Repository: metamath/set.mm
Pull request #4702: A quotient by a maximal ideal is a division ring.
Author: login: tirix
Created: 2025-03-09T12:41:09Z | Merged: 2025-03-10T13:08:10Z

This follows @benjub's [proposal](https://github.com/metamath/set.mm/pull/4675#issuecomment-2688289638).

The difficulty came from the opposite ring of the quotient ring / the quotient ring of the opposite ring, which are not exactly the same in `set.mm` because of structure technicalities.

I've renamed [`grpridd`](https://us.metamath.org/mpeuni/grpridd.html) to `grprida` and chose to use @icecream17's version [`sn-grpidd`](https://us.metamath.org/mpeuni/sn-grpridd.html) instead. Was that the correct way? Should I have revised the old `grpridd` instead? It has only one use.


Diff:
diff --git a/changes-set.txt b/changes-set.txt
index caa6610faf..29ad371ef3 100644
--- a/changes-set.txt
+++ b/changes-set.txt
@@ -84,6 +84,17 @@ make a github issue.)
 
 DONE:
 Date      Old       New         Notes
+10-Mar-25 brrici    [same]      Moved from SN's mathbox to main set.mm
+ 9-Mar-25 2rspcedvdw [same]     Moved from SN's mathbox to main set.mm
+ 9-Mar-25 elmapdd   [same]      Moved from SN's mathbox to main set.mm
+ 9-Mar-25 grpassd   [same]      Moved from SN's mathbox to main set.mm
+ 9-Mar-25 grplidd   [same]      Moved from SN's mathbox to main set.mm
+ 9-Mar-25 sn-grpridd grpridd    Moved from SN's mathbox to main set.mm
+ 9-Mar-25 grpridd   grprida
+ 9-Mar-25 grplinvd  [same]      Moved from SN's mathbox to main set.mm
+ 9-Mar-25 ringassd  [same]      Moved from SN's mathbox to main set.mm
+ 9-Mar-25 ringlidmd [same]      Moved from SN's mathbox to main set.mm
+ 9-Mar-25 ringridmd [same]      Moved from SN's mathbox to main set.mm
  5-Mar-25 nabbi     nabbib
  2-Mar-25 isassad   [same]      revised - scalars may be noncommutative
  2-Mar-25 assasca   [same]      revised - scalars may be noncommutative
diff --git a/set.mm b/set.mm
index 76fea1f852..6a046e58bf 100644
--- a/set.mm
+++ b/set.mm
@@ -34005,6 +34005,19 @@ general is seen either by substitution (when the variable ` V ` has no
       CEINZOZAEINZDHNUCUDBUFUDBOUEUCCBEGIKPQRUGUEDFHDSFTACEIJUAPUB $.
   $}
 
+  ${
+    $d A x y $.  $d B y $.  $d X x $.  $d Y x y $.  $d ch x $.  $d th y $.
+    2rspcedvdw.1 $e |- ( x = A -> ( ps <-> ch ) ) $.
+    2rspcedvdw.2 $e |- ( y = B -> ( ch <-> th ) ) $.
+    2rspcedvdw.a $e |- ( ph -> A e. X ) $.
+    2rspcedvdw.b $e |- ( ph -> B e. Y ) $.
+    2rspcedvdw.3 $e |- ( ph -> th ) $.
+    $( Double application of ~ rspcedvdw .  (Contributed by SN,
+       24-Aug-2024.) $)
+    2rspcedvdw $p |- ( ph -> E. x e. X E. y e. Y ps ) $=
+      ( wcel wrex rspc2ev syl3anc ) AGIPHJPDBFJQEIQMNOBDCEFGHIJKLRS $.
+  $}
+
   ${
     $d x y A $.  $d y B $.  $d x C $.  $d x y D $.  $d y ch $.  $d x th $.
     rspc2dv.1 $e |- ( x = A -> ( ps <-> th ) ) $.
@@ -91419,6 +91432,16 @@ the first case of his notation (simple exponentiation) and subscript it
       ( wcel cmap co wf wb elmapg syl2anc ) ABEICFIDBCJKICBDLMGHBCDEFNO $.
   $}
 
+  ${
+    elmapdd.a $e |- ( ph -> A e. V ) $.
+    elmapdd.b $e |- ( ph -> B e. W ) $.
+    elmapdd.c $e |- ( ph -> C : B --> A ) $.
+    $( Deduction associated with ~ elmapd .  (Contributed by SN,
+       29-Jul-2024.) $)
+    elmapdd $p |- ( ph -> C e. ( A ^m B ) ) $=
+      ( cmap co wcel wf elmapd mpbird ) ADBCJKLCBDMIABCDEFGHNO $.
+  $}
+
   ${
     $d B f $.  $d V f $.
     $( The empty set is the only map with empty domain.  (Contributed by Glauco
@@ -228310,7 +228333,7 @@ associative structure (such as a group).  (Contributed by NM,
     $( Deduce right identity from left inverse and left identity in an
        associative structure (such as a group).  (Contributed by NM,
        10-Aug-2013.)  (Proof shortened by Mario Carneiro, 6-Jan-2015.) $)
-    grpridd $p |- ( ( ph /\ x e. B ) -> ( x .+ O ) = x ) $=
+    grprida $p |- ( ( ph /\ x e. B ) -> ( x .+ O ) = x ) $=
       ( vn vu vv cv wcel wa co wceq vw oveq1 eqeq1d cbvrexvw sylib w3a caovassg
       adantlr simprl simprrl caovassd simprrr grprinvd oveq1d 3eqtr3d rexlimddv
       wrex oveq2d anassrs eqtr3d ) ABPZEQZRZGVAFSZVAGFSZVAVCMPZVAFSZGTZVDVETZME
@@ -232100,6 +232123,19 @@ unital rings ( ~ df-ring ) and fields ( ~ df-field ).  (Contributed by
       ZDNUCOUCUBDNUCOPACQBCRESABCDEGHTUA $.
   $}
 
+  ${
+    grpassd.b $e |- B = ( Base ` G ) $.
+    grpassd.p $e |- .+ = ( +g ` G ) $.
+    grpassd.g $e |- ( ph -> G e. Grp ) $.
+    grpassd.1 $e |- ( ph -> X e. B ) $.
+    grpassd.2 $e |- ( ph -> Y e. B ) $.
+    grpassd.3 $e |- ( ph -> Z e. B ) $.
+    $( A group operation is associative.  (Contributed by SN, 29-Jan-2025.) $)
+    grpassd $p |- ( ph -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) ) $=
+      ( cgrp wcel co wceq grpass syl13anc ) ADNOEBOFBOGBOEFCPGCPEFGCPCPQJKLMBCD
+      EFGHIRS $.
+  $}
+
   ${
     grpmndd.1 $e |- ( ph -> G e. Grp ) $.
     $( A group is a monoid.  (Contributed by SN, 1-Jun-2024.) $)
@@ -232259,7 +232295,7 @@ with the same (relevant) properties is also a group.  (Contributed by
          don't assume there is an expression for the inverse of ` x ` .
          (Contributed by NM, 6-Jan-2015.) $)
       isgrpde $p |- ( ph -> G e. Grp ) $=
-        ( grpridd grpidd ismndd isgrpd2e ) ABCEFGHIJABEFGHIJMNABCDEFHKMNLOPZQAB
+        ( grprida grpidd ismndd isgrpd2e ) ABCEFGHIJABEFGHIJMNABCDEFHKMNLOPZQAB
         CDEFGHIJKLMNTROS $.
     $}
 
@@ -232395,6 +232431,23 @@ with the same (relevant) properties is also a group.  (Contributed by
       P $.
   $}
 
+  ${
+    grplidd.b $e |- B = ( Base ` G ) $.
+    grplidd.p $e |- .+ = ( +g ` G ) $.
+    grplidd.u $e |- .0. = ( 0g ` G ) $.
+    grplidd.g $e |- ( ph -> G e. Grp ) $.
+    grplidd.1 $e |- ( ph -> X e. B ) $.
+    $( The identity element of a group is a left identity.  Deduction
+       associated with ~ grplid .  (Contributed by SN, 29-Jan-2025.) $)
+    grplidd $p |- ( ph -> ( .0. .+ X ) = X ) $=
+      ( cgrp wcel co wceq grplid syl2anc ) ADLMEBMFECNEOJKBCDEFGHIPQ $.
+
+    $( The identity element of a group is a right identity.  Deduction
+       associated with ~ grprid .  (Contributed by SN, 29-Jan-2025.) $)
+    grpridd $p |- ( ph -> ( X .+ .0. ) = X ) $=
+      ( cgrp wcel co wceq grprid syl2anc ) ADLMEBMEFCNEOJKBCDEFGHIPQ $.
+  $}
+
   $( A group is not empty.  (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
      (Revised by Mario Carneiro, 2-Dec-2014.) $)
   grpn0 $p |- ( G e. Grp -> G =/= (/) ) $=
@@ -232690,6 +232743,20 @@ with the same (relevant) properties is also a group.  (Contributed by
       WCWBCWBFEVMVNUNVOVPVQVS $.
   $}
 
+  ${
+    grplinvd.b $e |- B = ( Base ` G ) $.
+    grplinvd.p $e |- .+ = ( +g ` G ) $.
+    grplinvd.u $e |- .0. = ( 0g ` G ) $.
+    grplinvd.n $e |- N = ( invg ` G ) $.
+    grplinvd.g $e |- ( ph -> G e. Grp ) $.
+    grplinvd.1 $e |- ( ph -> X e. B ) $.
+    $( The left inverse of a group element.  Deduction associated with
+       ~ grplinv .  (Contributed by SN, 29-Jan-2025.) $)
+    grplinvd $p |- ( ph -> ( ( N ` X ) .+ X ) = .0. ) $=
+      ( cgrp wcel cfv co wceq grplinv syl2anc ) ADNOFBOFEPFCQGRLMBCDEFGHIJKST
+      $.
+  $}
+
   ${
     $d B y $.  $d G x y $.  $d .+ y $.  $d .0. y $.
     grplrinv.b $e |- B = ( Base ` G ) $.
@@ -254573,6 +254640,20 @@ the additive structure must be abelian (see ~ ringcom ), care must be
       HFUCDEUEUHGUDUAUB $.
   $}
 
+  ${
+    ringassd.b $e |- B = ( Base ` R ) $.
+    ringassd.t $e |- .x. = ( .r ` R ) $.
+    ringassd.r $e |- ( ph -> R e. Ring ) $.
+    ringassd.x $e |- ( ph -> X e. B ) $.
+    ringassd.y $e |- ( ph -> Y e. B ) $.
+    ringassd.z $e |- ( ph -> Z e. B ) $.
+    $( Associative law for multiplication in a ring.  (Contributed by SN,
+       14-Aug-2024.) $)
+    ringassd $p |- ( ph -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) ) $=
+      ( crg wcel co wceq ringass syl13anc ) ACNOEBOFBOGBOEFDPGDPEFGDPDPQJKLMBCD
+      EFGHIRS $.
+  $}
+
   ${
     ringcld.b $e |- B = ( Base ` R ) $.
     ringcld.t $e |- .x. = ( .r ` R ) $.
@@ -254660,6 +254741,23 @@ the additive structure must be abelian (see ~ ringcom ), care must be
       UKULHUEUJJRZARZDSUNTUNUMDSUNTUAABUBZJBUCUOJBUDAJBCDGHUFUOJBUGUHUI $.
   $}
 
+  ${
+    ringlidmd.b $e |- B = ( Base ` R ) $.
+    ringlidmd.t $e |- .x. = ( .r ` R ) $.
+    ringlidmd.u $e |- .1. = ( 1r ` R ) $.
+    ringlidmd.r $e |- ( ph -> R e. Ring ) $.
+    ringlidmd.x $e |- ( ph -> X e. B ) $.
+    $( The unity element of a ring is a left multiplicative identity.
+       (Contributed by SN, 14-Aug-2024.) $)
+    ringlidmd $p |- ( ph -> ( .1. .x. X ) = X ) $=
+      ( crg wcel co wceq ringlidm syl2anc ) ACLMFBMEFDNFOJKBCDEFGHIPQ $.
+
+    $( The unity element of a ring is a right multiplicative identity.
+       (Contributed by SN, 14-Aug-2024.) $)
+    ringridmd $p |- ( ph -> ( X .x. .1. ) = X ) $=
+      ( crg wcel co wceq ringridm syl2anc ) ACLMFBMFEDNFOJKBCDEFGHIPQ $.
+  $}
+
   ${
     $d B u $.  $d R u $.  $d X u $.  $d .x. u $.
     ringid.b $e |- B = ( Base ` R ) $.
@@ -257364,6 +257462,11 @@ irreducible elements (see ~ df-irred ).  (Contributed by Mario Carneiro,
       HCDEUDUEUFUG $.
   $}
 
+  $( Prove isomorphic by an explicit isomorphism.  (Contributed by SN,
+     10-Jan-2025.) $)
+  brrici $p |- ( F e. ( R RingIso S ) -> R ~=r S ) $=
+    ( crs co wcel c0 wne cric wbr ne0i brric sylibr ) CABDEZFNGHABIJNCKABLM $.
+
   ${
     $d R f $.  $d S f $.
     $( The relation "is isomorphic to" for (unital) rings.  This theorem
@@ -266525,6 +266628,22 @@ left ideal which is also a right ideal (or a left ideal over the opposite
       STUTAKUIVHUTCUJCTOUTTOCDUKVHCVGTFAPUIULUTCUMUNUOUQUR $.
   $}
 
+  ${
+    2idllidld.1 $e |- ( ph -> I e. ( 2Ideal ` R ) ) $.
+    $( A two-sided ideal is a left ideal.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    2idllidld $p |- ( ph -> I e. ( LIdeal ` R ) ) $=
+      ( clidl cfv coppr c2idl cin eqid 2idlval eleqtrdi elin1d ) ABEFZBGFZEFZCA
+      CBHFZNPIDBQNPONJOJPJQJKLM $.
+
+    2idlridld.o $e |- O = ( oppR ` R ) $.
+    $( A two-sided ideal is a right ideal.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    2idlridld $p |- ( ph -> I e. ( LIdeal ` O ) ) $=
+      ( clidl cfv c2idl cin eqid 2idlval eleqtrdi elin2d ) ABGHZDGHZCACBIHZOPJE
+      BQOPDOKFPKQKLMN $.
+  $}
+
   ${
     2idlcpbl.x $e |- X = ( Base ` R ) $.
     2idlcpbl.r $e |- E = ( R ~QG S ) $.
@@ -266604,6 +266723,31 @@ left ideal which is also a right ideal (or a left ideal over the opposite
       WKCBWAMZLEBDWDOLWKCXBYMXJWKBWELZYMXBSWIYNWJBWBVRBWCVDWFAEBDGCJHKWGVDWH $.
   $}
 
+  ${
+    $d .X. t x y z $.  $d .x. p q $.  $d .x. q t y $.  $d B p q $.
+    $d B t x y z $.  $d I t x y z $.  $d R t x y z $.  $d X t y $.  $d Y t y $.
+    $d p ph q y $.  $d ph t x y z $.
+    qusmul2.h $e |- Q = ( R /s ( R ~QG I ) ) $.
+    qusmul2.v $e |- B = ( Base ` R ) $.
+    qusmul2.p $e |- .x. = ( .r ` R ) $.
+    qusmul2.a $e |- .X. = ( .r ` Q ) $.
+    qusmul2.1 $e |- ( ph -> R e. Ring ) $.
+    qusmul2.2 $e |- ( ph -> I e. ( 2Ideal ` R ) ) $.
+    qusmul2.3 $e |- ( ph -> X e. B ) $.
+    qusmul2.4 $e |- ( ph -> Y e. B ) $.
+    $( Value of the ring operation in a quotient ring.  (Contributed by Thierry
+       Arnoux, 1-Sep-2024.) $)
+    qusmul2 $p |- ( ph -> ( [ X ] ( R ~QG I ) .X. [ Y ] ( R ~QG I ) )
+                           = [ ( X .x. Y ) ] ( R ~QG I ) ) $=
+      ( wcel co cv vt vy vx vz vp cqg cec wceq crg cqus a1i cbs cfv csubg clidl
+      vq wer 2idllidld eqid lidlsubg syl2anc eqger syl c2idl wbr wa wi 2idlcpbl
+      ringcl 3expb sylan caovclg qusmulval mpd3an23 ) AHBRIBRHDGUFSZUGIVOUGFSHI
+      ESVOUGUHPQAVODFECBHIUIUAUBUCUDCDVOUJSUHAJUKBDULUMUHAKUKAGDUNUMRZBVOUQADUI
+      RZGDUOUMZRVPNADGOURDVRGVRUSUTVAVODBGKVOUSZVBVCNAVQGDVDUMZRUCTZUBTZVOVEUDT
+      ZUATZVOVEVFWAWCESWBWDESVOVEVGNOWAWCWBWDDGEVOVTBKVSVTUSLVHVAAUEUPWBWDBBBEA
+      VQUETZBRZUPTZBRZVFWEWGESBRZNVQWFWHWIBDEWEWGKLVIVJVKVLLMVMVN $.
+  $}
+
   ${
     $d x y O $.  $d x y R $.
     crng2idl.i $e |- I = ( LIdeal ` R ) $.
@@ -496903,6 +497047,32 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
 -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
 $)
 
+  ${
+    $d B e p x y $.  $d S e p x y $.  $d T e p x y $.  $d e p ph x y $.
+    urpropd.b $e |- B = ( Base ` S ) $.
+    urpropd.s $e |- ( ph -> S e. V ) $.
+    urpropd.t $e |- ( ph -> T e. W ) $.
+    urpropd.1 $e |- ( ph -> B = ( Base ` T ) ) $.
+    urpropd.2 $e |- ( ( ( ph /\ x e. B ) /\ y e. B )
+                                -> ( x ( .r ` S ) y ) = ( x ( .r ` T ) y ) ) $.
+    $( Sufficient condition for ring unities to be equal.  (Contributed by
+       Thierry Arnoux, 9-Mar-2025.) $)
+    urpropd $p |- ( ph -> ( 1r ` S ) = ( 1r ` T ) ) $=
+      ( ve vp cfv co wceq wa eqid cmgp c0g cur cv wcel cmulr wral adantr anasss
+      cio cbs ralrimivva weq oveq1 eqeq12d oveq2 simplr eqidd simpr rspc2vd mpd
+      ad2antrr eqeq1d anbi12d raleqbidva pm5.32da eleq2d anbi1d iotabidv mgpbas
+      bitrd mgpplusg grpidval 3eqtr4g ringidval ) AEUAPZUBPZFUAPZUBPZEUCPZFUCPZ
+      ANUDZDUEZWBOUDZEUFPZQZWDRZWDWBWEQZWDRZSZODUGZSZNUJWBFUKPZUEZWBWDFUFPZQZWD
+      RZWDWBWOQZWDRZSZOWMUGZSZNUJVQVSAWLXBNAWLWCXASXBAWCWKXAAWCSZWJWTODWMADWMRW
+      CLUHXCWDDUEZSZWGWQWIWSXEWFWPWDXEBUDZCUDZWEQZXFXGWOQZRZCDUGBDUGZWFWPRZAXKW
+      CXDAXJBCDDAXFDUEXGDUEXJMUIULVBZXEXLWBXGWEQZWBXGWOQZRXJBCWBWDDDDBNUMZXHXNX
+      IXOXFWBXGWEUNXFWBXGWOUNUOCOUMXNWFXOWPXGWDWBWEUPXGWDWBWOUPUOAWCXDUQZXEXPSD
+      URXCXDUSZUTVAVCXEWHWRWDXEXKWHWRRZXMXEXSWDXGWEQZWDXGWOQZRXJBCWDWBDDDBOUMZX
+      HXTXIYAXFWDXGWEUNXFWDXGWOUNUOCNUMXTWHYAWRXGWBWDWEUPXGWBWDWOUPUOXRXEYBSDUR
+      XQUTVAVCVDVEVFAWCWNXAADWMWBLVGVHVKVIODWENVPVQDEVPVPTZIVJEWEVPYCWETVLVQTVM
+      OWMWONVRVSWMFVRVRTZWMTVJFWOVRYDWOTVLVSTVMVNEVTVPYCVTTVOFWAVRYDWATVOVN $.
+  $}
+
   ${
     0ringsubrg.1 $e |- B = ( Base ` R ) $.
     0ringsubrg.2 $e |- ( ph -> R e. Ring ) $.
@@ -497170,6 +497340,61 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
 -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
 $)
 
