -> o e. ( 0 ... B ) ) $.
+      aks6d1c2p3.4 $e |- ( ph -> J = ( r E o ) ) $.
+      aks6d1c2p3.5 $e |- ( ph -> p e. ( 0 ... B ) ) $.
+      aks6d1c2p3.6 $e |- ( ph -> q e. ( 0 ... B ) ) $.
+      aks6d1c2p3.7 $e |- ( ph -> I = ( p E q ) ) $.
+      aks6d1c2p3.8 $e |- ( ph -> I e. NN0 ) $.
+      $( Lemma for ~ aks6d1c2 to simplify context.  (Contributed by metakunt,
+         1-May-2025.) $)
+      aks6d1c2lem3 $p |- ( ph -> ( J ( .g ` ( mulGrp ` K ) ) ( (
+      ( eval1 ` K ) ` ( G ` s ) ) ` M ) ) = ( I ( .g ` ( mulGrp ` K ) ) ( (
+        ( eval1 ` K ) ` ( G ` s ) ) ` M ) ) ) $=
+        ( vv cv cfv ce1 cmgp cmg co cexp cdiv cmul cn0 cvv cmpo wceq a1i simprl
+        wa oveq2d simprr oveq12d cc0 cfz wcel elfznn0 ovexd ovmpod eqtrd oveq1d
+        syl oveq2 fveq2d eqeq12d wbr wral syl2anc eqid aks6d1c1rh aks6d1c1p1rcl
+        mpbid cbs cn simpld aks6d1c1p1 rspcdva eqcomd w3a nnnn0d 3jca eqtr2d wf
+        cprimroots fveq2 cmap wb nn0ex elmapg cpl1 simprd caddc cplusg cmnd crg
+        ccrg fldcrngd crngring ringmgp nn0mulcld c0g wi ccmn crngmgp isprimroot
+        cdvds biimpd mpd simp1d mulgnn0dir mulgnn0ass simp2d mulgnn0cld mndrid
+        mulgnn0z ) AUEUHUKWMZUBWNZUFWOWNZWNZWNZUFWPWNZWQWNZWRGULWMZWSWRZUIGWTWR
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+        UVTHILMUWFUVTUVOUVSUVPUQVVTYIYMYNYJVLYOYPXRXRYTXR $.
+    $}
+
+    ${
+      $d .~ a $.  $d A a o p q r s $.  $d A g i o p q r s $.  $d A h s $.
+      $d A k l o p q r s $.  $d A o p q r s x $.  $d B a o p q r $.
+      $d B g i o p q r $.  $d B k l o p q r $.  $d B o p q r x $.
+      $d E a o p q r $.  $d E g i o p q r $.  $d E k l o p q r $.
+      $d E o p q r x $.  $d G e f o p q r y $.  $d G h $.  $d H s $.
+      $d I a o p q r $.  $d I g i o p q r $.  $d I k l o p q r $.
+      $d I o p q r x $.  $d J a o p q r $.  $d J g i o p q r $.
+      $d J k l o p q r $.  $d J o p q r x $.  $d K a o p q r s $.
+      $d K e f o p q r s y $.  $d K g i o p q r s $.  $d K h s $.  $d K v $.
+      $d K o p q r s x $.  $d M h $.  $d M o p q r y $.  $d M v $.
+      $d N a o r $.  $d N e f o r y $.  $d N k l o r $.  $d N o r x $.
+      $d P e f y $.  $d P k l $.  $d P x $.  $d R e f y $.  $d R v $.
+      $d R x $.  $d S s $.  $d a o p ph q r s $.  $d g i o p ph q r s $.
+      $d h ph s $.  $d k l o p ph q r s $.  $d ph v $.  $d ph q r s x $.
+      $( Claim 2 of Theorem 6.1 AKS, Preparation for injectivity proof.
+         (Contributed by metakunt, 1-May-2025.) $)
+      aks6d1c2lem4 $p |- ( ph -> ( # ` ( H " ( NN0 ^m ( 0 ... A ) ) ) ) <_
+       ( N ^ B ) ) $=
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+        YWQWXKAYWPUHAUHGAUHXUEYSYWNXUCVXJYOXGXGVXKWXNGWWCEWXNGYVPYSWXNWWCYVTYSA
+        XVCWXKWXMXVPVUGVXNVXLWXNVYFYWGVXMYTYRYTXGYTXVSYMYTYTYTYRYR $.
+    $}
+  $}
+
+  ${
+    $d A f $.  $d A x y $.  $d B f $.  $d F f $.  $d F x y $.
+    hashnexinj.1 $e |- ( ph -> A e. Fin ) $.