+  ${
+    $d .0. x $.  $d .1. x y z $.  $d .x. x y z $.  $d B x y z $.  $d R x y $.
+    $d U x z $.  $d U y z $.  $d ph x z $.  $d ph y z $.
+    isdrng4.b $e |- B = ( Base ` R ) $.
+    isdrng4.0 $e |- .0. = ( 0g ` R ) $.
+    isdrng4.1 $e |- .1. = ( 1r ` R ) $.
+    isdrng4.x $e |- .x. = ( .r ` R ) $.
+    isdrng4.u $e |- U = ( Unit ` R ) $.
+    isdrng4.r $e |- ( ph -> R e. Ring ) $.
+    ${
+      ringinveu.1 $e |- ( ph -> X e. B ) $.
+      ringinveu.2 $e |- ( ph -> Y e. B ) $.
+      ringinveu.3 $e |- ( ph -> Z e. B ) $.
+      ringinveu.4 $e |- ( ph -> ( Y .x. X ) = .1. ) $.
+      ringinveu.5 $e |- ( ph -> ( X .x. Z ) = .1. ) $.
+      $( If a ring unit element ` X ` admits both a left inverse ` Y ` and a
+         right inverse ` Z ` , they are equal.  (Contributed by Thierry Arnoux,
+         9-Mar-2025.) $)
+      ringinveu $p |- ( ph -> Z = Y ) $=
+        ( co oveq2d oveq1d ringassd ringlidmd 3eqtr3d ringridmd ) AHGJDUBZDUBZH
+        FDUBJHAUIFHDUAUCAHGDUBZJDUBFJDUBUJJAUKFJDTUDABCDHGJKNPRQSUEABCDFJKNMPSU
+        FUGABCDFHKNMPRUHUG $.
+    $}
+
+    $( A division ring is a ring in which ` 1 =/= 0 ` and every nonzero element
+       has a left and right inverse.  (Contributed by Thierry Arnoux,
+       2-Mar-2025.) $)
+    isdrng4 $p |- ( ph -> ( R e. DivRing <-> ( .1. =/= .0.
+       /\ A. x e. ( B \ { .0. } ) E. y e. B ( ( x .x. y ) = .1.
+                                                /\ ( y .x. x ) = .1. ) ) ) ) $=
+      ( wcel wceq wa wrex syl2anc vz cdr csn cdif wne wral crg isdrng biantrurd
+      cv bitr4id 1unit syl adantr simpr eleqtrd eldifsni simpll biimpar ad5antr
+      eleq2d unitcl ad5antlr simp-4r simplr simpllr ringinveu oveq2d eqtr3d cfv
+      coppr cdsr wbr ad3antlr eqid isunit simprbi opprbas dvdsr2 biimpa opprmul
+      cmulr eqeq1i rexbii sylib oveq2 eqeq1d cbvrexvw r19.29a jca anasss adantl
+      co simplbi reximddv ralrimiva wss wn unitss a1i simprl necon3bbid ssdifsn
+      0unit sylanbrc eldifad reximi simpl sylibr ex ralimdva impr dfss3 impbida
+      eqssd bitrd ) AEUBPZGDIUCZUDZQZHIUEZBUJZCUJZFWMZHQZYCYBFWMHQZRZCDSZBXSUFZ
+      RZAXQEUGPZXTRXTDEGIJNKUHAYKXTOUIUKAXTYJAXTRZYAYIYLHXSPYAYLHGXSAHGPZXTAYKY
+      MOEGHNLULUMUNAXTUOZUPHDIUQUMYLYHBXSYLYBXSPZRAYBGPZYHAXTYOURYLYPYOYLGXSYBY
+      NVAUSAYPRZYFYGCDYQYCDPZYFYGYQYRRZYFRZYEYFYTYBUAUJZFWMZHQZYEUADYTUUADPZRZU
+      UCRZUUBYDHUUFUUAYCYBFUUFDEFGHYBYCIUUAJKLMNAYKYPYRYFUUDUUCOUTYPYBDPZAYRYFU
+      UDUUCDEGYBJNVBZVCYQYRYFUUDUUCVDYTUUDUUCVEYSYFUUDUUCVFUUEUUCUOZVGVHUUIVIYT
+      UUGYBHEVKVJZVLVJZVMZUUCUADSZYPUUGAYRYFUUHVNYPUULAYRYFYPYBHEVLVJZVMZUULUUN
+      EUUJGHUUKYBNLUUNVOZUUJVOZUUKVOZVPZVQVNUUGUULRZYECDSZUUMUUTYCYBUUJWBVJZWMZ
+      HQZCDSZUVAUUGUULUVECDUUKUUJUVBYBHDEUUJUUQJVRUURUVBVOZVSZVTUVDYECDUVCYDHDE
+      UVBFUUJYCYBJMUUQUVFWAWCWDZWEYEUUCCUADYCUUAQYDUUBHYCUUAYBFWFWGWHWETWIYSYFU
+      OWJWKYQUUGUUOYFCDSZYPUUGAUUHWLYPUUOAYPUUOUULUUSWNWLUUGUUOUVICDUUNEFYBHJUU
+      PMVSZVTTWOTWPWJAYJRZGXSUVKGDWQZIGPZWRZGXSWQUVLUVKDEGJNWSWTUVKYKYAUVNAYKYJ
+      OUNAYAYIXAYKUVNYAYKUVMHIEGHINKLXDXBUSTGDIXCXEUVKYPBXSUFZXSGWQAYAYIUVOAYAR
+      ZYHYPBXSUVPYORZYHYPUVQYHRZUUOUULYPUVRUUGUVIUUOUVRYBDXRUVPYOYHVEXFZYHUVIUV
+      QYGYFCDYEYFUOXGWLUUGUUOUVIUVJUSTUVRUUGUVEUULUVSUVRUVAUVEYHUVAUVQYGYECDYEY
+      FXHXGWLUVHXIUUGUULUVEUVGUSTUUSXEXJXKXLBXSGXMXIXOXNXP $.
+  $}
+
   ${
     rndrhmcl.r $e |- R = ( N |`s ran F ) $.
     rndrhmcl.1 $e |- .0. = ( 0g ` N ) $.
@@ -498570,7 +498795,6 @@ then this is also a division ring (see ~ fldgensdrg ).  If the base
       CUGUHUIURUTVGAMVCVIAVERUTAVEUJURVECADEVEUOUKULVJUMTUTURVHDMVCDUSUNUPUQ $.
   $}
 
-
 $(
 -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
   The ring of integers modulo ` N `
@@ -499820,6 +500044,23 @@ The sumset operation can be used for both group (additive) operations and
       XDVXEVXCYGUXFUCUXCUYRCYTRUVAYSOUVBUVCUVDUVHUVIBEFUVMUVNUVEUVF $.
   $}
 
+  ${
+    $d F p q $.  $d G q $.  $d H q $.  $d K q $.  $d Q p q $.  $d ph q $.
+    gicqusker.1 $e |- .0. = ( 0g ` H ) $.
+    gicqusker.f $e |- ( ph -> F e. ( G GrpHom H ) ) $.
+    gicqusker.k $e |- K = ( `' F " { .0. } ) $.
+    gicqusker.q $e |- Q = ( G /s ( G ~QG K ) ) $.
+    gicqusker.s $e |- ( ph -> ran F = ( Base ` H ) ) $.
+    $( The image ` H ` of a group homomorphism ` F ` is isomorphic with the
+       quotient group ` Q ` over ` F ` 's kernel ` K ` .  Together with
+       ~ ghmker and ~ ghmima , this is sometimes called the first isomorphism
+       theorem for groups.  (Contributed by Thierry Arnoux, 10-Mar-2025.) $)
+    gicqusker $p |- ( ph -> Q ~=g H ) $=
+      ( vp vq cbs cfv cv cima cuni cmpt cgim co wcel cgic wbr weq imaeq2 unieqd
+      cbvmptv ghmqusker brgici syl ) AMBOPZCMQZRZSZTZBEUAUBUCBEUDUEABCDEUQFGNHI
+      JKMNUMUPCNQZRZSMNUFUOUSUNURCUGUHUILUJBEUQUKUL $.
+  $}
+
   ${
     $d F q x $.  $d G q x $.  $d H k q r x $.  $d J k q r x $.  $d K q x $.
     $d Q k q r x $.  $d k ph q r x $.  $d .0. x $.
@@ -499827,39 +500068,51 @@ The sumset operation can be used for both group (additive) operations and
     lmhmqusker.f $e |- ( ph -> F e. ( G LMHom H ) ) $.
     lmhmqusker.k $e |- K = ( `' F " { .0. } ) $.
     lmhmqusker.q $e |- Q = ( G /s ( G ~QG K ) ) $.
-    lmhmqusker.j $e |- J = ( q e. ( Base ` Q ) |-> U. ( F " q ) ) $.
     lmhmqusker.s $e |- ( ph -> ran F = ( Base ` H ) ) $.
-    $( A surjective module homomorphism ` F ` from ` G ` to ` H ` induces an
-       isomorphism ` J ` from ` Q ` to ` H ` , where ` Q ` is the factor group
-       of ` G ` by ` F ` 's kernel ` K ` .  (Contributed by Thierry Arnoux,
-       25-Feb-2025.) $)
-    lmhmqusker $p |- ( ph -> J e. ( Q LMIso H ) ) $=
-      ( co wcel cfv eqid wceq vk vr clmhm wf1o clmim cvsca csca clmod lmhmlmod1
-      cbs syl clss lmhmkerlss quslmod lmhmlmod2 lmhmsca cqg cvv cqus ovexd quss
-      vx a1i eqtrd cgim cghm lmghm ghmqusker gimghm cv wa cec wer cqs cnsg ccnv
-      csubg csn cima ghmker eqeltrid nsgsubg eqger 3syl ad4antr qusbas eleqtrrd
-      simpllr simplr qsel syl3anc oveq2d simp-4r fveq2d wss cpw eqsstrrd sselda
-      qsss elpwid ad5ant13 sseldd qusvsval lmodvscl ghmquskerlem1 lmhmlin simpr
-      3eqtrd eqtr4d ad2antrr ghmquskerlem2 r19.29a anasss islmhmd gimf1o islmim
-      sylanbrc ) AFBEUCPQBUJRZEUJRZFUDZFBEUEPQAUAUBBEBUFRZEUFRZFEUGRZBUGRZYDUJR
-      ZXRXRSZYASZYBSZYDSYCSZYESAGDBDUJRZMYJSZACDEUCPQZDUHQZKDECUIUKZAYLGDULRZQZ
-      KDEYOCGHLJYOSUMUKZUNAYLEUHQKDECUOUKAYCDUGRZYDAYLYCYRTKDECYRYCYRSZYIUPUKAD
-      GUQPZDBYRYJURUHBDYTUSPTAMVCZYJYJTAYKVCZADGUQUTZYNYSVAZVDAFBEVEPQZFBEVFPQA
-      BCDEFGHIJAYLCDEVFPQZKDECVGUKZLMNOVHZBEFVIUKAUAVJZYEQZUBVJZXRQZUUIUUKYAPZF
-      RZUUIUUKFRZYBPZTZAUUJVKZUULVKZUUOVBVJZCRZTZUUQVBUUKUUSUUTUUKQZVKZUVBVKZUU
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-      KYJAXRYJWPZUUKAXRUVMUWEUVRAYJYTUVQWSWQWRWTXAUVSXBZXCVDWNUVEBCDEFGUVHHIJAU
-      UFUUJUULUVCUVBUUGWELMNUVEYMUUIUVTQZUUTYJQZUVHYJQUWCUWDUWFUUIUVGYRUVTYJDUU
-      TYKYSUWBUWAXDWKXEUVEYLUWGUWHUVJUVFTAYLUUJUULUVCUVBKWEUWDUWFUVTDEUVGYBYJCY
-      RUUIUUTYSUWAYKUWBYHXFWKXHUVEUUOUVAUUIYBUVDUVBXGWLXIUUSVBBCDEFGUUKHIJAUUFU
-      UJUULUUGXJLMNUURUULXGXKXLXMXNAUUEXTUUHXRXSBEFYFXSSZXOUKXRXSBEFYFUWIXPXQ
-      $.
+    ${
+      lmhmqusker.j $e |- J = ( q e. ( Base ` Q ) |-> U. ( F " q ) ) $.
+      $( A surjective module homomorphism ` F ` from ` G ` to ` H ` induces an
+         isomorphism ` J ` from ` Q ` to ` H ` , where ` Q ` is the factor
+         group of ` G ` by ` F ` 's kernel ` K ` .  (Contributed by Thierry
+         Arnoux, 25-Feb-2025.) $)
+      lmhmqusker $p |- ( ph -> J e. ( Q LMIso H ) ) $=
+        ( co wcel cfv eqid wceq vk vr clmhm cbs wf1o clmim cvsca csca lmhmlmod1
+        vx clmod syl clss lmhmkerlss quslmod lmhmlmod2 lmhmsca cqg cvv cqus a1i
+        ovexd quss eqtrd cgim cghm lmghm ghmqusker gimghm cv cec wer cnsg csubg
+        wa cqs ccnv csn cima ghmker eqeltrid nsgsubg eqger 3syl ad4antr simpllr
+        qusbas eleqtrrd simplr qsel syl3anc oveq2d simp-4r fveq2d qsss eqsstrrd
+        wss sselda elpwid ad5ant13 sseldd qusvsval ghmquskerlem1 lmhmlin 3eqtrd
+        cpw lmodvscl simpr eqtr4d ad2antrr ghmquskerlem2 r19.29a anasss islmhmd
+        gimf1o islmim sylanbrc ) AFBEUCPQBUDRZEUDRZFUEZFBEUFPQAUAUBBEBUGRZEUGRZ
+        FEUHRZBUHRZYDUDRZXRXRSZYASZYBSZYDSYCSZYESAGDBDUDRZMYJSZACDEUCPQZDUKQZKD
+        ECUIULZAYLGDUMRZQZKDEYOCGHLJYOSUNULZUOAYLEUKQKDECUPULAYCDUHRZYDAYLYCYRT
+        KDECYRYCYRSZYIUQULADGURPZDBYRYJUSUKBDYTUTPTAMVAZYJYJTAYKVAZADGURVBZYNYS
+        VCZVDAFBEVEPQZFBEVFPQABCDEFGHIJAYLCDEVFPQZKDECVGULZLMONVHZBEFVIULAUAVJZ
+        YEQZUBVJZXRQZUUIUUKYAPZFRZUUIUUKFRZYBPZTZAUUJVOZUULVOZUUOUJVJZCRZTZUUQU
+        JUUKUUSUUTUUKQZVOZUVBVOZUUNUUIUVAYBPZUUPUVEUUNUUIUUTDUGRZPZYTVKZFRUVHCR
+        ZUVFUVEUUMUVIFUVEUUMUUIUUTYTVKZYAPUVIUVEUUKUVKUUIYAUVEYJYTVLZUUKYJYTVPZ
+        QUVCUUKUVKTAUVLUUJUULUVCUVBAGDVMRZQGDVNRQUVLAGCVQHVRVSZUVNLAUUFUVOUVNQU
+        UGDECHJVTULWAGDWBYTDYJGYKYTSZWCWDZWEUVEUUKXRUVMUURUULUVCUVBWFAUVMXRTUUJ
+        UULUVCUVBAYTDBYJUSUKUUAUUBUUCYNWGZWEWHUUSUVCUVBWIZYJUUKUUTYTYJWJWKWLUVE
+        YJYTYRUDRZYAUVGGUUIDBUUTYKUVPUVTSZUVGSZAYMUUJUULUVCUVBYNWEZAYPUUJUULUVC
+        UVBYQWEUVEUUIYEUVTAUUJUULUVCUVBWMAUVTYETUUJUULUVCUVBAYRYDUDUUDWNWEWHZMY
+        GUVEUUKYJUUTAUULUUKYJWQUUJUVCUVBAUULVOUUKYJAXRYJXFZUUKAXRUVMUWEUVRAYJYT
+        UVQWOWPWRWSWTUVSXAZXBVDWNUVEBCDEFGUVHHIJAUUFUUJUULUVCUVBUUGWELMOUVEYMUU
+        IUVTQZUUTYJQZUVHYJQUWCUWDUWFUUIUVGYRUVTYJDUUTYKYSUWBUWAXGWKXCUVEYLUWGUW
+        HUVJUVFTAYLUUJUULUVCUVBKWEUWDUWFUVTDEUVGYBYJCYRUUIUUTYSUWAYKUWBYHXDWKXE
+        UVEUUOUVAUUIYBUVDUVBXHWLXIUUSUJBCDEFGUUKHIJAUUFUUJUULUUGXJLMOUURUULXHXK
+        XLXMXNAUUEXTUUHXRXSBEFYFXSSZXOULXRXSBEFYFUWIXPXQ $.
+    $}
+
+    $d F p q $.  $d G q $.  $d H q $.  $d K q $.  $d Q p q $.  $d ph q $.
+    $( The image ` H ` of a module homomorphism ` F ` is isomorphic with the
+       quotient module ` Q ` over ` F ` 's kernel ` K ` .  This is part of what
+       is sometimes called the first isomorphism theorem for modules.
+       (Contributed by Thierry Arnoux, 10-Mar-2025.) $)
+    lmicqusker $p |- ( ph -> Q ~=m H ) $=
+      ( vp vq cbs cfv cv cima cuni cmpt clmim wcel clmic wbr weq imaeq2 cbvmptv
+      co unieqd lmhmqusker brlmici syl ) AMBOPZCMQZRZSZTZBEUAUHUBBEUCUDABCDEUQF
+      GNHIJKLMNUMUPCNQZRZSMNUEUOUSUNURCUFUIUGUJBEUQUKUL $.
   $}
 