+    hashnexinj.2 $e |- ( ph -> B e. Fin ) $.
+    hashnexinj.3 $e |- ( ph -> ( # ` B ) < ( # ` A ) ) $.
+    hashnexinj.4 $e |- ( ph -> F : A --> B ) $.
+    $( If the number of elements of the domain are greater than the number of
+       elements in a codomain, then there are two different values that map to
+       the same.  (Contributed by metakunt, 2-May-2025.) $)
+    hashnexinj $p |- ( ph -> E. x e. A E. y e. A
+     ( ( F ` x ) = ( F ` y ) /\ x =/= y ) ) $=
+      ( vf cfv wa wn cfn wcel syl notbid mpd wral cv wceq wne wrex wf1 wal cdom
+      wf wex wbr chash cle clt cn0 hashcl nn0red ltnled mpbid wb hashdom biimpd
+      syl2anc wi brdomg alnex sylibr cmap elmapdd f1eq1 spcgv dff13 iman anbi2i
+      co df-ne xchbinxr 2ralbii ralnex2 bitri a1i mpnanrd notnotrd ) ABUAZFLCUA
+      ZFLUBZWCWDUCZMZCDUDBDUDZADEFUHZWHNZJADEFUEZNZWIWJMZNZADEKUAZUEZNZKUFZWLAW
+      PKUIZNZWRADEUGUJZNZWTADUKLZEUKLZULUJZNZXBAXDXCUMUJXFIAXDXCAXDAEOPZXDUNPHE
+      UOQUPAXCADOPZXCUNPGDUOQUPUQURAXFXBAXEXAAXHXGXEXAUSGHDEOUTVBRVASAXGXBWTVCH
+      XGXBWTXGXAWSDEOKVDRVAQSWPKVEVFAFEDVGVNZPWRWLVCAEDFOOHGJVHWQWLKFXIWOFUBWPW
+      KDEWOFVIRVJQSAWLWNAWKWMWKWMUSAWKWIWEWCWDUBZVCZCDTBDTZMWMBCDEFVKXLWJWIXLWG
+      NZCDTBDTWJXKXMBCDDXKWEXJNZMWGWEXJVLWFXNWEWCWDVOVMVPVQWGBCDDVRVSVMVSVTRVAS
+      WAWB $.
+  $}
+
+  ${
+    $d A w x y z $.  $d F w x y z $.  $d ph x y $.
+    hashnexinjle.1 $e |- ( ph -> A e. Fin ) $.
+    hashnexinjle.2 $e |- ( ph -> B e. Fin ) $.
+    hashnexinjle.3 $e |- ( ph -> ( # ` B ) < ( # ` A ) ) $.
+    hashnexinjle.4 $e |- ( ph -> F : A --> B ) $.
+    hashnexinjle.5 $e |- ( ph -> A C_ RR ) $.
+    $( If the number of elements of the domain are greater than the number of
+       elements in a codomain, then there are two different values that map to
+       the same.  Also we introduce a one sided inequality to simplify a
+       duplicateable proof.  (Contributed by metakunt, 2-May-2025.) $)
+    hashnexinjle $p |- ( ph -> E. x e. A E. y e. A
+     ( ( F ` x ) = ( F ` y ) /\ x < y ) ) $=
+      ( vw vz cv cfv wceq clt wbr wa wrex simpr wb fveq2 eqeq2d anbi12d fveqeq2
+      breq2 breq1 cbvrex2vw a1i biimpd imp sylib rexcom hashnexinj wcel simplrl
+      wne wo jca orcd eqcomd olcd simprr simpl simprl sselda syl adantr lttri2d
+      cr mpbid mpjaodan ex reximdvva r19.43 rexbii mpd ) ABNZFOZCNZFOZPZVSWAQRZ
+      SZCDTZBDTZWGWBVTPZWAVSQRZSZCDTZBDTZAWGUAAWLSZWEBDTCDTZWGWMLNZFOZMNZFOZPZW
+      OWQQRZSZLDTMDTZWNAWLXBAWLXBWLXBUBAWJXAWBWRPZWAWQQRZSBCMLDDVSWQPZWHXCWIXDX
+      EVTWRWBVSWQFUCUDVSWQWAQUGUEWAWOPXCWSXDWTWAWOWRFUFWAWOWQQUHUEUIUJUKULXAWEW
+      PWBPZWOWAQRZSMLCBDDWQWAPZWSXFWTXGXHWRWBWPWQWAFUCUDWQWAWOQUGUEWOVSPXFWCXGW
+      DWOVSWBFUFWOVSWAQUHUEUIUMWECBDDUNUMAWCVSWAURZSZCDTBDTZWGWLUSZABCDEFGHIJUO
+      AXKXLAXKSZWFWKUSZBDTZXLXMWEWJUSZCDTZBDTZXOAXKXRAXJXPBCDDAVSDUPZWADUPZSZSZ
+      XJXPYBXJSZWDXPWIYCWDSZWEWJYDWCWDYBWCXIWDUQYCWDUAUTVAYCWISZWJWEYEWHWIYEVTW
+      BYBWCXIWIUQVBYCWIUAUTVCYCXIWDWIUSYBWCXIVDYCVSWAYBVSVKUPZXJYBAXSSYFYBAXSAY
+      AVEZAXSXTVFUTADVKVSKVGVHVIYBWAVKUPZXJYBAXTSYHYBAXTYGAXSXTVDUTADVKWAKVGVHV
+      IVJVLVMVNVOULXQXNBDWEWJCDVPVQUMWFWKBDVPUMVNVRVM $.