 
@@ -499955,47 +500208,60 @@ The sumset operation can be used for both group (additive) operations and
     rhmqusker.f $e |- ( ph -> F e. ( G RingHom H ) ) $.
     rhmqusker.k $e |- K = ( `' F " { .0. } ) $.
     rhmqusker.q $e |- Q = ( G /s ( G ~QG K ) ) $.
-    rhmqusker.j $e |- J = ( q e. ( Base ` Q ) |-> U. ( F " q ) ) $.
     rhmqusker.s $e |- ( ph -> ran F = ( Base ` H ) ) $.
     rhmqusker.2 $e |- ( ph -> G e. CRing ) $.
-    $( A surjective ring homomorphism ` F ` from ` G ` to ` H ` induces an
-       isomorphism ` J ` from ` Q ` to ` H ` , where ` Q ` is the factor group
-       of ` G ` by ` F ` 's kernel ` K ` .  (Contributed by Thierry Arnoux,
-       25-Feb-2025.) $)
-    rhmqusker $p |- ( ph -> J e. ( Q RingIso H ) ) $=
-      ( co wcel cfv eqid vr vs vx crh cbs wf1o crs cmulr cur crg cqg wceq c2idl
-      vy cec rhmrcl1 syl clidl ccnv cima kerlidl eqeltrid ccrg crng2idl eleqtrd
-      csn qus1 syl2anc simpld rhmrcl2 cghm rhmghm ringidcl ghmquskerlem1 simprd
-      wa fveq2d rhm1 3eqtr3d cv ad6antr wss cpw cqs cvv cqus eqidd ovexd qusbas
-      a1i cnsg csubg wer ghmker nsgsubg eqger 3syl qsss eqsstrrd sselda ad5antr
-      elpwid simp-4r sseldd adantlr ad4antr simplr rhmmul syl3anc eleqtrrd qsel
-      simp-6r simp-5r oveq12d qusmul eqtr2d ringcld simpllr simpr ghmquskerlem2
-      eqtr3d 3eqtr4d r19.29a ad2antrr anasss cgim ghmqusker gimghm gimf1o isrim
-      isrhm2d sylanbrc ) AFBEUDQRBUESZEUESZFUFZFBEUGQRAUAUBYMBEBUHSZEUHSZBUISZF
-      EUISZYMTZYRTYSTZYPTZYQTZABUJRZDUISZDGUKQZUOZYRULZADUJRZGDUMSZRUUDUUHVPACD
-      EUDQRZUUIKDECUPZUQZAGDURSZUUJAGCUSHVFUTZUUNLAUUKUUOUUNRKDECUUNHUUNTZJVAUQ
-      VBZADVCRZUUNUUJULPDUUNUUPVDUQVEDGBUUEUUJMUUJTUUETZVGVHZVIAUUKEUJRKDECVJUQ
-      AUUGFSUUECSZYRFSYSABCDEFGUUEHIJAUUKCDEVKQRZKDECVLZUQZLMNAUUIUUEDUESZRUUMU
-      VEDUUEUVETZUUSVMUQVNAUUGYRFAUUDUUHUUTVOVQAUUKUVAYSULKDEUUECYSUUSUUAVRUQVS
-      AUAVTZYMRZUBVTZYMRZUVGUVIYPQZFSZUVGFSZUVIFSZYQQZULZAUVHVPZUVJVPZUVMUCVTZC
-      SZULZUVPUCUVGUVRUVSUVGRZVPZUWAVPZUVNUNVTZCSZULZUVPUNUVIUWDUWEUVIRZVPZUWGV
-      PZUVSUWEDUHSZQZCSZUVTUWFYQQZUVLUVOUWJUUKUVSUVERUWEUVERUWMUWNULAUUKUVHUVJU
-      WBUWAUWHUWGKWAZUWJUVGUVEUVSUVQUVGUVEWBUVJUWBUWAUWHUWGUVQUVGUVEAYMUVEWCZUV
-      GAYMUVEUUFWDZUWPAUUFDBUVEWEVCBDUUFWFQULAMWJAUVEWGADGUKWHPWIZAUVEUUFAGDWKS
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-      HUWGPWAAGUUNRUVHUVJUWBUWAUWHUWGUUQWAUXDUXGXOXPVQUWJBCDEFGUWLHIJUWJUUKUVBU
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-      QRZFBEVKQRABCDEFGHIJUVDLMNOYGZBEFYHUQYKAUXNYOUXOYMYNBEFYTYNTZYIUQYMYNBEFY
-      TUXPYJYL $.
+    ${
+      rhmqusker.j $e |- J = ( q e. ( Base ` Q ) |-> U. ( F " q ) ) $.
+      $( A surjective ring homomorphism ` F ` from ` G ` to ` H ` induces an
+         isomorphism ` J ` from ` Q ` to ` H ` , where ` Q ` is the factor
+         group of ` G ` by ` F ` 's kernel ` K ` .  (Contributed by Thierry
+         Arnoux, 25-Feb-2025.) $)
+      rhmqusker $p |- ( ph -> J e. ( Q RingIso H ) ) $=
+        ( co wcel cfv eqid vr vs vx crh cbs wf1o crs cmulr cur crg cqg cec wceq
+        vy c2idl rhmrcl1 syl clidl ccnv csn cima kerlidl eqeltrid ccrg crng2idl
+        wa eleqtrd qus1 syl2anc simpld rhmrcl2 cghm rhmghm ghmquskerlem1 simprd
+        ringidcl fveq2d rhm1 3eqtr3d cv ad6antr wss cpw cqs cvv a1i eqidd ovexd
+        cqus qusbas cnsg csubg wer ghmker nsgsubg eqger 3syl qsss sselda elpwid
+        eqsstrrd ad5antr simp-4r sseldd adantlr ad4antr simplr syl3anc eleqtrrd
+        rhmmul simp-6r qsel simp-5r oveq12d qusmul eqtr2d ringcld simpllr simpr
+        eqtr3d 3eqtr4d ghmquskerlem2 r19.29a ad2antrr anasss cgim gimghm gimf1o
+        ghmqusker isrhm2d isrim sylanbrc ) AFBEUDQRBUESZEUESZFUFZFBEUGQRAUAUBYM
+        BEBUHSZEUHSZBUISZFEUISZYMTZYRTYSTZYPTZYQTZABUJRZDUISZDGUKQZULZYRUMZADUJ
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+        VMZUQZLMPAUUIUUEDUESZRUUMUVEDUUEUVETZUUSVPUQVNAUUGYRFAUUDUUHUUTVOVQAUUK
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+        CSZUMZUVPUNUVIUWDUWEUVIRZVFZUWGVFZUVSUWEDUHSZQZCSZUVTUWFYQQZUVLUVOUWJUU
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+        IQUMAMWFAUVEWGADGUKWHOWJZAUVEUUFAGDWKSZRGDWLSRUVEUUFWMZAGUUOUWSLAUVBUUO
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+        XBWSWTXEXFUWDUWHUWGXGZXDZUVSUWEDEUWKYQCUVEUVFUWKTZUUCXJXHUWJUWLUUFULZFS
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+        GUWQRUWBUVGUXJUMAUWTUVHUVJUWBUWAUWHUWGUXAWAZUWJUVGYMUWQAUVHUVJUWBUWAUWH
+        UWGXKAUWQYMUMUVHUVJUWBUWAUWHUWGUWRWAZXIUXCUVEUVGUVSUUFUVEXLXHUWJUWTUVIU
+        WQRUWHUVIUXKUMUXLUWJUVIYMUWQUVQUVJUWBUWAUWHUWGXMUXMXIUXFUVEUVIUWEUUFUVE
+        XLXHXNUWJUVEBDUWKYPGUVSUWEMUVFUXHUUBAUURUVHUVJUWBUWAUWHUWGOWAAGUUNRUVHU
+        VJUWBUWAUWHUWGUUQWAUXDUXGXOXPVQUWJBCDEFGUWLHIJUWJUUKUVBUWOUVCUQLMPUWJUV
+        EDUWKUVSUWEUVFUXHUWJUUKUUIUWOUULUQUXDUXGXQVNXTUWJUVMUVTUVNUWFYQUWCUWAUW
+        HUWGXRUWIUWGXSXNYAUWDUNBCDEFGUVIHIJAUVBUVHUVJUWBUWAUVDXFLMPUVQUVJUWBUWA
+        XRYBYCUVRUCBCDEFGUVGHIJAUVBUVHUVJUVDYDLMPAUVHUVJXGYBYCYEAFBEYFQRZFBEVLQ
+        RABCDEFGHIJUVDLMPNYIZBEFYGUQYJAUXNYOUXOYMYNBEFYTYNTZYHUQYMYNBEFYTUXPYKY
+        L $.
+    $}
+
+    $d F p q $.  $d G q $.  $d H q $.  $d K q $.  $d Q p q $.  $d ph q $.
+    $( The image ` H ` of a ring homomorphism ` F ` is isomorphic with the
+       quotient ring ` Q ` over ` F ` 's kernel ` K ` .  This a part of what is
+       sometimes called the first isomorphism theorem for rings.  (Contributed
+       by Thierry Arnoux, 10-Mar-2025.) $)
+    ricqusker $p |- ( ph -> Q ~=r H ) $=
+      ( vp vq cbs cfv cv cima cuni cmpt crs co wcel cric wbr weq imaeq2 cbvmptv
+      unieqd rhmqusker brrici syl ) ANBPQZCNRZSZTZUAZBEUBUCUDBEUEUFABCDEURFGOHI
+      JKLMNOUNUQCORZSZTNOUGUPUTUOUSCUHUJUIUKBEURULUM $.
   $}
 
   ${
@@ -500009,141 +500275,251 @@ The sumset operation can be used for both group (additive) operations and
     elrspunidl.1 $e |- .0. = ( 0g ` R ) $.
     elrspunidl.x $e |- .x. = ( .r ` R ) $.
     elrspunidl.r $e |- ( ph -> R e. Ring ) $.
-    elrspunidl.i $e |- ( ph -> S C_ ( LIdeal ` R ) ) $.
-    $( Elementhood to the span of a union of ideals.  (Contributed by Thierry
-       Arnoux, 30-Jun-2024.) $)
-    elrspunidl $p |- ( ph -> ( X e. ( N ` U. S ) <-> E. a e. ( B ^m S )
+    ${
+      elrspunidl.i $e |- ( ph -> S C_ ( LIdeal ` R ) ) $.
+      $( Elementhood in the span of a union of ideals.  (Contributed by Thierry
+         Arnoux, 30-Jun-2024.) $)
+      elrspunidl $p |- ( ph -> ( X e. ( N ` U. S ) <-> E. a e. ( B ^m S )
        ( a finSupp .0. /\ X = ( R gsum a ) /\ A. k e. S ( a ` k ) e. k ) ) ) $=
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-      disjpreima iunid imaeq2i iunpreima ad2antlr 3eqtr3a resmptd sneqd mpteq1d
-      gsumpart ovexd fvmptd2 cnvex imaex csubmnd csubg lidlsubg eleq12d rspcdva
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-      XKUYSVXLVXM $.
+        ( vj wcel wa cvv vb vi vf vl vm cfv cfsupp wbr cmpt cgsu wceq cmap wrex
+        cv co wral w3a wss clidl sselda syl ralrimiva fvexd ssexd biimpi adantl
+        eqid wf ccnv csn adantr a1i ad7antr ad7antlr cnvimass sseqtrid ad3antlr
+        cima 3syl ffvelcdmd sseldd ringcl syl3anc weq fveq2 cbvmptv fmptd csupp
+        wfun mptexd ffund simp-5r nfv nfcv nfan cdif ffnd ad2antrr fvexi ssdifd
+        fvdifsupp oveq1d eldifad syl2anc eqtrd suppss2f syl22anc elmapd biimpar
+        fsuppsssupp wb breq1 eqeq2d fveq1 funmpt simplr simp-4r simpllr sseqtrd
+        cfn cin sstrd oveq2d ad6antr sylancl ffvelcdmda simpr wdisj ciun 3eqtrd
+        mpteq2dva imaeq2d wfn biimpa sylancom syl2an2r rspcedvd chash ad4antr
+        cz cuni lidlss unissb sylibr elrsp wex uniexd eluni2 eleq2 sylc simp-5l
+        ac6sg ad3antrrr crg ccmn ringcmn vex cnvexg imaexg mp2b cdm fdm oveq12d
+        elmapi id nfmpt1 nfov nfeq2 nfra1 c0g ringlz gsumcl fmpttd oveq2 eleq1d
+        cbs ralbidv 3anbi123d fsuppimpd imafi snssi sspreima difpreima suppssdm
+        ad6antlr fssdm sseqtrrd sseqin2 dminss eqsstrrdi sscond fimacnv difeq1d
+        fdmd ssidd suppssr syldan cmnd cmnmndd gsumz suppss2 ssfid isfsupp cres
+        mpteq2da ad5antlr fmptdf disjpreima imaeq2i iunpreima ad2antlr gsumpart
+        sndisj iunid 3eqtr3a resmptd sneqd mpteq1d ovexd fvmptd2 cnvex lidlsubg
+        imaex csubmnd csubg subgsubm eleq12d elpreima elsni simpl2im gsumsubmcl
+        rspcdva eleqtrd lidlmcl eqeltrd 3jca exlimddv r19.29an crn czrh cxp cun
+        anasss czring crh zringbas rhmf cn0 ssv ssdif ax-mp sseli fsuppinisegfi
+        zrhrhm nn0zd ring0cl fconst6g c0 disjdif fun2d simplll elssuni ralimdva
+        hashcl sseld fnfvrnss ssdifssd undif sylib feq2d mpbid mpteq2dv anbi12d
+        imp mpbird elexd simprd fsupprnfi diffi fdmfifsupp fczfsuppd funisfsupp
+        fsuppun syl31anc fvex fnmpti fnconstg fvun1d sneq fvmptd3 fvun2 mp3an12
+        fveq2d fvconst2 gsummptres2 cmg gsumhashmul sselid cur zrhmulg ringidcl
+        simpll mulgass2 syl13anc ringlidm syl21anc 3eqtr4d 3eqtr4rd exp41 3imp2
+        jca impbida bitrd ) AHDUUAZGUFRUAUNZIUGUHZHCQVXNQUNZVXOUFZVXQEUOZUIZUJU
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+    $}
+
+    ${
+      $( TODO - shorten using ~ elrspunidl $)
+      $d .+ a i r x y $.  $d .0. a i r x y $.  $d .0. b $.  $d .x. a i r x y $.
+      $d .x. b x y $.  $d A a i r x y $.  $d A b $.  $d B a i r x y $.
+      $d B b $.  $d I a i r x y $.  $d I b $.  $d N a b $.  $d R a i r x y $.
+      $d R b $.  $d X a i r x y $.  $d X b $.  $d a i ph r x y $.  $d b ph $.
+      elrspunsn.p $e |- .+ = ( +g ` R ) $.
+      elrspunsn.i $e |- ( ph -> I e. ( LIdeal ` R ) ) $.
+      elrspunsn.x $e |- ( ph -> X e. ( B \ I ) ) $.
+      $( Membership to the span of an ideal ` R ` and a single element ` X ` .
+         (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+      elrspunsn $p |- ( ph -> ( A e. ( N ` ( I u. { X } ) )
+                        <-> E. r e. B E. i e. I A = ( ( r .x. X ) .+ i ) ) ) $=
+        ( va vx vb vy csn cun cfv wcel cv cfsupp co cmpt cgsu wceq wa cmap wrex
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+        syl oveq2 elmapi ad3antlr ad3antrrr snidg elun2 3syl ffvelcdmd cres cvv
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+        fveq1 adantl simplr weq fveq2 id oveq12d mpteq2dva jca rspcedvd syl2anc
+        simpr ad2antrr unexd sseldd ringcld mptexd wfun funmpt fsuppimpd adantr
+        csupp fvdifsupp eqtrd suppss2 ssfid isfsuppd wn sylibr eqidd gsumsplit2
+        cdif cmnmndd sselid gsumsnd fmpttd cmncom syl3anc simp-4r ad4antr ifcld
+        cif eqeltrd iftrued iffalsed fvmptd2 wne eqtr3d ring0cl fsuppres fvresd
+        crg cbvmptv eqtr4id mpbird rspidlid eleqtrd ccmn ringcmnd snex ffnd cin
+        ringlz eldifbd disjsn ad2antlr ssun2 fveq2d ssdifd sselda syldan 3eqtrd
+        c0 gsumcl2 2rspcedvdw anasss r19.29an cpr cur ringidcl prfi mptiffisupp
+        cfn simpllr eleqtrdi gsummptif1n0 prid2g ad5antlr nelneq 3eqtrrd eleq1w
+        ringlidmd eqeq1 ifbid ad5antr fvmptd3 nelne2 neqne nelprd eqtr2d ifeqda
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+        XLWUCEUMWUQUBUXFUXKWUBWUQUXJWUAUXIFUXIUXGVYSWMVHWJWKVIWLWNVYLVYTWUEVYLU
+        DUXFVYMVYQVTVYSCCKVYSVAZWUHVYMUVOUHVYLJUXTUVMWDVYLVYNUOVYOUXRVYPCAVYHVY
+        JUYBVYNYKAWUKVYHVYJUYBVYNWUNYLYMAVVPVYHVYJUYBVWRVNZUVNZVYLUYAEUBHWUBULZ
+        UMUKZEUBUXEWUBULUMUKZDUKZBWUDVYLUXTUXSDUKZUYAWVDVYLVWGUXTCUHZUXSCUHWVEU
+        YAUNAVWGVYHVYJUYBVWHVNZVYLHCUXTVYLUYEUYFWUGUYHVJZVYIVYJUYBWOZXGZVYLCEFU
+        XRJNPAVVOVYHVYJUYBQVNZAVYHVYJUYBUVPZAVUFVYHVYJUYBUYJVNZXHZCDEUXTUXSNRYI
+        YJVYLUXTWVBUXSWVCDVYLEUBHUBGWPZUXTKYNZULZUMUKUXTWVBVYLUXTUBWVQEHUYDUXTK
+        OVYLEWVGYEZWUGWVIWVQVAVYLUXTCEWBUGWVJNUVQUVRVYLWVQWVAEUMVYLUBHWVPWUBVYL
+        VYBUOZWVOUXTKWUBWVSWVOUOZWUBUXTVYSUGZUXTFUKZVYPUXTFUKUXTWVOWUBWWBUNWVSW
+        VOWUAWWAUXIUXTFUXIUXTVYSWQWVOWRWSWNWVTWWAVYPUXTFWVTUDUXTVYRVYPUXFVYSCWU
+        RWVTUDGWPZUOZVYRVYQVYPWWDVYNVYQKWWDVUNUXTVYMWVTWWCXDZVYJUXTVYMUHVYIUYBV
+        YBWVOWWCJUXTHUVSUVTYOYPWWDVYOUXRVYPWWDVWBVXLVYOXTWWDVUNUXTHWWEWVTVYJWWC
+        VYLVYJVYBWVOWVIXEZXMYOVYLVXLVYBWVOWWCAVXLVYHVYJUYBVXNVNZVNVUNJHUWAXCYQX
+        PWVTHUXFUXTVVMWWFYFWVTVVOWUKVYLVVOVYBWVOWVKXEZWUMVJYRVHWVTCEFVYPUXTNPWU
+        LWWHVYLWVFVYBWVOWVJXEUWDUWBWVSWVOXTZUOZWUBVXDKWWJWUAKUXIFWWJWUAUXIVYMUH
+        ZVXSUXRVYPYNZKYNZKWWJUDUXIVYRWWMUXFVYSCWURUDUBWPZVYNWWKVYQWWLKUDUBVYMUW
+        CWWNVYOVXSUXRVYPVUNUXIJUWEUWFUXDWWJHUXFUXIVVMVYLVYBWWIWOZYFWWJWWKWWLKCW
+        WJVXSUXRVYPCVYLVYHVYBWWIWVLXEAWUKVYHVYJUYBVYBWWIWUNUWGYMVYLVVPVYBWWIWUS
+        XEYMUWHWWJWWKWWLKWWJUXIJUXTWWJVYBVXLUXIJYSWWOVYLVXLVYBWWIWWGXEUXIJHUWIX
+        CWWIUXIUXTYSWVSUXIUXTUWJWNUWKYQXPVHWWJVVOVXIVXJVYLVVOVYBWWIWVKXEWWJHCUX
+        IVYLUYFVYBWWIWVHXEWWOXGVXKXCUWLUWMWTWKYTVYLWVCUXSVYLWUBCUXSUBEJCNWVRWVM
+        WVNVYLVXSUOZWUAUXRUXIJFWWPUDUXIVYRUXRUXFVYSCWURWWPWWNUOZVYRVYQUXRWWQVYN
+        VYQKWWQVUNUXIVYMWWPWWNXDZWWQUXIJVYMVYLVXSWWNWOZWWQVUFJVYMUHVYLVUFVXSWWN
+        WVMXEJUXTCUWNVJYOYOYPWWQVYOUXRVYPWWQVUNUXIJWWRWWSXPYPXPWWPUXIJUXFVYLVXS
+        XDZWWPVUGVUHAVUGVYHVYJUYBVXSVXQYLVUJVJYOVYLVYHVXSWVLXMYRWWTWSYGUWOWSYTV
+        YKUYBXDVYLUXFCHUXEDUBEVTWUBKNORWVGWUHVYLVWNUOZCEFWUAUXINPAVVOVYHVYJUYBV
+        WNQYLVYLUXFCUXIVYSWUPUWPWXAUXFCUXIAVWQVYHVYJUYBVWNUYKYLVYLVWNXDXGXHVYLW
+        UCVTCKVYLUBUXFWUBVTWUHXIWUSWUCXJVYLUBUXFWUBXKWDVYLVYSKXNUKZWUCKXNUKVYLV
+        YSKWUTXLVYLUXFWUBUBVTWXBKVYLUXIUXFWXBYDUHZUOZWUBVXDKWXDWUAKUXIFWXDUXFVY
+        SVTCUXIKVYLVYSUXFUWQWXCVYLUDUXFVYRVYSCVYLUDUWRWUOWURUWSXMVYLVXEWXCWUHXM
+        VYLVVPWXCWUSXMVYLWXCXDZXOVHWXDVVOVXIVXJAVVOVYHVYJUYBWXCQYLWXDUXFCUXIAVW
+        QVYHVYJUYBWXCUYKYLWXDUXIUXFWXBWXEVCXGVXKXCXPWUHXQXRXSVYLVXLVXMWWGVXOYAV
+        YLUXFYBYCUWTXAXBUXAUXBUXC $.
+    $}
   $}
 
   ${
@@ -500624,6 +501000,31 @@ The sumset operation can be used for both group (additive) operations and
       wa ) BEFZCBGHFZOAIFZCBJHFZABCDKPSRQABCDLMN $.
   $}
 
+  ${
+    qsnzr.q $e |- Q = ( R /s ( R ~QG I ) ) $.
+    qsnzr.1 $e |- B = ( Base ` R ) $.
+    qsnzr.r $e |- ( ph -> R e. Ring ) $.
+    qsnzr.z $e |- ( ph -> R e. NzRing ) $.
+    qsnzr.i $e |- ( ph -> I e. ( 2Ideal ` R ) ) $.
+    qsnzr.2 $e |- ( ph -> I =/= B ) $.
+    $( A quotient of a non-zero ring by a proper ideal is a non-zero ring.
+       (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+    qsnzr $p |- ( ph -> Q e. NzRing ) $=
+      ( crg wcel cfv eqid syl2anc co wceq syl wa cur c0g wne cnzr c2idl qusring
+      cqg cec cminusg cplusg cgrp ringgrp grpinvid 3syl oveq1d ringidcl grplidd
+      wbr eqtrd clidl wn 2idllidld pridln1 syl3anc eqneltrd adantr cnsg lidlnsg
+      wss csubg nsgsubg subgss wer eqger simpr ersym w3a eqgval biimpa syl21anc
+      simp3d mtand erth mtbid neqned qus1 simprd qus0 3netr3d isnzr sylanbrc )
+      ACLMZCUANZCUBNZUCCUDMADLMZEDUENZMZWLHJDECWPFWPOZUFPADUANZDEUGQZUHZDUBNZWT
+      UHZWMWNAXAXCAWSXBWTURZXAXCRAXDXBDUINZNZWSDUJNZQZEMZAXHWSEAXHXBWSXGQWSAXFX
+      BWSXGAWODUKMZXFXBRHDULZDXEXBXBOZXEOZUMUNUOABXGDWSXBGXGOZXLAWOXJHXKSAWOWSB
+      MZHBDWSGWSOZUPSZUQUSAWOEDUTNMZEBUCWSEMVAHADEJVBZKBDWSEGXPVCVDVEAXDTZWOEBV
+      IZXBWSWTURZXIAWOXDHVFAYAXDAEDVJNMZYAAEDVGNMZYCAWOXRYDHXSDEVHPZEDVKSZBEDGV
+      LSVFXTWSXBWTBABWTVMZXDAYCYGYFWTDBEGWTOZVNSZVFAXDVOVPWOYATZYBTXBBMZXOXIYJY
+      BYKXOXIVQXBWSXGWTEDXELBGXMXNYHVRVSWAVTWBAWSXBWTBYIXQWCWDWEAWLXAWMRZAWOWQW
+      LYLTHJDECWSWPFWRXPWFPWGAYDXCWNRYEEDCXBFXLWHSWICWMWNWMOWNOWJWK $.
+  $}
+
 
 $(
 -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
@@ -500721,6 +501122,21 @@ The sumset operation can be used for both group (additive) operations and
       UIUHCUAUBUCUKUFUJMBUHUIUPUOUDUER $.
   $}
 
+  ${
+    mxidlmaxv.1 $e |- B = ( Base ` R ) $.
+    mxidlmaxv.2 $e |- ( ph -> R e. Ring ) $.
+    mxidlmaxv.3 $e |- ( ph -> M e. ( MaxIdeal ` R ) ) $.
+    mxidlmaxv.4 $e |- ( ph -> I e. ( LIdeal ` R ) ) $.
+    mxidlmaxv.5 $e |- ( ph -> M C_ I ) $.
+    mxidlmaxv.6 $e |- ( ph -> X e. ( I \ M ) ) $.
+    $( An ideal ` I ` strictly containing a maximal ideal ` M ` is the whole
+       ring ` B ` .  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+    mxidlmaxv $p |- ( ph -> I = B ) $=
+      ( wceq crg wcel cmxidl cfv clidl wss wo mxidlmax syl22anc eldifad eldifbd
+      wn wne nelne1 syl2anc neneqd orcnd ) ADEMZDBMZACNOECPQODCRQOEDSUKULTHIJKB
+      CDEGUAUBADEAFDOFEOUEDEUFAFDELUCAFDELUDFDEUGUHUIUJ $.
+  $}
+
   ${
     $d .X. a k u $.  $d .X. b $.  $d M a k u x y $.  $d M b $.  $d R a b x y $.
     $d R a k u x y $.
@@ -500880,6 +501296,249 @@ The sumset operation can be used for both group (additive) operations and
       EUABANHZAUBUFOQPRS $.
   $}
 
+  ${
+    opprabs.o $e |- O = ( oppR ` R ) $.
+    opprabs.m $e |- .x. = ( .r ` R ) $.
+    opprabs.1 $e |- ( ph -> R e. V ) $.
+    opprabs.2 $e |- ( ph -> Fun R ) $.
+    opprabs.3 $e |- ( ph -> ( .r ` ndx ) e. dom R ) $.
+    opprabs.4 $e |- ( ph -> .x. Fn ( B X. B ) ) $.
+    $( The opposite ring of the opposite ring is the original ring.  Note the
+       conditions on this theorem, which makes it unpractical in case we only
+       have e.g. ` R e. Ring ` as a premise.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    opprabs $p |- ( ph -> R = ( oppR ` O ) ) $=
+      ( cmulr cfv ctpos cop csts co wceq eqid cnx coppr cxp wfn cbs opprmulfval
+      tposeqi wrel cdm fnrel fndm releqd mpbiri tpostpos2 syl2anc eqtrid eqtrdi
+      relxp syl opeq2d oveq2d wcel opprbas opprval oveq1i eqtri cvv fvex tposex
+      setsabs mpan2 mulridx setsidvald 3eqtr4rd ) ACUAMNZEMNZOZPZQRZCVOCMNZPZQR
+      EUBNZCAVRWACQAVQVTVOAVQDVTADBBUCZUDZVQDSLWDVQDOZOZDVPWECUENZCVPDEWGTZHGVP
+      TZUFUGWDDUHDUIZUHZWFDSWCDUJWDWKWCUHBBURWDWJWCWCDUKULUMDUNUOUPUSHUQUTVAACF
+      VBZWBVSSIWLWBCVOWEPQRZVRQRZVSWBEVRQRWNWGEVPWBWGCEGWHVCWIWBTVDEWMVRQWGCDEW
+      HHGVDVEVFWLVQVGVBWNVSSVPEMVHVIVOWEVQCFVGVJVKUPUSACMVOFVLIJKVMVN $.
+  $}
+
+  ${
+    $d B x y $.  $d I x y $.  $d O x y $.  $d R x y $.
+    oppreqg.o $e |- O = ( oppR ` R ) $.
+    ${
+      oppreqg.b $e |- B = ( Base ` R ) $.
+      $( Group coset equivalence relation for the opposite ring.  (Contributed
+         by Thierry Arnoux, 9-Mar-2025.) $)
+      oppreqg $p |- ( ( R e. V /\ I C_ B ) -> ( R ~QG I ) = ( O ~QG I ) ) $=
+        ( vx vy wcel wss wa cqg co cv cpr cfv eqid eqgfval cvv copab wceq coppr
+        cminusg cplusg fvexi opprbas opprneg oppradd mpan adantl eqtr4d ) BEJZC
+        AKZLBCMNZHOZIOZPAKUPBUDQZQUQBUEQZNCJLHIUAZDCMNZHIUSUOCBUREAGURRZUSRZUOR
+        SUNVAUTUBZUMDTJUNVDDBUCFUFHIUSVACDURTAABDFGUGBURDFVBUHUSBDFVCUIVARSUJUK
+        UL $.
+    $}
+
+    $d O a b i x $.  $d R a b i x $.  $d a b i ph x $.
+    oppr2idl.2 $e |- ( ph -> R e. Ring ) $.
+    $( The ideals of the opposite ring's opposite ring are the ideals of the
+       original ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+    opprlidlabs $p |- ( ph -> ( LIdeal ` R ) = ( LIdeal ` ( oppR ` O ) ) ) $=
+      ( vi vx va vb clidl cfv cv cbs cmulr co wcel wral w3a wa eqid coppr c0 wb
+      wss wne cplusg opprmul eqtr2i a1i oveq1d eleq1d ralbidva anasss 2ralbidva
+      wceq 3anbi3d islidl opprbas oppradd 3bitr4g eqrdv ) AFBJKZCUAKZJKZAFLZBMK
+      ZUDZVEUBUEZGLZHLZBNKZOZILZBUFKZOZVEPZIVEQZHVEQGVFQZRVGVHVIVJVCNKZOZVMVNOZ
+      VEPZIVEQZHVEQGVFQZRVEVBPVEVDPAVRWDVGVHAVQWCGHVFVEAVIVFPZVJVEPZVQWCUCAWESW
+      FSZVPWBIVEWGVMVEPSZVOWAVEWHVLVTVMVNVLVTUOWHVTVJVICNKZOVLCMKZCVSWIVCVIVJWJ
+      TWITZVCTZVSTZUGVFBWIVKCVJVIVFTZVKTZDWKUGUHUIUJUKULUMUNUPGVFVNBVKVBVEHIVBT
+      WNVNTZWOUQGVFVNVCVSVDVEHIVDTVFCVCWLVFBCDWNURURVNCVCWLVNBCDWPUSUSWMUQUTVA
+      $.
+
+    $( Two sided ideal of the opposite ring.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    oppr2idl $p |- ( ph -> ( 2Ideal ` R ) = ( 2Ideal ` O ) ) $=
+      ( clidl cfv cin coppr c2idl incom opprlidlabs ineq2d eqid 2idlval 3eqtr4g
+      eqtrid ) ABFGZCFGZHZSCIGZFGZHZBJGZCJGZATSRHUCRSKARUBSABCDELMQBUDRSCRNDSNZ
+      UDNOCUESUBUAUFUANUBNUENOP $.
+
+    $d M j $.  $d O j $.  $d j ph $.
+    opprmxidl.3 $e |- ( ph -> M e. ( MaxIdeal ` R ) ) $.
+    $( The maximal ideal of the opposite ring's opposite ring.  (Contributed by
+       Thierry Arnoux, 9-Mar-2025.) $)
+    opprmxidlabs $p |- ( ph -> M e. ( MaxIdeal ` ( oppR ` O ) ) ) $=
+      ( vj cfv wcel clidl wceq cmxidl opprring eqid syl2anc wa ad2antrr opprbas
+      crg coppr cbs wne cv wo wi wral 3syl mxidlidl opprlidlabs eleqtrd mxidlnr
+      wss simplr eleqtrrd simpr mxidlmax syl22anc ralrimiva w3a ismxidl biimpar
+      ex syl13anc ) ADUAIZTJZCVEKIZJZCBUBIZUCZCHUDZUMZVKCLVKVILUEZUFZHVGUGZCVEM
+      IJZABTJZDTJVFFBDENDVEVEOZNUHACBKIZVGAVQCBMIJZCVSJFGVIBCVIOZUIPABDEFUJZUKA
+      VQVTVJFGVIBCWAULPAVNHVGAVKVGJZQZVLVMWDVLQZVQVTVKVSJVLVMAVQWCVLFRAVTWCVLGR
+      WEVKVGVSAWCVLUNAVSVGLWCVLWBRUOWDVLUPVIBVKCWAUQURVCUSVFVPVHVJVOUTVIVEHCVID
+      VEVRVIBDEWASSVAVBVD $.
+  $}
+
+  ${
+    $d B p q $.  $d I p q x y $.  $d O p q x y $.  $d Q p q x y $.  $d R p q $.
+    $d X p q $.  $d Y p q $.  $d p ph q x y $.
+    opprqus.b $e |- B = ( Base ` R ) $.
+    opprqus.o $e |- O = ( oppR ` R ) $.
+    opprqus.q $e |- Q = ( R /s ( R ~QG I ) ) $.
+    ${
+      opprqusbas.r $e |- ( ph -> R e. V ) $.
+      opprqusbas.i $e |- ( ph -> I C_ B ) $.
+      $( The base of the quotient of the opposite ring is the same as the base
+         of the opposite of the quotient ring.  (Contributed by Thierry Arnoux,
+         9-Mar-2025.) $)
+      opprqusbas $p |- ( ph -> ( Base ` ( oppR ` Q ) )
+                               = ( Base ` ( O /s ( O ~QG I ) ) ) ) $=
+        ( coppr cfv cbs cqg co wceq cvv a1i cqus opprbas cqs wss oppreqg qseq2d
+        eqid wcel syl2anc ovexd qusbas eqidd fvexi 3eqtr3d eqtr3id ) ACMNZONCON
+        ZFFEPQZUAQZONZUQCUPUPUGUQUGUBABDEPQZUCBURUCUQUTAVAURBADGUHEBUDVAURRKLBD
+        EFGIHUEUIUFAVADCBSGCDVAUAQRAJTBDONRAHTADEPUJKUKAURFUSBSSAUSULBFONRABDFI
+        HUBTAFEPUJFSUHAFDMIUMTUKUNUO $.
+    $}
+
+    opprqus1r.r $e |- ( ph -> R e. Ring ) $.
+    opprqus1r.i $e |- ( ph -> I e. ( 2Ideal ` R ) ) $.
+    ${
+      opprqusmulr.e $e |- E = ( Base ` Q ) $.
+      opprqusmulr.x $e |- ( ph -> X e. E ) $.
+      opprqusmulr.y $e |- ( ph -> Y e. E ) $.
+      $( The multiplication operation of the quotient of the opposite ring is
+         the same as the multiplication operation of the opposite of the
+         quotient ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+      opprqusmulr $p |- ( ph -> ( X ( .r ` ( oppR ` Q ) ) Y )
+                                = ( X ( .r ` ( O /s ( O ~QG I ) ) ) Y ) ) $=
+        ( cfv co wcel vp vq coppr cmulr cqg cqus eqid opprmul cv cec wa ad4antr
+        crg c2idl simplr simp-4r qusmul2 simpr simpllr oveq12d opprbas opprring
+        wceq syl oppr2idl eleqtrd clidl 2idllidld lidlss oppreqg syl2anc eceq2d
+        wss eqtrd a1i eceq1d eqtr3d 3eqtr4d cqs cbs eleqtrdi ad2antrr cvv ovexd
+        wrex qusbas eqtr3i eqtr2di elqsi r19.29a eqtrid ) AHICUCRZUDRZSIHCUDRZS
+        ZHIGGFUESZUFSZUDRZSZECWMWNWLHIOWNUGZWLUGZWMUGUHAHUAUIZDFUESZUJZVCZWOWSV
+        CZUABAXBBTZUKZXEUKZIUBUIZXCUJZVCZXFUBBXIXJBTZUKZXLUKZXKXDWNSXJXBDUDRZSZ
+        XCUJZWOWSXOBCDXPWNFXJXBLJXPUGZWTADUMTZXGXEXMXLMULAFDUNRZTXGXEXMXLNULXIX
+        MXLUOZAXGXEXMXLUPZUQXOIXKHXDWNXNXLURZXHXEXMXLUSZUTXOXBWPUJZXJWPUJZWRSXB
+        XJGUDRZSZWPUJZWSXRXOBWQGYHWRFXBXJWQUGBDGKJVAYHUGZWRUGAGUMTZXGXEXMXLAXTY
+        LMDGKVBVDULAFGUNRZTXGXEXMXLAFYAYMNADGKMVEVFULYCYBUQXOHYFIYGWRXOHXDYFYEX
+        OXCWPXBAXCWPVCZXGXEXMXLAXTFBVMZYNMAFDVGRZTYOADFNVHBFYPDJYPUGVIVDBDFGUMK
+        JVJVKULZVLVNXOIXKYGYDXOXCWPXJYQVLVNUTXOYIXCUJXRYJXOYIXQXCYIXQVCXOBDYHXP
+        GXBXJJXSKYKUHVOVPXOXCWPYIYQVLVQVRVRXIIBXCVSZTXLUBBWEXIIWLVTRZYRAIYSTXGX
+        EAIEYSQECWLXAOVAZWAWBAYSYRVCXGXEAYRCVTRZYSAXCDCBWCUMCDXCUFSVCALVOBDVTRV
+        CAJVOADFUEWDMWFEUUAYSOYTWGWHZWBVFUBBIXCWIVDWJAHYRTXEUABWEAHYSYRAHEYSPYT
+        WAUUBVFUABHXCWIVDWJWK $.
+    $}
+
+    $( The ring unity of the quotient of the opposite ring is the same as the
+       ring unity of the opposite of the quotient ring.  (Contributed by
+       Thierry Arnoux, 9-Mar-2025.) $)
+    opprqus1r $p |- ( ph -> ( 1r ` ( oppR ` Q ) ) = ( 1r ` ( O /s ( O ~QG I ) )
+       ) ) $=
+      ( vx vy coppr cfv cbs co eqid wcel wa cqg cvv fvexd ovexd clidl 2idllidld
+      cqus crg wss lidlss syl opprqusbas ad2antrr c2idl simpr opprbas eleqtrrdi
+      cv adantr adantlr opprqusmulr urpropd ) ALMCNOZPOZVCFFEUAQZUGQUBUBVDRACNU
+      CAFVEUGUDABCDEFUHGHIJAEDUEOZSEBUIADEKUFBEVFDGVFRUJUKULALURZVDSZTZMURZVDSZ
+      TBCDCPOZEFVGVJGHIADUHSVHVKJUMAEDUNOSVHVKKUMVLRZVIVGVLSVKVIVGVDVLAVHUOVLCV
+      CVCRVMUPZUQUSAVKVJVLSVHAVKTVJVDVLAVKUOVNUQUTVAVB $.
+  $}
+
+  ${
+    $d M m r u v x $.  $d M n s $.  $d M u v x y z $.  $d M w $.  $d O r s w $.
+    $d Q m r u $.  $d Q n s $.  $d Q u v x y z $.  $d Q w $.  $d R m r u v x $.
+    $d R n u x $.  $d R s u v w x $.  $d R u v x y z $.  $d X m r $.
+    $d X v y z $.  $d m ph r u $.  $d n ph s x $.  $d ph u v x y z $.
+    $d ph w $.
+    qsdrng.0 $e |- O = ( oppR ` R ) $.
+    qsdrng.q $e |- Q = ( R /s ( R ~QG M ) ) $.
+    qsdrng.r $e |- ( ph -> R e. NzRing ) $.
+    qsdrngi.1 $e |- ( ph -> M e. ( MaxIdeal ` R ) ) $.
+    qsdrngi.2 $e |- ( ph -> M e. ( MaxIdeal ` O ) ) $.
+    ${
+      qsdrngilem.1 $e |- ( ph -> X e. ( Base ` R ) ) $.
+      qsdrngilem.2 $e |- ( ph -> -. X e. M ) $.
+      $( Lemma for ~ qsdrngi .  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+      qsdrngilem $p |- ( ph -> E. v e. ( Base ` Q )
+                           ( v ( .r ` Q ) [ X ] ( R ~QG M ) ) = ( 1r ` Q ) ) $=
+        ( cfv co wceq wcel eqid ad3antrrr vr vm cur cv cmulr cplusg cqg cec cbs
+        wrex wa cqs simpllr ovex ecelqsi syl cvv cnzr cqus ovexd qusbas eleqtrd
+        a1i wb oveq1 eqeq1d adantl crg nzrring c2idl clidl cin mxidlidl syl2anc
+        opprring elind 2idlval eleqtrrdi qusmul2 wer cnsg csubg lidlnsg nsgsubg
+        cmxidl eqger 3syl wss cminusg lidlss ringcld ringidcl simpr oveq2d cgrp
+        wbr c0g ringgrpd oveq1d grpinvcld simplr sseldd grpassd grplidd 3eqtr3d
+        grplinvd eqtrd eqeltrd w3a eqgval biimpar syl23anc qus1 simprd rspcedvd
+        erthi csn cun crsp snssd unssd rspcl rspssid unssad unssbd snssg eldifd
+        mxidlmaxv eleqtrrd elrspunsn mpbid r19.29vva ) ADUCOZUAUDZGDUEOZPZUBUDZ
+        DUFOZPZQZBUDZGDEUGPZUHZCUEOZPZCUCOZQZBCUIOZUJUAUBDUIOZEAYNUUIRZUKZYQERZ
+        UKZYTUKZUUGYNUUBUHZUUCUUDPZUUFQZBUUOUUHUUNUUOUUIUUBULZUUHUUNUUJUUOUURRA
+        UUJUULYTUMZUUIYNUUBDEUGUNUOUPAUURUUHQUUJUULYTAUUBDCUUIUQURCDUUBUSPQAIVC
+        UUIUUIQAUUISZVCADEUGUTJVATVBUUAUUOQZUUGUUQVDUUNUVAUUEUUPUUFUUAUUOUUCUUD
+        VEVFVGUUNUUPYMUUBUHZUUFUUNUUPYPUUBUHUVBUUNUUICDYOUUDEYNGIUUTYOSZUUDSADV
+        HRZUUJUULYTADURRUVDJDVIUPZTZAEDVJOZRZUUJUULYTAEDVKOZFVKOZVLUVGAUVIUVJEA
+        UVDEDWEOREUVIRZUVEKUUIDEUUTVMVNZAFVHRZEFWEOREUVJRAUVDUVMUVEDFHVOUPLFUIO
+        ZFEUVNSVMVNVPDUVGUVIUVJFUVISZHUVJSUVGSZVQVRTZUUSAGUUIRZUUJUULYTMTZVSUUN
+        YPYMUUBUUIAUUIUUBVTZUUJUULYTAEDWAORZEDWBORUVTAUVDUVKUWAUVEUVLDEWCVNEDWD
+        UUBDUUIEUUTUUBSZWFWGTUUNUVDEUUIWHZYPUUIRZYMUUIRZYPDWIOZOZYMYRPZERZYPYMU
+        UBWPZUVFAUWCUUJUULYTAUVKUWCUVLUUIEUVIDUUTUVOWJUPZTZUUNUUIDYOYNGUUTUVCUV
+        FUUSUVSWKZAUWEUUJUULYTAUVDUWEUVEUUIDYMUUTYMSZWLUPZTUUNUWHYQEUUNUWHUWGYS
+        YRPZYQUUNYMYSUWGYRUUMYTWMWNUUNUWGYPYRPZYQYRPDWQOZYQYRPUWPYQUUNUWQUWRYQY
+        RUUNUUIYRDUWFYPUWRUUTYRSZUWRSZUWFSZADWORUUJUULYTADUVEWRTZUWMXFWSUUNUUIY
+        RDUWGYPYQUUTUWSUXBUUNUUIDUWFYPUUTUXAUXBUWMWTUWMUUNEUUIYQUWLUUKUULYTXAZX
+        BZXCUUNUUIYRDYQUWRUUTUWSUWTUXBUXDXDXEXGUXCXHUVDUWCUKUWJUWDUWEUWIXIYPYMY
+        RUUBEDUWFVHUUIUUTUXAUWSUWBXJXKXLXPXGUUNUVDUVHUVBUUFQZUVFUVQUVDUVHUKCVHR
+        UXEDECYMUVGIUVPUWNXMXNVNXGXOAYMEGXQZXRZDXSOZOZRYTUBEUJUAUUIUJAYMUUIUXIU
+        WOAUUIDUXIEGUUTUVEKAUVDUXGUUIWHZUXIUVIRUVEAEUXFUUIUWKAGUUIMXTYAZUUIDUVI
+        UXGUXHUXHSZUUTUVOYBVNAEUXFUXIAUVDUXJUXGUXIWHUVEUXKUUIDUXGUXHUXLUUTYCVNZ
+        YDAGUXIEAUVRUXFUXIWHZGUXIRZMAEUXFUXIUXMYEUVRUXOUXNGUXIUUIYFXKVNNYGYHYIA
+        YMUUIYRDYOUBEUXHGUWRUAUXLUUTUWTUVCUVEUWSUVLAGUUIEMNYGYJYKYL $.
+    $}
+
+    $( A quotient by a maximal left and maximal right ideal is a division ring.
+       (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+    qsdrngi $p |- ( ph -> Q e. DivRing ) $=
+      ( vv wcel cfv cv co wceq wa eqid syl syl2anc vu vx vw vr vs cdr cur cmulr
+      c0g wne cbs wrex csn cdif wral cnzr crg nzrring clidl cin cmxidl mxidlidl
+      c2idl opprring elind 2idlval eleqtrrdi mxidlnr qsnzr cqg cec cqus qusring
+      nzrnz cui ad10antr adantr eldifi adantl cqs ovex ecelqsi ad4antlr cvv a1i
+      eqidd ovexd qusbas eleqtrd ad2antlr simpllr simp-9r eqcomd oveq12d eqtr3d
+      simp-7r coppr opprmul simp-5r ad3antrrr ad8antr opprqusmulr simpr oppreqg
+      wss lidlss eceq2d eqtr2d eqtr4d oppr1 opprqus1r 3eqtr4d eqtr3id ringinveu
+      eqtrid 3eqtr4rd oveq2d 3eqtrd simp-4r ad9antr opprbas fvexi elqsi r19.29a
+      qseq2d ad6antr eleqtrrd opprnzr opprmxidlabs simplr cgrp ringgrpd ad4antr
+      csubg cnsg lidlnsg nsgsubg eqg0el biimpar qsdrngilem syl21anc wn pm2.65da
+      eqgid qus0 eldifsnneq ad2antrr eqtrd reximddv ralrimiva isdrng4 mpbir2and
+      jca anasss ) ABUFLBUGMZBUIMZUJZUANZKNZBUHMZOZUUOPZUUSUURUUTOZUUOPZQZKBUKM
+      ZULZUAUVFUUPUMZUNZUOABUPLUUQACUKMZBCDGUVJRZACUPLZCUQLZHCURSZHADCUSMZEUSMZ
+      UTCVCMZAUVOUVPDAUVMDCVAMLZDUVOLZUVNIUVJCDUVKVBTZAEUQLZDEVAMLZDUVPLAUVMUWA
+      UVNCEFVDSJEUKMZEDUWCRVBTVECUVQUVOUVPEUVORZFUVPRUVQRZVFVGZAUVMUVRDUVJUJUVN
+      IUVJCDUVKVHTVIBUUOUUPUUORZUUPRZVNSAUVGUAUVIAUURUVILZQZUURUBNZCDVJOZVKZPZU
+      VGUBUVJUWJUWKUVJLZQZUWNQZUUSUWMUUTOZUUOPZUVEKUVFUWQUUSUVFLZUWSUVEUWQUWTQZ
+      UWSQZUVBUVDUXBUCNZUWKEDVJOZVKZEUXDVLOZUHMZOZUXFUGMZPZUVBUCUXFUKMZUXBUXCUX
+      KLZQZUXJQZUUSUDNZUWLVKZPZUVBUDUVJUXNUXOUVJLZQZUXQQZUXCUENZUWLVKZPZUVBUEUV
+      JUXTUYAUVJLZQZUYCQZUVAUURUXCUUTOUURUYBUUTOZUUOUYFUUSUXCUURUUTUYFUYBUXPUXC
+      UUSUYFUVFBUUTBVOMZUUOUURUXPUUPUYBUVFRZUWHUWGUUTRZUYHRZUYEBUQLZUYCAUYLUWIU
+      WOUWNUWTUWSUXLUXJUXRUXQUYDAUVMDUVQLZUYLUVNUWFCDBUVQGUWEVMTZVPVQUWJUURUVFL
+      ZUWOUWNUWTUWSUXLUXJUXRUXQUYDUYCUWIUYOAUURUVFUVHVRVSZVPZUYFUXPUVJUWLVTZUVF
+      UXRUXPUYRLUXNUXQUYDUYCUVJUXOUWLCDVJWAZWBWCUWJUYRUVFPZUWOUWNUWTUWSUXLUXJUX
+      RUXQUYDUYCAUYTUWIAUWLCBUVJWDUPBCUWLVLOPAGWEAUVJWFACDVJWGHWHVQZVPZWIUYFUYB
+      UYRUVFUYDUYBUYRLUXTUYCUVJUYAUWLUYSWBWJVUBWIZUYFUWRUXPUURUUTOUUOUYFUUSUXPU
+      WMUURUUTUXSUXQUYDUYCWKZUYFUURUWMUWPUWNUWTUWSUXLUXJUXRUXQUYDUYCWLZWMWNUXAU
+      WSUXLUXJUXRUXQUYDUYCWPWOUYFUYGUYBUURBWQMZUHMZOZUUOUVFBVUGUUTVUFUYBUURUYIU
+      YJVUFRZVUGRWRUYFUXHUXIVUHUUOUXMUXJUXRUXQUYDUYCWSUYFVUHUYBUURUXGOUXHUYFUVJ
+      BCUVFDEUYBUURUVKFGUWQUVMUWTUWSUXLUXJUXRUXQUYDUYCAUVMUWIUWOUWNUVNWTZXAUWQU
+      YMUWTUWSUXLUXJUXRUXQUYDUYCAUYMUWIUWOUWNUWFWTXAUYIVUCUYQXBUYFUXCUYBUXEUURU
+      XGUYEUYCXCZUYFUURUWMUXEVUEUYFUWLUXDUWKUYEUWLUXDPZUYCAVULUWIUWOUWNUWTUWSUX
+      LUXJUXRUXQUYDAUVMDUVJXEZVULUVNAUVSVUMUVTUVJDUVOCUVKUWDXFSUVJCDEUQFUVKXDTZ
+      VPVQXGXHWNXIUYEUUOUXIPZUYCAVUOUWIUWOUWNUWTUWSUXLUXJUXRUXQUYDAUUOVUFUGMUXI
+      BUUOVUFVUIUWGXJAUVJBCDEUVKFGUVNUWFXKXOVPVQXLXMZXNVUKVUDXPXQUYFUXCUYBUURUU
+      TVUKXQVUPXRUXTUXCUYRLUYCUEUVJULUXTUXCUXKUYRUXBUXLUXJUXRUXQXSUXTUYRUVJUXDV
+      TZUXKAUYRVUQPUWIUWOUWNUWTUWSUXLUXJUXRUXQAUWLUXDUVJVUNYEXTUXTUXDEUXFUVJWDW
+      DUXTUXFWFUVJUWCPZUXTUVJCEFUVKYAZWEUXTEDVJWGEWDLUXTECWQFYBWEWHXHWIUEUVJUXC
+      UWLYCSYDUXNUUSUYRLUXQUDUVJULUXNUUSUVFUYRUWQUWTUWSUXLUXJXSUWJUYTUWOUWNUWTU
+      WSUXLUXJVUAYFYGUDUVJUUSUWLYCSYDUWQUXJUCUXKULUWTUWSUWQUCUXFEDEWQMZUWKVUTRU
+      XFRUWQUVLEUPLAUVLUWIUWOUWNHWTZCEFYHSAUWBUWIUWOUWNJWTZUWQCDEFVUJAUVRUWIUWO
+      UWNIWTZYIUWQUWKUVJUWCUWJUWOUWNYJZVURUWQVUSWEWIUWQUWKDLZUURUUPPZUWQVVEQZUU
+      RUWMCUIMZUWLVKZUUPUWPUWNVVEYJVVGUWMDVVIVVGCYKLZDCYNMLZVVEUWMDPZAVVJUWIUWO
+      UWNVVEACUVNYLYMAVVKUWIUWOUWNVVEADCYOMLZVVKAUVMUVSVVMUVNUVTCDYPTZDCYQSYMZU
+      WQVVEXCVVJVVKQVVLVVEUWLCDUWKUWLRZYRYSUUAVVGVVKVVIDPVVOUWLCUVJDVVHUVKVVPVV
+      HRZUUDSXIAVVIUUPPZUWIUWOUWNVVEAVVMVVRVVNDCBVVHGVVQUUESYMXRUWIVVFUUBAUWOUW
+      NVVEUURUVFUUPUUFWCUUCZYTUUGYDUXBUVCUWRUUOUXBUURUWMUUSUUTUWPUWNUWTUWSWKXQU
+      XAUWSXCUUHUUMUUNUWQKBCDEUWKFGVVAVVCVVBVVDVVSYTUUIUWJUURUYRLUWNUBUVJULUWJU
+      URUVFUYRUYPVUAYGUBUVJUURUWLYCSYDUUJAUAKUVFBUUTUYHUUOUUPUYIUWHUWGUYJUYKUYN
+      UUKUUL $.
+  $}
+
 