+  $}
+
+  ${
+    $d .~ a $.  $d A a b c d $.  $d A b c d g i $.  $d A b c d h $.
+    $d A b c d k l $.  $d A b c d x $.  $d B a b c d $.  $d B b c d g i $.
+    $d B b c d k l $.  $d B u w $.  $d B b c d x $.  $d C a b c d $.
+    $d C b c d g i $.  $d C b c d h $.  $d C b c d k l $.  $d C b c s t $.
+    $d C b c d x $.  $d E a $.  $d E g i $.  $d E k l $.  $d E o $.  $d E t $.
+    $d E u w $.  $d E x $.  $d G e f y $.  $d G h $.  $d H b c d $.  $d K a $.
+    $d K e f y $.  $d K g i $.  $d K h $.  $d K x $.  $d L b c d t $.
+    $d L b c s t $.  $d M h $.  $d M y $.  $d N a b c d $.  $d N e f y $.
+    $d N b c d k l $.  $d N b c d x $.  $d P e f y $.  $d P k l $.  $d P x $.
+    $d R a d $.  $d R e f y $.  $d R d g i $.  $d R d h $.  $d R d k l $.
+    $d R d x $.  $d a b c d ph $.  $d g i ph $.  $d h ph $.  $d k l ph $.
+    $d o ph $.  $d ph s t $.  $d ph w $.  $d ph x $.
+    aks6d1c2a.1 $e |- .~ = { <. e , f >. | ( e e. NN /\ f e.
+     ( Base ` ( Poly1 ` K ) ) /\ A. y e. ( ( mulGrp ` K ) PrimRoots R )
+     ( e ( .g ` ( mulGrp ` K ) ) ( ( ( eval1 ` K ) ` f ) ` y ) ) =
+     ( ( ( eval1 ` K ) ` f ) ` ( e ( .g ` ( mulGrp ` K ) ) y ) ) ) } $.
+    aks6d1c2a.2 $e |- P = ( chr ` K ) $.
+    aks6d1c2a.3 $e |- ( ph -> K e. Field ) $.
+    aks6d1c2a.4 $e |- ( ph -> P e. Prime ) $.
+    aks6d1c2a.5 $e |- ( ph -> R e. NN ) $.
+    aks6d1c2a.6 $e |- ( ph -> N e. NN ) $.
+    aks6d1c2a.7 $e |- ( ph -> P || N ) $.
+    aks6d1c2a.8 $e |- ( ph -> ( N gcd R ) = 1 ) $.
+    aks6d1c2a.9 $e |- ( ph -> F : ( 0 ... A ) --> NN0 ) $.
+    aks6d1c2a.10 $e |- G = ( g e. ( NN0 ^m ( 0 ... A ) ) |->
+    ( ( mulGrp ` ( Poly1 ` K ) ) gsum ( i e. ( 0 ... A ) |-> ( ( g ` i )
+    ( .g ` ( mulGrp ` ( Poly1 ` K ) ) ) ( ( var1 ` K ) ( +g ` ( Poly1 ` K ) )
+    ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` i ) ) ) ) ) ) ) $.
+    aks6d1c2a.11 $e |- ( ph -> A e. NN0 ) $.
+    aks6d1c2a.12 $e |- E = ( k e. NN0 , l e. NN0 |->
+    ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $.
+    aks6d1c2a.13 $e |- L = ( ZRHom ` ( Z/nZ ` R ) ) $.
+    aks6d1c2a.14 $e |- ( ph -> A. a e. ( 1 ... A ) N .~ ( ( var1 ` K )
+     ( +g ` ( Poly1 ` K ) )
+     ( ( algSc ` ( Poly1 ` K ) ) ` ( ( ZRHom ` K ) ` a ) ) ) ) $.