 $(
 -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
@@ -662947,19 +663606,6 @@ D Fn ( I ... ( M - 1 ) ) /\
       ( wcel wrex rspcev syl2anc ) AEFJCBDFKHIBCDEFGLM $.
   $}
 
-  ${
-    $d A x y $.  $d B y $.  $d X x $.  $d Y x y $.  $d ch x $.  $d th y $.
-    2rspcedvdw.1 $e |- ( x = A -> ( ps <-> ch ) ) $.
-    2rspcedvdw.2 $e |- ( y = B -> ( ch <-> th ) ) $.
-    2rspcedvdw.a $e |- ( ph -> A e. X ) $.
-    2rspcedvdw.b $e |- ( ph -> B e. Y ) $.
-    2rspcedvdw.3 $e |- ( ph -> th ) $.
-    $( Double application of ~ rspcedvdw .  (Contributed by SN,
-       24-Aug-2024.) $)
-    2rspcedvdw $p |- ( ph -> E. x e. X E. y e. Y ps ) $=
-      ( wcel wrex rspc2ev syl3anc ) AGIPHJPDBFJQEIQMNOBDCEFGHIJKLRS $.
-  $}
-
   ${
     $d ch x $.  $d th y $.  $d ta z $.  $d A x y z $.  $d B y z $.  $d C z $.
     $d X x $.  $d Y x y $.  $d Z x y z $.
@@ -663281,16 +663927,6 @@ D Fn ( I ... ( M - 1 ) ) /\
       VTVLVQPZFVLERZVTVLFEDWAWLVDVTVSFDPZWMWNTWJAWOVSIQVLFEDDVEVFVGVJVHVIVK $.
   $}
 
-  ${
-    elmapdd.a $e |- ( ph -> A e. V ) $.
-    elmapdd.b $e |- ( ph -> B e. W ) $.
-    elmapdd.c $e |- ( ph -> C : B --> A ) $.
-    $( Deduction associated with ~ elmapd .  (Contributed by SN,
-       29-Jul-2024.) $)
-    elmapdd $p |- ( ph -> C e. ( A ^m B ) ) $=
-      ( cmap co wcel wf elmapd mpbird ) ADBCJKLCBDMIABCDEFGHNO $.
-  $}
-
   ${
     $d B k $.  $d M k $.  $d N k $.  $d ph k $.
     fzosumm1.1 $e |- ( ph -> ( N - 1 ) e. ( ZZ>= ` M ) ) $.
@@ -663518,48 +664154,12 @@ D Fn ( I ... ( M - 1 ) ) /\
   $}
 
   ${
-    grplidd.b $e |- B = ( Base ` G ) $.
-    grplidd.p $e |- .+ = ( +g ` G ) $.
-    grplidd.u $e |- .0. = ( 0g ` G ) $.
-    grplidd.g $e |- ( ph -> G e. Grp ) $.
-    grplidd.1 $e |- ( ph -> X e. B ) $.
-    $( The identity element of a group is a left identity.  Deduction
-       associated with ~ grplid .  (Contributed by SN, 29-Jan-2025.) $)
-    sn-grplidd $p |- ( ph -> ( .0. .+ X ) = X ) $=
-      ( cgrp wcel co wceq grplid syl2anc ) ADLMEBMFECNEOJKBCDEFGHIPQ $.
-
-    $( The identity element of a group is a right identity.  Deduction
-       associated with ~ grprid .  (Contributed by SN, 29-Jan-2025.) $)
-    sn-grpridd $p |- ( ph -> ( X .+ .0. ) = X ) $=
-      ( cgrp wcel co wceq grprid syl2anc ) ADLMEBMEFCNEOJKBCDEFGHIPQ $.
-  $}
-
-  ${
-    grpassd.b $e |- B = ( Base ` G ) $.
-    grpassd.p $e |- .+ = ( +g ` G ) $.
-    grpassd.g $e |- ( ph -> G e. Grp ) $.
-    grpassd.1 $e |- ( ph -> X e. B ) $.
-    grpassd.2 $e |- ( ph -> Y e. B ) $.
-    grpassd.3 $e |- ( ph -> Z e. B ) $.
-    $( A group operation is associative.  (Contributed by SN, 29-Jan-2025.) $)
-    grpassd $p |- ( ph -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) ) $=
-      ( cgrp wcel co wceq grpass syl13anc ) ADNOEBOFBOGBOEFCPGCPEFGCPCPQJKLMBCD
-      EFGHIRS $.
-  $}
-
-  ${
-    grplinvd.b $e |- B = ( Base ` G ) $.
-    grplinvd.p $e |- .+ = ( +g ` G ) $.
-    grplinvd.u $e |- .0. = ( 0g ` G ) $.
-    grplinvd.n $e |- N = ( invg ` G ) $.
-    grplinvd.g $e |- ( ph -> G e. Grp ) $.
-    grplinvd.1 $e |- ( ph -> X e. B ) $.
-    $( The left inverse of a group element.  Deduction associated with
-       ~ grplinv .  (Contributed by SN, 29-Jan-2025.) $)
-    grplinvd $p |- ( ph -> ( ( N ` X ) .+ X ) = .0. ) $=
-      ( cgrp wcel cfv co wceq grplinv syl2anc ) ADNOFBOFEPFCQGRLMBCDEFGHIJKST
-      $.
-
+    sn-grprinvd.b $e |- B = ( Base ` G ) $.
+    sn-grprinvd.p $e |- .+ = ( +g ` G ) $.
+    sn-grprinvd.u $e |- .0. = ( 0g ` G ) $.
+    sn-grprinvd.n $e |- N = ( invg ` G ) $.
+    sn-grprinvd.g $e |- ( ph -> G e. Grp ) $.
+    sn-grprinvd.1 $e |- ( ph -> X e. B ) $.
     $( The right inverse of a group element.  Deduction associated with
        ~ grprinv .  (Contributed by SN, 29-Jan-2025.) $)
     sn-grprinvd $p |- ( ph -> ( X .+ ( N ` X ) ) = .0. ) $=
@@ -663593,12 +664193,12 @@ D Fn ( I ... ( M - 1 ) ) /\
        (case of the first element commuting with the inverse of the second
        element).  (Contributed by SN, 29-Jan-2025.) $)
     grpcominv1 $p |- ( ph -> ( X .+ ( N ` Y ) ) = ( ( N ` Y ) .+ X ) ) $=
-      ( cfv co wceq grpassd 3eqtr4rd wcel grpinvcld eqid grplinvd oveq1d eqtr2d
-      c0g sn-grplidd oveq2d grpasscan2d cgrp wb grpcld grprcan syl13anc mpbid )
-      AFGEOZCPZGCPZUPFCPZGCPZQZUQUSQZAUPFGCPZCPZFUTURAUPGCPZFCPZUPGFCPZCPFVDABC
-      DUPGFHIKABDEGHJKMUAZMLRAVFDUFOZFCPFAVEVIFCABCDEGVIHIVIUBZJKMUCUDABCDFVIHI
-      VJKLUGUEAVCVGUPCNUHSABCDUPFGHIKVHLMRABCDEFGHIJKLMUISADUJTUQBTUSBTGBTVAVBU
-      KKABCDFUPHIKLVHULABCDUPFHIKVHLULMBCDUQUSGHIUMUNUO $.
+      ( cfv co wceq grpassd 3eqtr4rd wcel grpinvcld c0g grplinvd oveq1d grplidd
+      eqid eqtr2d oveq2d grpasscan2d cgrp wb grpcld grprcan syl13anc mpbid ) AF
+      GEOZCPZGCPZUPFCPZGCPZQZUQUSQZAUPFGCPZCPZFUTURAUPGCPZFCPZUPGFCPZCPFVDABCDU
+      PGFHIKABDEGHJKMUAZMLRAVFDUBOZFCPFAVEVIFCABCDEGVIHIVIUFZJKMUCUDABCDFVIHIVJ
+      KLUEUGAVCVGUPCNUHSABCDUPFGHIKVHLMRABCDEFGHIJKLMUISADUJTUQBTUSBTGBTVAVBUKK
+      ABCDFUPHIKLVHULABCDUPFHIKVHLULMBCDUQUSGHIUMUNUO $.
 
     $( If two elements commute, then they commute with each other's inverses
        (case of the second element commuting with the inverse of the first
@@ -663628,44 +664228,6 @@ D Fn ( I ... ( M - 1 ) ) /\
       UH $.
   $}
 
-  ${
-    ringassd.b $e |- B = ( Base ` R ) $.
-    ringassd.t $e |- .x. = ( .r ` R ) $.
-    ringassd.r $e |- ( ph -> R e. Ring ) $.
-    ringassd.x $e |- ( ph -> X e. B ) $.
-    ringassd.y $e |- ( ph -> Y e. B ) $.
-    ringassd.z $e |- ( ph -> Z e. B ) $.
-    $( Associative law for multiplication in a ring.  (Contributed by SN,
-       14-Aug-2024.) $)
-    ringassd $p |- ( ph -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) ) $=
-      ( crg wcel co wceq ringass syl13anc ) ACNOEBOFBOGBOEFDPGDPEFGDPDPQJKLMBCD
-      EFGHIRS $.
-  $}
-
-  ${
-    ringlidmd.b $e |- B = ( Base ` R ) $.
-    ringlidmd.t $e |- .x. = ( .r ` R ) $.
-    ringlidmd.u $e |- .1. = ( 1r ` R ) $.
-    ringlidmd.r $e |- ( ph -> R e. Ring ) $.
-    ringlidmd.x $e |- ( ph -> X e. B ) $.
-    $( The unity element of a ring is a left multiplicative identity.
-       (Contributed by SN, 14-Aug-2024.) $)
-    ringlidmd $p |- ( ph -> ( .1. .x. X ) = X ) $=
-      ( crg wcel co wceq ringlidm syl2anc ) ACLMFBMEFDNFOJKBCDEFGHIPQ $.
-  $}
-
-  ${
-    ringridmd.b $e |- B = ( Base ` R ) $.
-    ringridmd.t $e |- .x. = ( .r ` R ) $.
-    ringridmd.u $e |- .1. = ( 1r ` R ) $.
-    ringridmd.r $e |- ( ph -> R e. Ring ) $.
-    ringridmd.x $e |- ( ph -> X e. B ) $.
-    $( The unity element of a ring is a right multiplicative identity.
-       (Contributed by SN, 14-Aug-2024.) $)
-    ringridmd $p |- ( ph -> ( X .x. .1. ) = X ) $=
-      ( crg wcel co wceq ringridm syl2anc ) ACLMFBMFEDNFOJKBCDEFGHIPQ $.
-  $}
-
   ${
     $d C a b x y $.  $d F a b $.  $d M a b $.  $d S a b $.  $d ph a b x y $.
     imacrhmcl.c $e |- C = ( N |`s ( F " S ) ) $.
@@ -663720,11 +664282,6 @@ D Fn ( I ... ( M - 1 ) ) /\
     KZCBJGHZSEABJGHZEKZBAJGHZSUIUCBCDLABELUJUMULUOUIUJUMSUEULUOSZUHABCDEMUPUFUN
     UKIZUGDENUOULUQUGHCBAUNUKMOPQRTACUDLUA $.
 
-  $( Prove isomorphic by an explicit isomorphism.  (Contributed by SN,
-     10-Jan-2025.) $)
-  brrici $p |- ( F e. ( R RingIso S ) -> R ~=r S ) $=
-    ( crs co wcel c0 wne cric wbr ne0i brric sylibr ) CABDEZFNGHABIJNCKABLM $.
-
   ${
     $d R f $.  $d S f $.
     $( Ring isomorphism is symmetric.  (Contributed by SN, 10-Jan-2025.) $)


Reviews:
user: login: avekens
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-09T18:40:52Z
user: login: avekens
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-09T18:44:11Z
user: login: avekens
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-09T18:45:08Z
user: login: avekens
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-09T18:46:23Z
user: login: avekens
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-09T18:51:41Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-09T19:13:09Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-09T19:19:18Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-09T19:19:37Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-09T19:32:10Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-09T19:36:10Z
user: login: benjub
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-09T21:58:45Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-09T22:28:51Z
user: login: benjub
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-09T23:05:38Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-09T23:41:55Z
user: login: icecream17
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-10T01:15:29Z
user: login: icecream17
state: APPROVED
author_association: CONTRIBUTOR
submitted_at: 2025-03-10T01:36:20Z
user: login: avekens
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-10T04:56:19Z
user: login: avekens
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-10T05:00:22Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-10T09:40:58Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-10T09:42:08Z
user: login: avekens
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-10T10:57:04Z
user: login: avekens
state: APPROVED
author_association: CONTRIBUTOR
submitted_at: 2025-03-10T11:00:43Z
user: login: tirix
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-10T13:03:45Z

Review comments:
diff_hunk: @@ -496903,6 +497001,32 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
 -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
 $)
 
+  ${
+    $d B e p x y $.  $d S e p x y $.  $d T e p x y $.  $d e p ph x y $.
+    urpropd.b $e |- B = ( Base ` S ) $.
+    urpropd.s $e |- ( ph -> S e. V ) $.
+    urpropd.t $e |- ( ph -> T e. W ) $.
+    urpropd.1 $e |- ( ph -> B = ( Base ` T ) ) $.
+    urpropd.2 $e |- ( ( ( ph /\ x e. B ) /\ y e. B )
+                                -> ( x ( .r ` S ) y ) = ( x ( .r ` T ) y ) ) $.
+    $( Sufficient condition for ring identities to be equal.  (Contributed by
path: set.mm
user: login: avekens
body: We wanted to call the multiplicative identities in a ring "ring unity".
created_at: 2025-03-09T18:40:51Z
author_association: CONTRIBUTOR
diff_hunk: @@ -498243,6 +498422,29 @@ then this is also a division ring (see ~ fldgensdrg ).  If the base
   $}
 
 
+$(
+-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
+  Ideals
+-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
+$)
+
+  ${
+    2idllidld.1 $e |- ( ph -> I e. ( 2Ideal ` R ) ) $.
+    $( A two-sided ideal is a left ideal.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    2idllidld $p |- ( ph -> I e. ( LIdeal ` R ) ) $=
+      ( clidl cfv coppr c2idl cin eqid 2idlval eleqtrdi elin1d ) ABEFZBGFZEFZCA
+      CBHFZNPIDBQNPONJOJPJQJKLM $.
+
+    2idllidld.o $e |- O = ( oppR ` R ) $.
+    $( A two-sided ideal is a right ideal.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    2idlridld $p |- ( ph -> I e. ( LIdeal ` O ) ) $=
+      ( clidl cfv c2idl cin eqid 2idlval eleqtrdi elin2d ) ABGHZDGHZCACBIHZOPJE
+      BQOPDOKFPKQKLMN $.
+  $}
+
+
path: set.mm
user: login: avekens
body: These theorems should be moved to main, so that I can use them, too. 
created_at: 2025-03-09T18:44:11Z
author_association: CONTRIBUTOR
diff_hunk: @@ -498243,6 +498422,29 @@ then this is also a division ring (see ~ fldgensdrg ).  If the base
   $}
 
 
+$(
+-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
+  Ideals
+-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
+$)
+
+  ${
+    2idllidld.1 $e |- ( ph -> I e. ( 2Ideal ` R ) ) $.
+    $( A two-sided ideal is a left ideal.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    2idllidld $p |- ( ph -> I e. ( LIdeal ` R ) ) $=
+      ( clidl cfv coppr c2idl cin eqid 2idlval eleqtrdi elin1d ) ABEFZBGFZEFZCA
+      CBHFZNPIDBQNPONJOJPJQJKLM $.
+
+    2idllidld.o $e |- O = ( oppR ` R ) $.
+    $( A two-sided ideal is a right ideal.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    2idlridld $p |- ( ph -> I e. ( LIdeal ` O ) ) $=
+      ( clidl cfv c2idl cin eqid 2idlval eleqtrdi elin2d ) ABGHZDGHZCACBIHZOPJE
+      BQOPDOKFPKQKLMN $.
+  $}
+
+
path: set.mm
user: login: avekens
body: ... and the following ~qusmul2, too!
created_at: 2025-03-09T18:45:07Z
author_association: CONTRIBUTOR
diff_hunk: @@ -500009,141 +500235,250 @@ The sumset operation can be used for both group (additive) operations and
     elrspunidl.1 $e |- .0. = ( 0g ` R ) $.
     elrspunidl.x $e |- .x. = ( .r ` R ) $.
     elrspunidl.r $e |- ( ph -> R e. Ring ) $.
-    elrspunidl.i $e |- ( ph -> S C_ ( LIdeal ` R ) ) $.
-    $( Elementhood to the span of a union of ideals.  (Contributed by Thierry
-       Arnoux, 30-Jun-2024.) $)
-    elrspunidl $p |- ( ph -> ( X e. ( N ` U. S ) <-> E. a e. ( B ^m S )
+    ${
+      elrspunidl.i $e |- ( ph -> S C_ ( LIdeal ` R ) ) $.
+      $( Elementhood to the span of a union of ideals.  (Contributed by Thierry
path: set.mm
user: login: avekens
body: "Elementhood in ..."
created_at: 2025-03-09T18:46:23Z
author_association: CONTRIBUTOR
diff_hunk: @@ -500881,6 +501256,228 @@ The sumset operation can be used for both group (additive) operations and
   $}
 
 
+  ${
+    $d B x y $.  $d I x y $.  $d O x y $.  $d R x y $.
+    oppreqg.o $e |- O = ( oppR ` R ) $.
+    ${
+      oppreqg.b $e |- B = ( Base ` R ) $.
+      $( Group coset equivalence relation for the opposite ring.  (Contributed
+         by Thierry Arnoux, 9-Mar-2025.) $)
+      oppreqg $p |- ( ( R e. V /\ I C_ B ) -> ( R ~QG I ) = ( O ~QG I ) ) $=
+        ( vx vy wcel wss wa cqg co cv cpr cfv eqid eqgfval cvv copab wceq coppr
+        cminusg cplusg fvexi opprbas opprneg oppradd mpan adantl eqtr4d ) BEJZC
+        AKZLBCMNZHOZIOZPAKUPBUDQZQUQBUEQZNCJLHIUAZDCMNZHIUSUOCBUREAGURRZUSRZUOR
+        SUNVAUTUBZUMDTJUNVDDBUCFUFHIUSVACDURTAABDFGUGBURDFVBUHUSBDFVCUIVARSUJUK
+        UL $.
+    $}
+
+    $d O a b i x $.  $d R a b i x $.  $d a b i ph x $.
+    oppr2idl.2 $e |- ( ph -> R e. Ring ) $.
+    $( The ideals of the opposite ring's opposite ring are the ideals of the
+       original ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
path: set.mm
user: login: avekens
body: The opposite ring's opposite ring is the original ring, isnt it?`This seems to be not proven yet. Maybe this is contained in your proof, an should be extracted.  
created_at: 2025-03-09T18:51:41Z
author_association: CONTRIBUTOR
diff_hunk: @@ -498243,6 +498422,29 @@ then this is also a division ring (see ~ fldgensdrg ).  If the base
   $}
 
 
+$(
+-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
+  Ideals
+-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
+$)
+
+  ${
+    2idllidld.1 $e |- ( ph -> I e. ( 2Ideal ` R ) ) $.
+    $( A two-sided ideal is a left ideal.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    2idllidld $p |- ( ph -> I e. ( LIdeal ` R ) ) $=
+      ( clidl cfv coppr c2idl cin eqid 2idlval eleqtrdi elin1d ) ABEFZBGFZEFZCA
+      CBHFZNPIDBQNPONJOJPJQJKLM $.
+
+    2idllidld.o $e |- O = ( oppR ` R ) $.
+    $( A two-sided ideal is a right ideal.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    2idlridld $p |- ( ph -> I e. ( LIdeal ` O ) ) $=
+      ( clidl cfv c2idl cin eqid 2idlval eleqtrdi elin2d ) ABGHZDGHZCACBIHZOPJE
+      BQOPDOKFPKQKLMN $.
+  $}
+
+
path: set.mm
user: login: tirix
body: Usually we move them to main in the same PR where we use them, but if you prefer I could move them in this PR?
created_at: 2025-03-09T19:13:09Z
author_association: CONTRIBUTOR
diff_hunk: @@ -496903,6 +497001,32 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
 -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
 $)
 
+  ${
+    $d B e p x y $.  $d S e p x y $.  $d T e p x y $.  $d e p ph x y $.
+    urpropd.b $e |- B = ( Base ` S ) $.
+    urpropd.s $e |- ( ph -> S e. V ) $.
+    urpropd.t $e |- ( ph -> T e. W ) $.
+    urpropd.1 $e |- ( ph -> B = ( Base ` T ) ) $.
+    urpropd.2 $e |- ( ( ( ph /\ x e. B ) /\ y e. B )
+                                -> ( x ( .r ` S ) y ) = ( x ( .r ` T ) y ) ) $.
+    $( Sufficient condition for ring identities to be equal.  (Contributed by
path: set.mm
user: login: tirix
body: Done in https://github.com/metamath/set.mm/pull/4702/commits/cba959dc98515e7a4ee85cdfab98c8e7eb598b54 (as in other places)
created_at: 2025-03-09T19:19:17Z
author_association: CONTRIBUTOR
diff_hunk: @@ -500009,141 +500235,250 @@ The sumset operation can be used for both group (additive) operations and
     elrspunidl.1 $e |- .0. = ( 0g ` R ) $.
     elrspunidl.x $e |- .x. = ( .r ` R ) $.
     elrspunidl.r $e |- ( ph -> R e. Ring ) $.
-    elrspunidl.i $e |- ( ph -> S C_ ( LIdeal ` R ) ) $.
-    $( Elementhood to the span of a union of ideals.  (Contributed by Thierry
-       Arnoux, 30-Jun-2024.) $)
-    elrspunidl $p |- ( ph -> ( X e. ( N ` U. S ) <-> E. a e. ( B ^m S )
+    ${
+      elrspunidl.i $e |- ( ph -> S C_ ( LIdeal ` R ) ) $.
+      $( Elementhood to the span of a union of ideals.  (Contributed by Thierry
path: set.mm
user: login: tirix
body: Done in https://github.com/metamath/set.mm/pull/4702/commits/cba959dc98515e7a4ee85cdfab98c8e7eb598b54
created_at: 2025-03-09T19:19:37Z
author_association: CONTRIBUTOR
diff_hunk: @@ -500881,6 +501256,228 @@ The sumset operation can be used for both group (additive) operations and
   $}
 
 
+  ${
+    $d B x y $.  $d I x y $.  $d O x y $.  $d R x y $.
+    oppreqg.o $e |- O = ( oppR ` R ) $.
+    ${
+      oppreqg.b $e |- B = ( Base ` R ) $.
+      $( Group coset equivalence relation for the opposite ring.  (Contributed
+         by Thierry Arnoux, 9-Mar-2025.) $)
+      oppreqg $p |- ( ( R e. V /\ I C_ B ) -> ( R ~QG I ) = ( O ~QG I ) ) $=
+        ( vx vy wcel wss wa cqg co cv cpr cfv eqid eqgfval cvv copab wceq coppr
+        cminusg cplusg fvexi opprbas opprneg oppradd mpan adantl eqtr4d ) BEJZC
+        AKZLBCMNZHOZIOZPAKUPBUDQZQUQBUEQZNCJLHIUAZDCMNZHIUSUOCBUREAGURRZUSRZUOR
+        SUNVAUTUBZUMDTJUNVDDBUCFUFHIUSVACDURTAABDFGUGBURDFVBUHUSBDFVCUIVARSUJUK
+        UL $.
+    $}
+
+    $d O a b i x $.  $d R a b i x $.  $d a b i ph x $.
+    oppr2idl.2 $e |- ( ph -> R e. Ring ) $.
+    $( The ideals of the opposite ring's opposite ring are the ideals of the
+       original ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
path: set.mm
user: login: tirix
body: Yes, but! An opposite ring is a `sSet`, so [`setsidvald`](https://us.metamath.org/mpeuni/setsidvald.html) is the applicable theorem. Its hypotheses are things like `Fun R` and ``( .r ` ndx ) e. dom R``, and for [`tpostpos2`](https://us.metamath.org/mpeuni/tpostpos2.html), we need `Rel .x. /\ Rel dom .x.`, and the usual `R e. Ring` will not be sufficient.

I made that attempt, but decided not to post this proof.
created_at: 2025-03-09T19:32:10Z
author_association: CONTRIBUTOR
diff_hunk: @@ -500881,6 +501256,228 @@ The sumset operation can be used for both group (additive) operations and
   $}
 
 
+  ${
+    $d B x y $.  $d I x y $.  $d O x y $.  $d R x y $.
+    oppreqg.o $e |- O = ( oppR ` R ) $.
+    ${
+      oppreqg.b $e |- B = ( Base ` R ) $.
+      $( Group coset equivalence relation for the opposite ring.  (Contributed
+         by Thierry Arnoux, 9-Mar-2025.) $)
+      oppreqg $p |- ( ( R e. V /\ I C_ B ) -> ( R ~QG I ) = ( O ~QG I ) ) $=
+        ( vx vy wcel wss wa cqg co cv cpr cfv eqid eqgfval cvv copab wceq coppr
+        cminusg cplusg fvexi opprbas opprneg oppradd mpan adantl eqtr4d ) BEJZC
+        AKZLBCMNZHOZIOZPAKUPBUDQZQUQBUEQZNCJLHIUAZDCMNZHIUSUOCBUREAGURRZUSRZUOR
+        SUNVAUTUBZUMDTJUNVDDBUCFUFHIUSVACDURTAABDFGUGBURDFVBUHUSBDFVCUIVARSUJUK
+        UL $.
+    $}
+
+    $d O a b i x $.  $d R a b i x $.  $d a b i ph x $.
+    oppr2idl.2 $e |- ( ph -> R e. Ring ) $.
+    $( The ideals of the opposite ring's opposite ring are the ideals of the
+       original ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
path: set.mm
user: login: tirix
body: Here is my attempt: [opprabs.mmp](https://gist.github.com/tirix/c3086f22e46ad2681f13460c73720566)

You can see the additional necessary hypotheses I could not get rid of. 

There are similar limitations with "the quotient set of the opposite ring" being "the opposite ring of the quotient set", which only works with hypotheses I don't find workable. That's I why I resorted to theorems like `qusopprmulr`, `qusopprbas`, `qusoppr1r`, which are only valid for single components.
created_at: 2025-03-09T19:36:10Z
author_association: CONTRIBUTOR
diff_hunk: @@ -500881,6 +501256,228 @@ The sumset operation can be used for both group (additive) operations and
   $}
 