+    aks6d1c2a.15 $e |- ( ph -> ( x e. ( Base ` K ) |->
+     ( P ( .g ` ( mulGrp ` K ) ) x ) ) e. ( K RingIso K ) ) $.
+    aks6d1c2a.16 $e |- ( ph -> M e. ( ( mulGrp ` K ) PrimRoots R ) ) $.
+    aks6d1c2a.17 $e |- H = ( h e. ( NN0 ^m ( 0 ... A ) ) |->
+     ( ( ( eval1 ` K ) ` ( G ` h ) ) ` M ) ) $.
+    aks6d1c2a.18 $e |- B = ( |_ ` ( sqrt `
+     ( # ` ( L " ( E " ( NN0 X. NN0 ) ) ) ) ) ) $.
+    aks6d1c2a.19 $e |- C = ( E " ( ( 0 ... B ) X. ( 0 ... B ) ) ) $.
+    aks6d1c2a.20 $e |- ( ph -> ( Q e. Prime /\ Q || N /\ P =/= Q ) ) $.
+    $( Claim 2 of Theorem 6.1 of ~ https://www3.nd.edu/andyp/notes/AKS.pdf
+       (Contributed by metakunt, 2-May-2025.) $)
+    aks6d1c2 $p |- ( ph -> ( # ` ( H " ( NN0 ^m ( 0 ... A ) ) ) ) <_
+       ( N ^ B ) ) $=
+      ( vb vc vd vt vs vo vu vw cv clt wbr cmul co caddc wceq cn wa cn0 cc0 cfz
+      wrex cima chash cfv cle wcel simprl jca simprr cmgp cmg ad5antr cprime c1
+      cdvds wf cbs cmpt simpllr eqid simplr simpr imp syl cz cvv nfcv a1i fvexd
+      fveq2d fvmptd eqcomd eqtrd wb adantr wss cxp nn0red nn0ge0d syl2anc mpbid
+      wne cr sylibr mpbird wf1 sstrd sseq1d ad2antrr syl3anc recnd cfn breqtrrd
+      crn cmap cexp simpl cv1 cpl1 csg cfield cgcd czrh cascl cplusg cprimroots
+      wral simp-5r simp-4r aks6d1c2lem4 ex rexlimdva cmin fveq2 cbvmpt fz0ssnn0
+      crs nnnn0d cuz csqrt cfl czn hashscontpowcl resqrtcld flcld sqrtge0d flge
+      c0 0zd elnn0z eleq1d nn0zd eluz fzn0 xpnz biimpi imass2 nfv simp1d simp2d
+      simp3d aks6d1c2p2 f1f ffnd fnfund ffvelcdmda funimassd sseldd nnzd zndvds
+      ssxpb zsubcld divides biimpd mpd cdiv nnred resubcld nnrpd posdifd rpgt0d
+      crp rpred divgt0d zred mulcomd gt0ne0d divmuld breqtrd cc eqeltrd subaddd
+      elnnz reximssdv wfun fzfid xpfi imafi czring crh crg ccrg zncrng crngring
+      zrhrhm imaexd hashclb hashcl sqrtmsq sqrtmul 1red flltp1 oveq1d ltmul12ad
+      readdcld eqbrtrd hashfz0 oveq12d hashxp wf1o wex ovexd mptexd cres sselda
+      xpexg fnfvimad fmptd feqresmpt feq1d f1resf1 eqidd f1eq123d df-ima eqtr2d
+      wfn rneqd dff1o5 f1oeq1 spcedv wi hasheqf1oi zringbas rhmf resss rnss frn
+      nnssz sseld fnfvima nnssre hashnexinjle reximddv2 r19.29vva ) AVGVOZVHVOZ
+      VPVQZVWBVWAVIVOZJVRVSZVTVSZWAZVIWBWGZWCZTWDWEDWFVSZUUAVSWHWIWJUDEUUBVSWKV
+      QZVGVHFFAVWAFWLZWCZVWBFWLZWCZVWIWCZVWOVWCWCZVWHWCVWKVWPVWQVWHVWPVWOVWCVWO