 
+  ${
+    $d B x y $.  $d I x y $.  $d O x y $.  $d R x y $.
+    oppreqg.o $e |- O = ( oppR ` R ) $.
+    ${
+      oppreqg.b $e |- B = ( Base ` R ) $.
+      $( Group coset equivalence relation for the opposite ring.  (Contributed
+         by Thierry Arnoux, 9-Mar-2025.) $)
+      oppreqg $p |- ( ( R e. V /\ I C_ B ) -> ( R ~QG I ) = ( O ~QG I ) ) $=
+        ( vx vy wcel wss wa cqg co cv cpr cfv eqid eqgfval cvv copab wceq coppr
+        cminusg cplusg fvexi opprbas opprneg oppradd mpan adantl eqtr4d ) BEJZC
+        AKZLBCMNZHOZIOZPAKUPBUDQZQUQBUEQZNCJLHIUAZDCMNZHIUSUOCBUREAGURRZUSRZUOR
+        SUNVAUTUBZUMDTJUNVDDBUCFUFHIUSVACDURTAABDFGUGBURDFVBUHUSBDFVCUIVARSUJUK
+        UL $.
+    $}
+
+    $d O a b i x $.  $d R a b i x $.  $d a b i ph x $.
+    oppr2idl.2 $e |- ( ph -> R e. Ring ) $.
+    $( The ideals of the opposite ring's opposite ring are the ideals of the
+       original ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+    opprlidlabs $p |- ( ph -> ( LIdeal ` R ) = ( LIdeal ` ( oppR ` O ) ) ) $=
+      ( vi vx va vb clidl cfv cv cbs cmulr co wcel wral w3a wa eqid coppr c0 wb
+      wss wne cplusg opprmul eqtr2i a1i oveq1d eleq1d ralbidva anasss 2ralbidva
+      wceq 3anbi3d islidl opprbas oppradd 3bitr4g eqrdv ) AFBJKZCUAKZJKZAFLZBMK
+      ZUDZVEUBUEZGLZHLZBNKZOZILZBUFKZOZVEPZIVEQZHVEQGVFQZRVGVHVIVJVCNKZOZVMVNOZ
+      VEPZIVEQZHVEQGVFQZRVEVBPVEVDPAVRWDVGVHAVQWCGHVFVEAVIVFPZVJVEPZVQWCUCAWESW
+      FSZVPWBIVEWGVMVEPSZVOWAVEWHVLVTVMVNVLVTUOWHVTVJVICNKZOVLCMKZCVSWIVCVIVJWJ
+      TWITZVCTZVSTZUGVFBWIVKCVJVIVFTZVKTZDWKUGUHUIUJUKULUMUNUPGVFVNBVKVBVEHIVBT
+      WNVNTZWOUQGVFVNVCVSVDVEHIVDTVFCVCWLVFBCDWNURURVNCVCWLVNBCDWPUSUSWMUQUTVA
+      $.
+
+    $( Two sided ideal of the opposite ring.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    oppr2idl $p |- ( ph -> ( 2Ideal ` R ) = ( 2Ideal ` O ) ) $=
+      ( clidl cfv cin coppr c2idl incom opprlidlabs ineq2d eqid 2idlval 3eqtr4g
+      eqtrid ) ABFGZCFGZHZSCIGZFGZHZBJGZCJGZATSRHUCRSKARUBSABCDELMQBUDRSCRNDSNZ
+      UDNOCUESUBUAUFUANUBNUENOP $.
+
+    $d M j $.  $d O j $.  $d j ph $.
+    opprmxidl.3 $e |- ( ph -> M e. ( MaxIdeal ` R ) ) $.
+    $( The maximal ideal of the opposite ring's opposite ring.  (Contributed by
+       Thierry Arnoux, 9-Mar-2025.) $)
+    opprmxidlabs $p |- ( ph -> M e. ( MaxIdeal ` ( oppR ` O ) ) ) $=
+      ( vj cfv wcel clidl wceq cmxidl opprring eqid syl2anc wa ad2antrr opprbas
+      crg coppr cbs wne cv wo wi wral 3syl mxidlidl opprlidlabs eleqtrd mxidlnr
+      wss simplr eleqtrrd simpr mxidlmax syl22anc ralrimiva w3a ismxidl biimpar
+      ex syl13anc ) ADUAIZTJZCVEKIZJZCBUBIZUCZCHUDZUMZVKCLVKVILUEZUFZHVGUGZCVEM
+      IJZABTJZDTJVFFBDENDVEVEOZNUHACBKIZVGAVQCBMIJZCVSJFGVIBCVIOZUIPABDEFUJZUKA
+      VQVTVJFGVIBCWAULPAVNHVGAVKVGJZQZVLVMWDVLQZVQVTVKVSJVLVMAVQWCVLFRAVTWCVLGR
+      WEVKVGVSAWCVLUNAVSVGLWCVLWBRUOWDVLUPVIBVKCWAUQURVCUSVFVPVHVJVOUTVIVEHCVID
+      VEVRVIBDEWASSVAVBVD $.
+  $}
+
+  ${
+    $d B p q $.  $d I p q x y $.  $d O p q x y $.  $d Q p q x y $.  $d R p q $.
+    $d X p q $.  $d Y p q $.  $d p ph q x y $.
+    opprqus.b $e |- B = ( Base ` R ) $.
+    opprqus.o $e |- O = ( oppR ` R ) $.
+    opprqus.q $e |- Q = ( R /s ( R ~QG I ) ) $.
+    ${
+      opprqusbas.r $e |- ( ph -> R e. V ) $.
+      opprqusbas.i $e |- ( ph -> I C_ B ) $.
+      $( The base of the quotient of the opposite ring is the same as the base
+         of the opposite of the quotient ring.  (Contributed by Thierry Arnoux,
+         9-Mar-2025.) $)
+      opprqusbas $p |- ( ph -> ( Base ` ( oppR ` Q ) )
+                               = ( Base ` ( O /s ( O ~QG I ) ) ) ) $=
+        ( coppr cfv cbs cqg co wceq cvv a1i cqus opprbas cqs wss oppreqg qseq2d
+        eqid wcel syl2anc ovexd qusbas eqidd fvexi 3eqtr3d eqtr3id ) ACMNZONCON
+        ZFFEPQZUAQZONZUQCUPUPUGUQUGUBABDEPQZUCBURUCUQUTAVAURBADGUHEBUDVAURRKLBD
+        EFGIHUEUIUFAVADCBSGCDVAUAQRAJTBDONRAHTADEPUJKUKAURFUSBSSAUSULBFONRABDFI
+        HUBTAFEPUJFSUHAFDMIUMTUKUNUO $.
+    $}
+
+    opprqus1r.r $e |- ( ph -> R e. Ring ) $.
+    opprqus1r.i $e |- ( ph -> I e. ( 2Ideal ` R ) ) $.
+    ${
+      opprqusmulr.e $e |- E = ( Base ` Q ) $.
+      opprqusmulr.x $e |- ( ph -> X e. E ) $.
+      opprqusmulr.y $e |- ( ph -> Y e. E ) $.
+      $( The multiplication operation of the quotient of the opposite ring is
+         the same as the multiplication operation of the opposite of the
+         quotient ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+      opprqusmulr $p |- ( ph -> ( X ( .r ` ( oppR ` Q ) ) Y )
+                                = ( X ( .r ` ( O /s ( O ~QG I ) ) ) Y ) ) $=
+        ( cfv co wcel vp vq coppr cmulr cqg cqus eqid opprmul cv cec wa ad4antr
+        crg c2idl simplr simp-4r qusmul2 simpr simpllr oveq12d opprbas opprring
+        wceq syl oppr2idl eleqtrd clidl 2idllidld lidlss oppreqg syl2anc eceq2d
+        wss eqtrd a1i eceq1d eqtr3d 3eqtr4d cqs cbs eleqtrdi ad2antrr cvv ovexd
+        wrex qusbas eqtr3i eqtr2di elqsi r19.29a eqtrid ) AHICUCRZUDRZSIHCUDRZS
+        ZHIGGFUESZUFSZUDRZSZECWMWNWLHIOWNUGZWLUGZWMUGUHAHUAUIZDFUESZUJZVCZWOWSV
+        CZUABAXBBTZUKZXEUKZIUBUIZXCUJZVCZXFUBBXIXJBTZUKZXLUKZXKXDWNSXJXBDUDRZSZ
+        XCUJZWOWSXOBCDXPWNFXJXBLJXPUGZWTADUMTZXGXEXMXLMULAFDUNRZTXGXEXMXLNULXIX
+        MXLUOZAXGXEXMXLUPZUQXOIXKHXDWNXNXLURZXHXEXMXLUSZUTXOXBWPUJZXJWPUJZWRSXB
+        XJGUDRZSZWPUJZWSXRXOBWQGYHWRFXBXJWQUGBDGKJVAYHUGZWRUGAGUMTZXGXEXMXLAXTY
+        LMDGKVBVDULAFGUNRZTXGXEXMXLAFYAYMNADGKMVEVFULYCYBUQXOHYFIYGWRXOHXDYFYEX
+        OXCWPXBAXCWPVCZXGXEXMXLAXTFBVMZYNMAFDVGRZTYOADFNVHBFYPDJYPUGVIVDBDFGUMK
+        JVJVKULZVLVNXOIXKYGYDXOXCWPXJYQVLVNUTXOYIXCUJXRYJXOYIXQXCYIXQVCXOBDYHXP
+        GXBXJJXSKYKUHVOVPXOXCWPYIYQVLVQVRVRXIIBXCVSZTXLUBBWEXIIWLVTRZYRAIYSTXGX
+        EAIEYSQECWLXAOVAZWAWBAYSYRVCXGXEAYRCVTRZYSAXCDCBWCUMCDXCUFSVCALVOBDVTRV
+        CAJVOADFUEWDMWFEUUAYSOYTWGWHZWBVFUBBIXCWIVDWJAHYRTXEUABWEAHYSYRAHEYSPYT
+        WAUUBVFUABHXCWIVDWJWK $.
+    $}
+
+    $( The ring unity of the quotient of the opposite ring is the same as the
+       ring unity of the opposite of the quotient ring.  (Contributed by
+       Thierry Arnoux, 9-Mar-2025.) $)
+    opprqus1r $p |- ( ph -> ( 1r ` ( oppR ` Q ) ) = ( 1r ` ( O /s ( O ~QG I ) )
+       ) ) $=
+      ( vx vy coppr cfv cbs co eqid wcel wa cqg cvv fvexd ovexd clidl 2idllidld
+      cqus crg wss lidlss syl opprqusbas ad2antrr c2idl simpr opprbas eleqtrrdi
+      cv adantr adantlr opprqusmulr urpropd ) ALMCNOZPOZVCFFEUAQZUGQUBUBVDRACNU
+      CAFVEUGUDABCDEFUHGHIJAEDUEOZSEBUIADEKUFBEVFDGVFRUJUKULALURZVDSZTZMURZVDSZ
+      TBCDCPOZEFVGVJGHIADUHSVHVKJUMAEDUNOSVHVKKUMVLRZVIVGVLSVKVIVGVDVLAVHUOVLCV
+      CVCRVMUPZUQUSAVKVJVLSVHAVKTVJVDVLAVKUOVNUQUTVAVB $.
+  $}
+
+  ${
+    $d M m r u v x $.  $d M n s $.  $d M u v x y z $.  $d M w $.  $d O r s w $.
+    $d Q m r u $.  $d Q n s $.  $d Q u v x y z $.  $d Q w $.  $d R m r u v x $.
+    $d R n u x $.  $d R s u v w x $.  $d R u v x y z $.  $d X m r $.
+    $d X v y z $.  $d m ph r u $.  $d n ph s x $.  $d ph u v x y z $.
+    $d ph w $.
+    qsdrng.0 $e |- O = ( oppR ` R ) $.
+    qsdrng.q $e |- Q = ( R /s ( R ~QG M ) ) $.
+    qsdrng.r $e |- ( ph -> R e. NzRing ) $.
+    qsdrngi.1 $e |- ( ph -> M e. ( MaxIdeal ` R ) ) $.
+    qsdrngi.2 $e |- ( ph -> M e. ( MaxIdeal ` O ) ) $.
+    ${
+      qsdrngilem.1 $e |- ( ph -> X e. ( Base ` R ) ) $.
+      qsdrngilem.2 $e |- ( ph -> -. X e. M ) $.
+      $( Lemma for ~ qsdrngi .  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+      qsdrngilem $p |- ( ph -> E. v e. ( Base ` Q )
+                           ( v ( .r ` Q ) [ X ] ( R ~QG M ) ) = ( 1r ` Q ) ) $=
+        ( cfv co wceq wcel eqid ad3antrrr vr vm cur cv cmulr cplusg cqg cec cbs
+        wrex wa cqs simpllr ovex ecelqsi syl cvv cnzr cqus ovexd qusbas eleqtrd
+        a1i wb oveq1 eqeq1d adantl crg nzrring c2idl clidl cin mxidlidl syl2anc
+        opprring elind 2idlval eleqtrrdi qusmul2 wer cnsg csubg lidlnsg nsgsubg
+        cmxidl eqger 3syl wss cminusg lidlss ringcld ringidcl simpr oveq2d cgrp
+        wbr c0g ringgrpd oveq1d grpinvcld simplr sseldd grpassd grplidd 3eqtr3d
+        grplinvd eqtrd eqeltrd w3a eqgval biimpar syl23anc qus1 simprd rspcedvd
+        erthi csn cun crsp snssd unssd rspcl rspssid unssad unssbd snssg eldifd
+        mxidlmaxv eleqtrrd elrspunsn mpbid r19.29vva ) ADUCOZUAUDZGDUEOZPZUBUDZ
+        DUFOZPZQZBUDZGDEUGPZUHZCUEOZPZCUCOZQZBCUIOZUJUAUBDUIOZEAYNUUIRZUKZYQERZ
+        UKZYTUKZUUGYNUUBUHZUUCUUDPZUUFQZBUUOUUHUUNUUOUUIUUBULZUUHUUNUUJUUOUURRA
+        UUJUULYTUMZUUIYNUUBDEUGUNUOUPAUURUUHQUUJUULYTAUUBDCUUIUQURCDUUBUSPQAIVC
+        UUIUUIQAUUISZVCADEUGUTJVATVBUUAUUOQZUUGUUQVDUUNUVAUUEUUPUUFUUAUUOUUCUUD
+        VEVFVGUUNUUPYMUUBUHZUUFUUNUUPYPUUBUHUVBUUNUUICDYOUUDEYNGIUUTYOSZUUDSADV
+        HRZUUJUULYTADURRUVDJDVIUPZTZAEDVJOZRZUUJUULYTAEDVKOZFVKOZVLUVGAUVIUVJEA
+        UVDEDWEOREUVIRZUVEKUUIDEUUTVMVNZAFVHRZEFWEOREUVJRAUVDUVMUVEDFHVOUPLFUIO
+        ZFEUVNSVMVNVPDUVGUVIUVJFUVISZHUVJSUVGSZVQVRTZUUSAGUUIRZUUJUULYTMTZVSUUN
+        YPYMUUBUUIAUUIUUBVTZUUJUULYTAEDWAORZEDWBORUVTAUVDUVKUWAUVEUVLDEWCVNEDWD
+        UUBDUUIEUUTUUBSZWFWGTUUNUVDEUUIWHZYPUUIRZYMUUIRZYPDWIOZOZYMYRPZERZYPYMU
+        UBWPZUVFAUWCUUJUULYTAUVKUWCUVLUUIEUVIDUUTUVOWJUPZTZUUNUUIDYOYNGUUTUVCUV
+        FUUSUVSWKZAUWEUUJUULYTAUVDUWEUVEUUIDYMUUTYMSZWLUPZTUUNUWHYQEUUNUWHUWGYS
+        YRPZYQUUNYMYSUWGYRUUMYTWMWNUUNUWGYPYRPZYQYRPDWQOZYQYRPUWPYQUUNUWQUWRYQY
+        RUUNUUIYRDUWFYPUWRUUTYRSZUWRSZUWFSZADWORUUJUULYTADUVEWRTZUWMXFWSUUNUUIY
+        RDUWGYPYQUUTUWSUXBUUNUUIDUWFYPUUTUXAUXBUWMWTUWMUUNEUUIYQUWLUUKUULYTXAZX
+        BZXCUUNUUIYRDYQUWRUUTUWSUWTUXBUXDXDXEXGUXCXHUVDUWCUKUWJUWDUWEUWIXIYPYMY
+        RUUBEDUWFVHUUIUUTUXAUWSUWBXJXKXLXPXGUUNUVDUVHUVBUUFQZUVFUVQUVDUVHUKCVHR
+        UXEDECYMUVGIUVPUWNXMXNVNXGXOAYMEGXQZXRZDXSOZOZRYTUBEUJUAUUIUJAYMUUIUXIU
+        WOAUUIDUXIEGUUTUVEKAUVDUXGUUIWHZUXIUVIRUVEAEUXFUUIUWKAGUUIMXTYAZUUIDUVI
+        UXGUXHUXHSZUUTUVOYBVNAEUXFUXIAUVDUXJUXGUXIWHUVEUXKUUIDUXGUXHUXLUUTYCVNZ
+        YDAGUXIEAUVRUXFUXIWHZGUXIRZMAEUXFUXIUXMYEUVRUXOUXNGUXIUUIYFXKVNNYGYHYIA
+        YMUUIYRDYOUBEUXHGUWRUAUXLUUTUWTUVCUVEUWSUVLAGUUIEMNYGYJYKYL $.
+    $}
+
+    $( A quotient by a maximal ideal is an division ring.  (Contributed by
path: set.mm
user: login: benjub
body: ```suggestion
    $( A quotient by a maximal left and maximal right ideal is a division ring.  (Contributed by
```
A maximal ideal need not be maximal among all proper left ideals.
created_at: 2025-03-09T21:58:44Z
author_association: CONTRIBUTOR
diff_hunk: @@ -500881,6 +501256,228 @@ The sumset operation can be used for both group (additive) operations and
   $}
 
 
+  ${
+    $d B x y $.  $d I x y $.  $d O x y $.  $d R x y $.
+    oppreqg.o $e |- O = ( oppR ` R ) $.
+    ${
+      oppreqg.b $e |- B = ( Base ` R ) $.
+      $( Group coset equivalence relation for the opposite ring.  (Contributed
+         by Thierry Arnoux, 9-Mar-2025.) $)
+      oppreqg $p |- ( ( R e. V /\ I C_ B ) -> ( R ~QG I ) = ( O ~QG I ) ) $=
+        ( vx vy wcel wss wa cqg co cv cpr cfv eqid eqgfval cvv copab wceq coppr
+        cminusg cplusg fvexi opprbas opprneg oppradd mpan adantl eqtr4d ) BEJZC
+        AKZLBCMNZHOZIOZPAKUPBUDQZQUQBUEQZNCJLHIUAZDCMNZHIUSUOCBUREAGURRZUSRZUOR
+        SUNVAUTUBZUMDTJUNVDDBUCFUFHIUSVACDURTAABDFGUGBURDFVBUHUSBDFVCUIVARSUJUK
+        UL $.
+    $}
+
+    $d O a b i x $.  $d R a b i x $.  $d a b i ph x $.
+    oppr2idl.2 $e |- ( ph -> R e. Ring ) $.
+    $( The ideals of the opposite ring's opposite ring are the ideals of the
+       original ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+    opprlidlabs $p |- ( ph -> ( LIdeal ` R ) = ( LIdeal ` ( oppR ` O ) ) ) $=
+      ( vi vx va vb clidl cfv cv cbs cmulr co wcel wral w3a wa eqid coppr c0 wb
+      wss wne cplusg opprmul eqtr2i a1i oveq1d eleq1d ralbidva anasss 2ralbidva
+      wceq 3anbi3d islidl opprbas oppradd 3bitr4g eqrdv ) AFBJKZCUAKZJKZAFLZBMK
+      ZUDZVEUBUEZGLZHLZBNKZOZILZBUFKZOZVEPZIVEQZHVEQGVFQZRVGVHVIVJVCNKZOZVMVNOZ
+      VEPZIVEQZHVEQGVFQZRVEVBPVEVDPAVRWDVGVHAVQWCGHVFVEAVIVFPZVJVEPZVQWCUCAWESW
+      FSZVPWBIVEWGVMVEPSZVOWAVEWHVLVTVMVNVLVTUOWHVTVJVICNKZOVLCMKZCVSWIVCVIVJWJ
+      TWITZVCTZVSTZUGVFBWIVKCVJVIVFTZVKTZDWKUGUHUIUJUKULUMUNUPGVFVNBVKVBVEHIVBT
+      WNVNTZWOUQGVFVNVCVSVDVEHIVDTVFCVCWLVFBCDWNURURVNCVCWLVNBCDWPUSUSWMUQUTVA
+      $.
+
+    $( Two sided ideal of the opposite ring.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    oppr2idl $p |- ( ph -> ( 2Ideal ` R ) = ( 2Ideal ` O ) ) $=
+      ( clidl cfv cin coppr c2idl incom opprlidlabs ineq2d eqid 2idlval 3eqtr4g
+      eqtrid ) ABFGZCFGZHZSCIGZFGZHZBJGZCJGZATSRHUCRSKARUBSABCDELMQBUDRSCRNDSNZ
+      UDNOCUESUBUAUFUANUBNUENOP $.
+
+    $d M j $.  $d O j $.  $d j ph $.
+    opprmxidl.3 $e |- ( ph -> M e. ( MaxIdeal ` R ) ) $.
+    $( The maximal ideal of the opposite ring's opposite ring.  (Contributed by
+       Thierry Arnoux, 9-Mar-2025.) $)
+    opprmxidlabs $p |- ( ph -> M e. ( MaxIdeal ` ( oppR ` O ) ) ) $=
+      ( vj cfv wcel clidl wceq cmxidl opprring eqid syl2anc wa ad2antrr opprbas
+      crg coppr cbs wne cv wo wi wral 3syl mxidlidl opprlidlabs eleqtrd mxidlnr
+      wss simplr eleqtrrd simpr mxidlmax syl22anc ralrimiva w3a ismxidl biimpar
+      ex syl13anc ) ADUAIZTJZCVEKIZJZCBUBIZUCZCHUDZUMZVKCLVKVILUEZUFZHVGUGZCVEM
+      IJZABTJZDTJVFFBDENDVEVEOZNUHACBKIZVGAVQCBMIJZCVSJFGVIBCVIOZUIPABDEFUJZUKA
+      VQVTVJFGVIBCWAULPAVNHVGAVKVGJZQZVLVMWDVLQZVQVTVKVSJVLVMAVQWCVLFRAVTWCVLGR
+      WEVKVGVSAWCVLUNAVSVGLWCVLWBRUOWDVLUPVIBVKCWAUQURVCUSVFVPVHVJVOUTVIVEHCVID
+      VEVRVIBDEWASSVAVBVD $.
+  $}
+
+  ${
+    $d B p q $.  $d I p q x y $.  $d O p q x y $.  $d Q p q x y $.  $d R p q $.
+    $d X p q $.  $d Y p q $.  $d p ph q x y $.
+    opprqus.b $e |- B = ( Base ` R ) $.
+    opprqus.o $e |- O = ( oppR ` R ) $.
+    opprqus.q $e |- Q = ( R /s ( R ~QG I ) ) $.
+    ${
+      opprqusbas.r $e |- ( ph -> R e. V ) $.
+      opprqusbas.i $e |- ( ph -> I C_ B ) $.
+      $( The base of the quotient of the opposite ring is the same as the base
+         of the opposite of the quotient ring.  (Contributed by Thierry Arnoux,