+      VWIUUCVWOVWCVWHWMWNVWOVWCVWHWOWNVWQVWHVWKVWQVWGVWKVIWBVWQVWDWBWLZWCZVWGVW
+      KVWSVWGWCBCDEFGIJVWBUAUUDWJZUAUUEWJZWPWJWQWJZVSVWAVWTVXBVSVXAUUFWJVSZVWDK
+      LMNOPQVXBRSTVWAVWBUAUBUCUDVWTUEUFUGUHAUAUUGWLVWLVWNVWCVWRVWGUIWRAGWSWLVWL
+      VWNVWCVWRVWGUJWRAJWBWLVWLVWNVWCVWRVWGUKWRAUDWBWLVWLVWNVWCVWRVWGULWRAGUDXA
+      VQVWLVWNVWCVWRVWGUMWRAUDJUUHVSWTWAVWLVWNVWCVWRVWGUNWRAVWJWDRXBVWLVWNVWCVW
+      RVWGUOWRUPADWDWLVWLVWNVWCVWRVWGUQWRURUSAUDVWTUEVOUAUUIWJWJVXAUUJWJWJVXAUU
+      KWJVSIVQUEWTDWFVSUUMVWLVWNVWCVWRVWGUTWRABUAXCWJGBVOUAWPWJZWQWJVSXDUAUAUVC
+      VSWLVWLVWNVWCVWRVWGVAWRAUCVXDJUULVSWLVWLVWNVWCVWRVWGVBWRVCVDVEAVWLVWNVWCV
+      WRVWGUUNVWMVWNVWCVWRVWGUUOVWOVWCVWRVWGXEVXBXFVWTXFVXCXFVWQVWRVWGXGVWSVWGX
+      HUUPUUQUURXIXJAVWAVJFVJVOZUBWJZXDZWJZVWBVXGWJZWAZVWCWCZVWIVGVHFFVWOVXKWCZ
+      VWCVWHVWOVXJVWCWOZVXLVWEVWBVWAUUSVSZWAZVWGVIWBXKVXLJVXNXAVQZVXOVIXKWGZVXL
+      VWBUBWJZVWAUBWJZWAZVXPVXLVXSVXRVXLVXSVXHVXRVXLVXHVXSVXLVKVWAVKVOZUBWJZVXS
+      FVXGXLVXGVKFVYBXDWAVXLVJVKFVXFVYBVKVXFXMVJVYBXMVXEVYAUBUUTUVAXNZVXLVYAVWA
+      WAZWCVYAVWAUBVXLVYDXHXPAVWLVWNVXKXEVXLVWAUBXOXQXRVXLVXHVXIVXRVWOVXJVWCWMV
+      XLVKVWBVYBVXRFVXGXLVYCVXLVYAVWBWAZWCVYAVWBUBVXLVYEXHXPVWMVWNVXKXGVXLVWBUB
+      XOXQXSXSXRVXLJWDWLZVWBXKWLZVWAXKWLZVXTVXPXTVWOVYFVXKVWMVYFVWNAVYFVWLAJUKU
+      VDZYAYAYAZVWOVYGVXKVWOVWBVWOFWBVWBAFWBYBZVWLVWNAVYKQWEEWFVSZVYLYCZWHZWBYB
+      AVYNQWDWDYCZWHZWBAVYMVYOYBZVYNVYPYBZAVYQVYLWDYBZVYSWCZAVYSVYSVYSAEUVBXNZW
+      UAWNAVYMUVNYHZVYQVYTXTAVYLUVNYHZWUCWCZWUBAWUCWUCAEWEUVEWJWLZWUCAWUEWEEWKV
+      QZAEAEWDWLZUBVYPWHZWIWJZUVFWJZUVGWJZWDWLZAWUKXKWLZWEWUKWKVQZWCZWULAWUMWUN
+      AWUJAWUIAWUIAGJPQUBUDJUVHWJZUFULUJUMUKUNURUSWUPXFZUVIZYDZAWUIWURYEZUVJZUV
+      KAWEWUJWKVQZWUNAWUIWUSWUTUVLAWUJYIWLZWEXKWLZWVBWUNXTZWVAAUVOZWUJWEUVMZYFY
+      GWNWUKUVPZYJAEWUKWDEWUKWAAVDXNZUVQZYKZYEAWVDEXKWLWUEWUFXTWVFAEWVKUVRWEEUV
+      SYFYKWEEUVTYJZWVLWNWUDWUBVYLVYLUWAUWBXJVYLVYLWDWDUWQXJYKZVYMVYOQUWCXJZAVL
+      VYOWBQAVLUWDAVYOQAVYOWBQAVYOWBQYLZVYOWBQXBZAGHPQUDUFULUJUMURAHWSWLZHUDXAV
+      QZGHYHZVFUWEAWVQWVRWVSVFUWFAWVQWVRWVSVFUWGUWHZVYOWBQUWIXJZUWJZUWKZAVYOWBV
+      LVOQWWAUWLUWMYMAFVYNWBFVYNWAAVEXNZYNYKZYOZVWMVWNXHUWNZUWOYAZVWOVYHVXKVWOV
+      WAVWOFWBVWAWWFAVWLVWNXGUWNZUWOYAZVWBVWAUBJWUPWUQUSUWPYPYGVXLVXPVXQVXLJXKW
+      LVXNXKWLVXPVXQXTVXLJVYJUVRVXLVWBVWAWWHWWJUWRVIJVXNUWSYFUWTUXAVXLVWDXKWLZV
+      XOWCZWCZWWKWEVWDVPVQZWCVWRWWMWWKWWNVXLWWKVXOWMZWWMWEVXNJUXBVSZVWDVPWWMVXN
+      JWWMVWBVWAWWMVWBVWOVWBWBWLVXKWWLWWGYOUXCZWWMVWAVWOVWAWBWLVXKWWLWWIYOUXCZU
+      XDZWWMJVXLJUXHWLZWWLVWOWWTVXKVWMWWTVWNAWWTVWLAJUKUXEYAYAYAYAZUXIZWWMVWCWE
+      VXNVPVQVXLVWCWWLVXMYAWWMVWAVWBWWRWWQUXFYGWWMJWXAUXGZUXJWWMWWPVWDWAJVWDVRV
+      SZVXNWAWWMWXDVWEVXNWWMJVWDWWMJWXBYQZWWMVWDWWMVWDWWOUXKYQZUXLVXLWWKVXOWOZX
+      SWWMVXNJVWDWWMVXNWWSYQZWXEWXFWWMJWXCUXMUXNYKUXOWNVWDUXSYJWWMVWFVWBWWMVXNV
+      WEWAVWFVWBWAWWMVWEVXNWXGXRWWMVWBVWAVWEWWMVWBWWQYQWWMVWAWWRYQWWMVWEVXNUXPW
+      XGWXHUXQUXRYGXRUXTWNAVGVHFWUHVXGAFYRWLVYNYRWLZAQUYAVYMYRWLZWXIWWCAVYLYRWL
+      ZWXKWXJAWEEUYBZWXLVYLVYLUYCYFQVYMUYDYFAFVYNYRWWDUVQYKAWUHYRWLZWUIWDWLZWUR
+      AWUHXLWLWXMWXNXTAUBVYPUYEWUPUYFVSZAWUPUYGWLZUBWXOWLZAWUPUYHWLZWXPAVYFWXRV
+      YIJWUPWUQUYIXJWUPUYJXJWUPUBUSUYKXJZUYLWUHXLUYMXJYKZAWUIVYNWIWJZFWIWJVPAWU
+      IVYMWIWJZWYAVPAWUIVYLWIWJZWYCVRVSZWYBVPAWUIEWTVTVSZWYEVRVSZWYDVPAWUIWUIWU
+      IVRVSUVFWJZWYFVPAWYGWUIAWUIYIWLZWEWUIWKVQZWYGWUIWAAWUIAWXMWXNWXTWUHUYNXJZ
+      YDZAWUIWYJYEZWUIUYOYFXRAWYGWUJWUJVRVSZWYFVPAWYHWYIWCZWYNWYGWYMWAAWYHWYIWY
+      KWYLWNZWYOWUIWUIUYPYFAWUJWYEWUJWYEAWUIWYKWYLUVJZAEWTAEAWUGWULAWUOWULAWUMW
+      UNAWUJWYPUVKAWVBWUNAWUIWYKWYLUVLZAWVCWVDWVEWYPWVFWVGYFYGWNWVHYJWVJYKZYDAU
+      YQVUAZWYPWYSWYQAWUJWUKWTVTVSZWYEVPAWVCWUJWYTVPVQWYPWUJUYRXJAEWUKWTVTWVIUY
+      SYSZWYQXUAUYTVUBVUBAWYCWYEWYCWYEVRAWUGWYCWYEWAWYREVUCXJZXUBVUDYSAWXKWXKWC
+      WYBWYDWAAWXKWXKWXLWXLWNVYLVYLVUEXJYSAVYMVYNVMVOZVUFZVMVUGZWYBWYAWAZAXUDVY
+      MVYNVNVYMVNVOZQWJZXDZVUFZVMXLXUIAVNVYMXUHXLAVYLXLWLZXUKWCVYMXLWLZAXUKXUKA
+      WEEWFVUHZXUMWNVYLVYLXLXLVULXJZVUIAVYMVYNXUIYLZXUIYTZVYNWAZWCXUJAXUOXUQAVY
+      MVYNQVYMVUJZYLZXUOAWVOVYQVYMVYNXURXBZXUSWVTWVMAXUTVYMVYNXUIXBAVNVYMXUHVYN
+      XUIAXUGVYMWLZWCVYOXUGVYMQAQVYOVVBXVAWWBYAAVYMVYOXUGWVMVUKAXVAXHVUMXUIXFVU
+      NAVYMVYNXURXUIAVNVYOWBVYMQWWAWVMVUOZVUPYKVYOWBVYMVYNQVUQYPAVYMVYMVYNVYNXU
+      RXUIXVBAVYMVURAVYNVURVUSYGAVYNXURYTZXUPVYNXVCWAAQVYMVUTXNAXURXUIXVBVVCVVA
+      WNVYMVYNXUIVVDYJVYMVYNXUCXUIVVEVVFAXULXUEXUFVVGXUNVYMVYNVMXLVVHXJUXAUXOAF
+      VYNWIWWDXPYSAVJFVXFWUHVXGAVXEFWLZWCUBXKVVBZVYPXKYBZVXEVYPWLZVXFWUHWLAXVEX
+      VDAXKWUPXCWJZUBAWXQXKXVHUBXBWXSXKXVHUYEWUPUBVVIXVHXFVVJXJUWJYAAXVFXVDAVYP
+      WBXKAVYPQYTZWBAVYPXVIYBQVYOVUJZYTZXVIYBZAXVJQYBZXVLXVMAQVYOVVKXNXVJQVVLXJ
+      AVYPXVKXVIVYPXVKWAAQVYOVUTXNYNYKAWVPXVIWBYBWWAVYOWBQVVMXJYMWBXKYBAVVNXNYM
+      YAAXVDXVGAFVYPVXEAFVYPYBVYRWVNAFVYNVYPWWDYNYKVVOXIXKVYPUBVXEVVPYPVXGXFVUN
+      AFWBYIWWEWBYIYBAVVQXNYMVVRVVSVVT $.
+  $}
+
+  ${
+    $d B x $.  $d D x $.
+    $( Special case related to ~ rspsbc .  (Contributed by metakunt,
+       5-May-2025.) $)
+    rspcsbnea $p |- ( ( A e. B /\ A. x e. B C =/= D ) ->
+     [_ A / x ]_ C =/= D ) $=
+      ( wcel wne wral csb wsbc rspsbc wceq wn wb df-ne sbcbii a1i sbcng sbceq1g
+      bitrd notbid biidd necon3bbid sylibd imp ) BCFZDEGZACHZABDIZEGZUFUHUGABJZ
+      UJUGABCKUFUKUIELZMZUJUFUKDELZMZABJZUMUKUPNUFUGUOABDEOPQUFUPUNABJZMUMUNABC
+      RUFUQULABDECSUATTUFULUIEUFULUBUCTUDUE $.
+  $}
+
+  ${
+    $d .^ x y $.  $d A x y $.  $d N x $.  $d R x y $.  $d ph x y $.
+    idomnnzpownz.1 $e |- ( ph -> R e. IDomn ) $.
+    idomnnzpownz.2 $e |- ( ph -> A e. ( Base ` R ) ) $.
+    idomnnzpownz.3 $e |- ( ph -> A =/= ( 0g ` R ) ) $.
+    idomnnzpownz.4 $e |- ( ph -> N e. NN0 ) $.
+    idomnnzpownz.5 $e |- .^ = ( .g ` ( mulGrp ` R ) ) $.
+    $( A non-zero power in an integral domain is non-zero.  (Contributed by
+       metakunt, 5-May-2025.) $)
+    idomnnzpownz $p |- ( ph -> ( N .^ A ) =/= ( 0g ` R ) ) $=
+      ( wcel wa co cfv wne wceq oveq1 eqid syl adantr vx vy cn0 c0g ancli cv c1
+      cc0 caddc neeq1d cur cmgp cbs eleqtrdi mulg0 ringidval eqtr4di cnzr cidom
+      mgpbas cdomn ccrg isidom simprbi domnnzr nzrnz eqnetrd cmnd crg idomringd
+      cplusg ringmgp simplr ad2antrr mulgnn0p1 syl3anc cmulr mgpplusg a1i oveqd
+      eqcomd mulgnn0cl eqcomi eleqtrd simpr jca domnmuln0 nn0indd ) AAEUCKZLEBD
+      MZCUDNZOZAWIIUEAUAUFZBDMZWKOUHBDMZWKOUBUFZBDMZWKOZWPUGUIMZBDMZWKOWLUAUBEW
+      MUHPWNWOWKWMUHBDQUJWMWPPWNWQWKWMWPBDQUJWMWSPWNWTWKWMWSBDQUJWMEPWNWJWKWMEB
+      DQUJAWOCUKNZWKAWOCULNZUDNZXAABXBUMNZKZWOXCPABCUMNZXDGXFCXBXBRZXFRZUTZUNZX
+      DDXBBXCXDRZXCRJUOSCXAXBXGXARZUPUQACURKZXAWKOACUSKZXMFXNCVAKZXMXNCVBKXOCVC
+      VDZCVESSCXAWKXLWKRZVFSVGAWPUCKZLZWRLZWTWQBXBVKNZMZWKXTXBVHKZXRXEWTYBPXSYC
+      WRAYCXRACVIKYCACFVJCXBXGVLSTTZAXRWRVMZAXEXRWRXJVNZXDYADXBWPBXKJYARVOVPXTY
+      BWQBCVQNZMZWKXTYAYGWQBXTYGYAYGYAPXTCYGXBXGYGRZVRVSWAVTXTXOWQXFKZWRLBXFKZB
+      WKOZLZYHWKOXSXOWRAXOXRAXNXOFXPSTTXTYJWRXTWQXDXFXTYCXRXEWQXDKYDYEYFXDDXBWP
+      BXKJWBVPXDXFPXTXFXDXIWCVSWDXSWRWEWFXSYMWRAYMXRAYKYLGHWFTTXFCYGWQBWKXHYIXQ
+      WGVPVGVGWHS $.
+  $}
+
+  ${
+    $d A m y z $.  $d A x y z $.  $d G m y z $.  $d G x y z $.  $d N m n y z $.
+    $d N n x y z $.  $d R m n y z $.  $d R n x y z $.  $d m n ph y z $.
+    $d ph x y z $.
+    idomnnzgmulnz.1 $e |- G = ( mulGrp ` R ) $.
+    idomnnzgmulnz.2 $e |- ( ph -> R e. IDomn ) $.
+    idomnnzgmulnz.3 $e |- ( ph -> N e. Fin ) $.
+    idomnnzgmulnz.4 $e |- ( ( ph /\ n e. N ) -> A e. ( Base ` R ) ) $.
+    idomnnzgmulnz.5 $e |- ( ( ph /\ n e. N ) -> A =/= ( 0g ` R ) ) $.
+    $( A finite product of non-zero elements in an integral domain is non-zero.
+       (Contributed by metakunt, 5-May-2025.) $)
+    idomnnzgmulnz $p |- ( ph -> ( G gsum ( n e. N |-> A ) )
path: set.mm
user: login: icecream17
body: ```diff
-idomnnzgmulnz
+idomnnzgprodnz
```
created_at: 2025-05-06T14:03:51Z
author_association: CONTRIBUTOR

PR comments:
user: login: metakunt
created_at: 2025-05-02T18:33:11Z
body: Ok, Claim 2 is proven now. It worked! What a glorious mess to formalise.
author_association: CONTRIBUTOR
user: login: metakunt
created_at: 2025-05-02T18:40:31Z
body: @tirix aren't you working on minimal polynomials? Is that close to proving the existence of a field with  p^n elements? I think I don't even need the algebraic closure. I think the existence of finite fields with size p^n for all primes p and positive integers n is enough.
author_association: CONTRIBUTOR
user: login: tirix
created_at: 2025-05-04T17:49:30Z
body: > @tirix aren't you working on minimal polynomials? Is that close to proving the existence of a field with p^n elements? I think I don't even need the algebraic closure. I think the existence of finite fields with size p^n for all primes p and positive integers n is enough.

I am, but I am not sure it is straightforward to deduce the existence of such finite fields. I'm would like to prove #4246, and next I will probably work on the "constructible" numbers.
author_association: CONTRIBUTOR
user: login: metakunt
created_at: 2025-05-04T23:21:33Z
body: Can this be merged? I want to open another PR for the next claim.
author_association: CONTRIBUTOR
user: login: tirix
created_at: 2025-05-04T23:41:21Z
body: > Can this be merged? 

I was expecting some kind of closure for @avekens's [remark](https://github.com/metamath/set.mm/pull/4801#discussion_r2072370575). He has not approved yet.
author_association: CONTRIBUTOR
user: login: metakunt
created_at: 2025-05-05T15:11:08Z
body: Should I just add the next claim to this PR? Because dealing with a 700000 line file is not very fun for VSCode, especially because I already work on the upcoming changes that build on it.
author_association: CONTRIBUTOR
user: login: metakunt
created_at: 2025-05-06T08:48:17Z
body: Yeah, let's get this merged so that I can work on claim 6, there are lots of missing lemmas that I need to formalise that are also missing in main. For example there needs to be a polynomial version of multiplicative degree in domains, I thought it was there but to my surprise the API is again very sparse.
author_association: CONTRIBUTOR
user: login: icecream17
created_at: 2025-05-06T14:06:28Z
body: late suggestions 😅 
author_association: CONTRIBUTOR