+         9-Mar-2025.) $)
+      opprqusbas $p |- ( ph -> ( Base ` ( oppR ` Q ) )
+                               = ( Base ` ( O /s ( O ~QG I ) ) ) ) $=
+        ( coppr cfv cbs cqg co wceq cvv a1i cqus opprbas cqs wss oppreqg qseq2d
+        eqid wcel syl2anc ovexd qusbas eqidd fvexi 3eqtr3d eqtr3id ) ACMNZONCON
+        ZFFEPQZUAQZONZUQCUPUPUGUQUGUBABDEPQZUCBURUCUQUTAVAURBADGUHEBUDVAURRKLBD
+        EFGIHUEUIUFAVADCBSGCDVAUAQRAJTBDONRAHTADEPUJKUKAURFUSBSSAUSULBFONRABDFI
+        HUBTAFEPUJFSUHAFDMIUMTUKUNUO $.
+    $}
+
+    opprqus1r.r $e |- ( ph -> R e. Ring ) $.
+    opprqus1r.i $e |- ( ph -> I e. ( 2Ideal ` R ) ) $.
+    ${
+      opprqusmulr.e $e |- E = ( Base ` Q ) $.
+      opprqusmulr.x $e |- ( ph -> X e. E ) $.
+      opprqusmulr.y $e |- ( ph -> Y e. E ) $.
+      $( The multiplication operation of the quotient of the opposite ring is
+         the same as the multiplication operation of the opposite of the
+         quotient ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+      opprqusmulr $p |- ( ph -> ( X ( .r ` ( oppR ` Q ) ) Y )
+                                = ( X ( .r ` ( O /s ( O ~QG I ) ) ) Y ) ) $=
+        ( cfv co wcel vp vq coppr cmulr cqg cqus eqid opprmul cv cec wa ad4antr
+        crg c2idl simplr simp-4r qusmul2 simpr simpllr oveq12d opprbas opprring
+        wceq syl oppr2idl eleqtrd clidl 2idllidld lidlss oppreqg syl2anc eceq2d
+        wss eqtrd a1i eceq1d eqtr3d 3eqtr4d cqs cbs eleqtrdi ad2antrr cvv ovexd
+        wrex qusbas eqtr3i eqtr2di elqsi r19.29a eqtrid ) AHICUCRZUDRZSIHCUDRZS
+        ZHIGGFUESZUFSZUDRZSZECWMWNWLHIOWNUGZWLUGZWMUGUHAHUAUIZDFUESZUJZVCZWOWSV
+        CZUABAXBBTZUKZXEUKZIUBUIZXCUJZVCZXFUBBXIXJBTZUKZXLUKZXKXDWNSXJXBDUDRZSZ
+        XCUJZWOWSXOBCDXPWNFXJXBLJXPUGZWTADUMTZXGXEXMXLMULAFDUNRZTXGXEXMXLNULXIX
+        MXLUOZAXGXEXMXLUPZUQXOIXKHXDWNXNXLURZXHXEXMXLUSZUTXOXBWPUJZXJWPUJZWRSXB
+        XJGUDRZSZWPUJZWSXRXOBWQGYHWRFXBXJWQUGBDGKJVAYHUGZWRUGAGUMTZXGXEXMXLAXTY
+        LMDGKVBVDULAFGUNRZTXGXEXMXLAFYAYMNADGKMVEVFULYCYBUQXOHYFIYGWRXOHXDYFYEX
+        OXCWPXBAXCWPVCZXGXEXMXLAXTFBVMZYNMAFDVGRZTYOADFNVHBFYPDJYPUGVIVDBDFGUMK
+        JVJVKULZVLVNXOIXKYGYDXOXCWPXJYQVLVNUTXOYIXCUJXRYJXOYIXQXCYIXQVCXOBDYHXP
+        GXBXJJXSKYKUHVOVPXOXCWPYIYQVLVQVRVRXIIBXCVSZTXLUBBWEXIIWLVTRZYRAIYSTXGX
+        EAIEYSQECWLXAOVAZWAWBAYSYRVCXGXEAYRCVTRZYSAXCDCBWCUMCDXCUFSVCALVOBDVTRV
+        CAJVOADFUEWDMWFEUUAYSOYTWGWHZWBVFUBBIXCWIVDWJAHYRTXEUABWEAHYSYRAHEYSPYT
+        WAUUBVFUABHXCWIVDWJWK $.
+    $}
+
+    $( The ring unity of the quotient of the opposite ring is the same as the
+       ring unity of the opposite of the quotient ring.  (Contributed by
+       Thierry Arnoux, 9-Mar-2025.) $)
+    opprqus1r $p |- ( ph -> ( 1r ` ( oppR ` Q ) ) = ( 1r ` ( O /s ( O ~QG I ) )
+       ) ) $=
+      ( vx vy coppr cfv cbs co eqid wcel wa cqg cvv fvexd ovexd clidl 2idllidld
+      cqus crg wss lidlss syl opprqusbas ad2antrr c2idl simpr opprbas eleqtrrdi
+      cv adantr adantlr opprqusmulr urpropd ) ALMCNOZPOZVCFFEUAQZUGQUBUBVDRACNU
+      CAFVEUGUDABCDEFUHGHIJAEDUEOZSEBUIADEKUFBEVFDGVFRUJUKULALURZVDSZTZMURZVDSZ
+      TBCDCPOZEFVGVJGHIADUHSVHVKJUMAEDUNOSVHVKKUMVLRZVIVGVLSVKVIVGVDVLAVHUOVLCV
+      CVCRVMUPZUQUSAVKVJVLSVHAVKTVJVDVLAVKUOVNUQUTVAVB $.
+  $}
+
+  ${
+    $d M m r u v x $.  $d M n s $.  $d M u v x y z $.  $d M w $.  $d O r s w $.
+    $d Q m r u $.  $d Q n s $.  $d Q u v x y z $.  $d Q w $.  $d R m r u v x $.
+    $d R n u x $.  $d R s u v w x $.  $d R u v x y z $.  $d X m r $.
+    $d X v y z $.  $d m ph r u $.  $d n ph s x $.  $d ph u v x y z $.
+    $d ph w $.
+    qsdrng.0 $e |- O = ( oppR ` R ) $.
+    qsdrng.q $e |- Q = ( R /s ( R ~QG M ) ) $.
+    qsdrng.r $e |- ( ph -> R e. NzRing ) $.
+    qsdrngi.1 $e |- ( ph -> M e. ( MaxIdeal ` R ) ) $.
+    qsdrngi.2 $e |- ( ph -> M e. ( MaxIdeal ` O ) ) $.
+    ${
+      qsdrngilem.1 $e |- ( ph -> X e. ( Base ` R ) ) $.
+      qsdrngilem.2 $e |- ( ph -> -. X e. M ) $.
+      $( Lemma for ~ qsdrngi .  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+      qsdrngilem $p |- ( ph -> E. v e. ( Base ` Q )
+                           ( v ( .r ` Q ) [ X ] ( R ~QG M ) ) = ( 1r ` Q ) ) $=
+        ( cfv co wceq wcel eqid ad3antrrr vr vm cur cv cmulr cplusg cqg cec cbs
+        wrex wa cqs simpllr ovex ecelqsi syl cvv cnzr cqus ovexd qusbas eleqtrd
+        a1i wb oveq1 eqeq1d adantl crg nzrring c2idl clidl cin mxidlidl syl2anc
+        opprring elind 2idlval eleqtrrdi qusmul2 wer cnsg csubg lidlnsg nsgsubg
+        cmxidl eqger 3syl wss cminusg lidlss ringcld ringidcl simpr oveq2d cgrp
+        wbr c0g ringgrpd oveq1d grpinvcld simplr sseldd grpassd grplidd 3eqtr3d
+        grplinvd eqtrd eqeltrd w3a eqgval biimpar syl23anc qus1 simprd rspcedvd
+        erthi csn cun crsp snssd unssd rspcl rspssid unssad unssbd snssg eldifd
+        mxidlmaxv eleqtrrd elrspunsn mpbid r19.29vva ) ADUCOZUAUDZGDUEOZPZUBUDZ
+        DUFOZPZQZBUDZGDEUGPZUHZCUEOZPZCUCOZQZBCUIOZUJUAUBDUIOZEAYNUUIRZUKZYQERZ
+        UKZYTUKZUUGYNUUBUHZUUCUUDPZUUFQZBUUOUUHUUNUUOUUIUUBULZUUHUUNUUJUUOUURRA
+        UUJUULYTUMZUUIYNUUBDEUGUNUOUPAUURUUHQUUJUULYTAUUBDCUUIUQURCDUUBUSPQAIVC
+        UUIUUIQAUUISZVCADEUGUTJVATVBUUAUUOQZUUGUUQVDUUNUVAUUEUUPUUFUUAUUOUUCUUD
+        VEVFVGUUNUUPYMUUBUHZUUFUUNUUPYPUUBUHUVBUUNUUICDYOUUDEYNGIUUTYOSZUUDSADV
+        HRZUUJUULYTADURRUVDJDVIUPZTZAEDVJOZRZUUJUULYTAEDVKOZFVKOZVLUVGAUVIUVJEA
+        UVDEDWEOREUVIRZUVEKUUIDEUUTVMVNZAFVHRZEFWEOREUVJRAUVDUVMUVEDFHVOUPLFUIO
+        ZFEUVNSVMVNVPDUVGUVIUVJFUVISZHUVJSUVGSZVQVRTZUUSAGUUIRZUUJUULYTMTZVSUUN
+        YPYMUUBUUIAUUIUUBVTZUUJUULYTAEDWAORZEDWBORUVTAUVDUVKUWAUVEUVLDEWCVNEDWD
+        UUBDUUIEUUTUUBSZWFWGTUUNUVDEUUIWHZYPUUIRZYMUUIRZYPDWIOZOZYMYRPZERZYPYMU
+        UBWPZUVFAUWCUUJUULYTAUVKUWCUVLUUIEUVIDUUTUVOWJUPZTZUUNUUIDYOYNGUUTUVCUV
+        FUUSUVSWKZAUWEUUJUULYTAUVDUWEUVEUUIDYMUUTYMSZWLUPZTUUNUWHYQEUUNUWHUWGYS
+        YRPZYQUUNYMYSUWGYRUUMYTWMWNUUNUWGYPYRPZYQYRPDWQOZYQYRPUWPYQUUNUWQUWRYQY
+        RUUNUUIYRDUWFYPUWRUUTYRSZUWRSZUWFSZADWORUUJUULYTADUVEWRTZUWMXFWSUUNUUIY
+        RDUWGYPYQUUTUWSUXBUUNUUIDUWFYPUUTUXAUXBUWMWTUWMUUNEUUIYQUWLUUKUULYTXAZX
+        BZXCUUNUUIYRDYQUWRUUTUWSUWTUXBUXDXDXEXGUXCXHUVDUWCUKUWJUWDUWEUWIXIYPYMY
+        RUUBEDUWFVHUUIUUTUXAUWSUWBXJXKXLXPXGUUNUVDUVHUVBUUFQZUVFUVQUVDUVHUKCVHR
+        UXEDECYMUVGIUVPUWNXMXNVNXGXOAYMEGXQZXRZDXSOZOZRYTUBEUJUAUUIUJAYMUUIUXIU
+        WOAUUIDUXIEGUUTUVEKAUVDUXGUUIWHZUXIUVIRUVEAEUXFUUIUWKAGUUIMXTYAZUUIDUVI
+        UXGUXHUXHSZUUTUVOYBVNAEUXFUXIAUVDUXJUXGUXIWHUVEUXKUUIDUXGUXHUXLUUTYCVNZ
+        YDAGUXIEAUVRUXFUXIWHZGUXIRZMAEUXFUXIUXMYEUVRUXOUXNGUXIUUIYFXKVNNYGYHYIA
+        YMUUIYRDYOUBEUXHGUWRUAUXLUUTUWTUVCUVEUWSUVLAGUUIEMNYGYJYKYL $.
+    $}
+
+    $( A quotient by a maximal ideal is an division ring.  (Contributed by
path: set.mm
user: login: tirix
body: Fixed in https://github.com/metamath/set.mm/pull/4702/commits/2b407945f95fcbc3bb20aefeeb27272ffc70c821
created_at: 2025-03-09T22:28:50Z
author_association: CONTRIBUTOR
diff_hunk: @@ -500881,6 +501256,228 @@ The sumset operation can be used for both group (additive) operations and
   $}
 
 
+  ${
+    $d B x y $.  $d I x y $.  $d O x y $.  $d R x y $.
+    oppreqg.o $e |- O = ( oppR ` R ) $.
+    ${
+      oppreqg.b $e |- B = ( Base ` R ) $.
+      $( Group coset equivalence relation for the opposite ring.  (Contributed
+         by Thierry Arnoux, 9-Mar-2025.) $)
+      oppreqg $p |- ( ( R e. V /\ I C_ B ) -> ( R ~QG I ) = ( O ~QG I ) ) $=
+        ( vx vy wcel wss wa cqg co cv cpr cfv eqid eqgfval cvv copab wceq coppr
+        cminusg cplusg fvexi opprbas opprneg oppradd mpan adantl eqtr4d ) BEJZC
+        AKZLBCMNZHOZIOZPAKUPBUDQZQUQBUEQZNCJLHIUAZDCMNZHIUSUOCBUREAGURRZUSRZUOR
+        SUNVAUTUBZUMDTJUNVDDBUCFUFHIUSVACDURTAABDFGUGBURDFVBUHUSBDFVCUIVARSUJUK
+        UL $.
+    $}
+
+    $d O a b i x $.  $d R a b i x $.  $d a b i ph x $.
+    oppr2idl.2 $e |- ( ph -> R e. Ring ) $.
+    $( The ideals of the opposite ring's opposite ring are the ideals of the
+       original ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+    opprlidlabs $p |- ( ph -> ( LIdeal ` R ) = ( LIdeal ` ( oppR ` O ) ) ) $=
+      ( vi vx va vb clidl cfv cv cbs cmulr co wcel wral w3a wa eqid coppr c0 wb
+      wss wne cplusg opprmul eqtr2i a1i oveq1d eleq1d ralbidva anasss 2ralbidva
+      wceq 3anbi3d islidl opprbas oppradd 3bitr4g eqrdv ) AFBJKZCUAKZJKZAFLZBMK
+      ZUDZVEUBUEZGLZHLZBNKZOZILZBUFKZOZVEPZIVEQZHVEQGVFQZRVGVHVIVJVCNKZOZVMVNOZ
+      VEPZIVEQZHVEQGVFQZRVEVBPVEVDPAVRWDVGVHAVQWCGHVFVEAVIVFPZVJVEPZVQWCUCAWESW
+      FSZVPWBIVEWGVMVEPSZVOWAVEWHVLVTVMVNVLVTUOWHVTVJVICNKZOVLCMKZCVSWIVCVIVJWJ
+      TWITZVCTZVSTZUGVFBWIVKCVJVIVFTZVKTZDWKUGUHUIUJUKULUMUNUPGVFVNBVKVBVEHIVBT
+      WNVNTZWOUQGVFVNVCVSVDVEHIVDTVFCVCWLVFBCDWNURURVNCVCWLVNBCDWPUSUSWMUQUTVA
+      $.
+
+    $( Two sided ideal of the opposite ring.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    oppr2idl $p |- ( ph -> ( 2Ideal ` R ) = ( 2Ideal ` O ) ) $=
+      ( clidl cfv cin coppr c2idl incom opprlidlabs ineq2d eqid 2idlval 3eqtr4g
+      eqtrid ) ABFGZCFGZHZSCIGZFGZHZBJGZCJGZATSRHUCRSKARUBSABCDELMQBUDRSCRNDSNZ
+      UDNOCUESUBUAUFUANUBNUENOP $.
+
+    $d M j $.  $d O j $.  $d j ph $.
+    opprmxidl.3 $e |- ( ph -> M e. ( MaxIdeal ` R ) ) $.
+    $( The maximal ideal of the opposite ring's opposite ring.  (Contributed by
+       Thierry Arnoux, 9-Mar-2025.) $)
+    opprmxidlabs $p |- ( ph -> M e. ( MaxIdeal ` ( oppR ` O ) ) ) $=
+      ( vj cfv wcel clidl wceq cmxidl opprring eqid syl2anc wa ad2antrr opprbas
+      crg coppr cbs wne cv wo wi wral 3syl mxidlidl opprlidlabs eleqtrd mxidlnr
+      wss simplr eleqtrrd simpr mxidlmax syl22anc ralrimiva w3a ismxidl biimpar
+      ex syl13anc ) ADUAIZTJZCVEKIZJZCBUBIZUCZCHUDZUMZVKCLVKVILUEZUFZHVGUGZCVEM
+      IJZABTJZDTJVFFBDENDVEVEOZNUHACBKIZVGAVQCBMIJZCVSJFGVIBCVIOZUIPABDEFUJZUKA
+      VQVTVJFGVIBCWAULPAVNHVGAVKVGJZQZVLVMWDVLQZVQVTVKVSJVLVMAVQWCVLFRAVTWCVLGR
+      WEVKVGVSAWCVLUNAVSVGLWCVLWBRUOWDVLUPVIBVKCWAUQURVCUSVFVPVHVJVOUTVIVEHCVID
+      VEVRVIBDEWASSVAVBVD $.
+  $}
+
+  ${
+    $d B p q $.  $d I p q x y $.  $d O p q x y $.  $d Q p q x y $.  $d R p q $.
+    $d X p q $.  $d Y p q $.  $d p ph q x y $.
+    opprqus.b $e |- B = ( Base ` R ) $.
+    opprqus.o $e |- O = ( oppR ` R ) $.
+    opprqus.q $e |- Q = ( R /s ( R ~QG I ) ) $.
+    ${
+      opprqusbas.r $e |- ( ph -> R e. V ) $.
+      opprqusbas.i $e |- ( ph -> I C_ B ) $.
+      $( The base of the quotient of the opposite ring is the same as the base
+         of the opposite of the quotient ring.  (Contributed by Thierry Arnoux,
+         9-Mar-2025.) $)
+      opprqusbas $p |- ( ph -> ( Base ` ( oppR ` Q ) )
+                               = ( Base ` ( O /s ( O ~QG I ) ) ) ) $=
+        ( coppr cfv cbs cqg co wceq cvv a1i cqus opprbas cqs wss oppreqg qseq2d
+        eqid wcel syl2anc ovexd qusbas eqidd fvexi 3eqtr3d eqtr3id ) ACMNZONCON
+        ZFFEPQZUAQZONZUQCUPUPUGUQUGUBABDEPQZUCBURUCUQUTAVAURBADGUHEBUDVAURRKLBD
+        EFGIHUEUIUFAVADCBSGCDVAUAQRAJTBDONRAHTADEPUJKUKAURFUSBSSAUSULBFONRABDFI
+        HUBTAFEPUJFSUHAFDMIUMTUKUNUO $.
+    $}
+
+    opprqus1r.r $e |- ( ph -> R e. Ring ) $.
+    opprqus1r.i $e |- ( ph -> I e. ( 2Ideal ` R ) ) $.
+    ${
+      opprqusmulr.e $e |- E = ( Base ` Q ) $.
+      opprqusmulr.x $e |- ( ph -> X e. E ) $.
+      opprqusmulr.y $e |- ( ph -> Y e. E ) $.
+      $( The multiplication operation of the quotient of the opposite ring is
+         the same as the multiplication operation of the opposite of the
+         quotient ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+      opprqusmulr $p |- ( ph -> ( X ( .r ` ( oppR ` Q ) ) Y )
+                                = ( X ( .r ` ( O /s ( O ~QG I ) ) ) Y ) ) $=
+        ( cfv co wcel vp vq coppr cmulr cqg cqus eqid opprmul cv cec wa ad4antr
+        crg c2idl simplr simp-4r qusmul2 simpr simpllr oveq12d opprbas opprring
+        wceq syl oppr2idl eleqtrd clidl 2idllidld lidlss oppreqg syl2anc eceq2d
+        wss eqtrd a1i eceq1d eqtr3d 3eqtr4d cqs cbs eleqtrdi ad2antrr cvv ovexd
+        wrex qusbas eqtr3i eqtr2di elqsi r19.29a eqtrid ) AHICUCRZUDRZSIHCUDRZS
+        ZHIGGFUESZUFSZUDRZSZECWMWNWLHIOWNUGZWLUGZWMUGUHAHUAUIZDFUESZUJZVCZWOWSV
+        CZUABAXBBTZUKZXEUKZIUBUIZXCUJZVCZXFUBBXIXJBTZUKZXLUKZXKXDWNSXJXBDUDRZSZ
+        XCUJZWOWSXOBCDXPWNFXJXBLJXPUGZWTADUMTZXGXEXMXLMULAFDUNRZTXGXEXMXLNULXIX
+        MXLUOZAXGXEXMXLUPZUQXOIXKHXDWNXNXLURZXHXEXMXLUSZUTXOXBWPUJZXJWPUJZWRSXB
+        XJGUDRZSZWPUJZWSXRXOBWQGYHWRFXBXJWQUGBDGKJVAYHUGZWRUGAGUMTZXGXEXMXLAXTY
+        LMDGKVBVDULAFGUNRZTXGXEXMXLAFYAYMNADGKMVEVFULYCYBUQXOHYFIYGWRXOHXDYFYEX
+        OXCWPXBAXCWPVCZXGXEXMXLAXTFBVMZYNMAFDVGRZTYOADFNVHBFYPDJYPUGVIVDBDFGUMK
+        JVJVKULZVLVNXOIXKYGYDXOXCWPXJYQVLVNUTXOYIXCUJXRYJXOYIXQXCYIXQVCXOBDYHXP
+        GXBXJJXSKYKUHVOVPXOXCWPYIYQVLVQVRVRXIIBXCVSZTXLUBBWEXIIWLVTRZYRAIYSTXGX
+        EAIEYSQECWLXAOVAZWAWBAYSYRVCXGXEAYRCVTRZYSAXCDCBWCUMCDXCUFSVCALVOBDVTRV
+        CAJVOADFUEWDMWFEUUAYSOYTWGWHZWBVFUBBIXCWIVDWJAHYRTXEUABWEAHYSYRAHEYSPYT
+        WAUUBVFUABHXCWIVDWJWK $.
+    $}
+
+    $( The ring unity of the quotient of the opposite ring is the same as the
+       ring unity of the opposite of the quotient ring.  (Contributed by
+       Thierry Arnoux, 9-Mar-2025.) $)
+    opprqus1r $p |- ( ph -> ( 1r ` ( oppR ` Q ) ) = ( 1r ` ( O /s ( O ~QG I ) )
+       ) ) $=
+      ( vx vy coppr cfv cbs co eqid wcel wa cqg cvv fvexd ovexd clidl 2idllidld
+      cqus crg wss lidlss syl opprqusbas ad2antrr c2idl simpr opprbas eleqtrrdi
+      cv adantr adantlr opprqusmulr urpropd ) ALMCNOZPOZVCFFEUAQZUGQUBUBVDRACNU
+      CAFVEUGUDABCDEFUHGHIJAEDUEOZSEBUIADEKUFBEVFDGVFRUJUKULALURZVDSZTZMURZVDSZ
+      TBCDCPOZEFVGVJGHIADUHSVHVKJUMAEDUNOSVHVKKUMVLRZVIVGVLSVKVIVGVDVLAVHUOVLCV
+      CVCRVMUPZUQUSAVKVJVLSVHAVKTVJVDVLAVKUOVNUQUTVAVB $.
+  $}
+
+  ${
+    $d M m r u v x $.  $d M n s $.  $d M u v x y z $.  $d M w $.  $d O r s w $.
+    $d Q m r u $.  $d Q n s $.  $d Q u v x y z $.  $d Q w $.  $d R m r u v x $.
+    $d R n u x $.  $d R s u v w x $.  $d R u v x y z $.  $d X m r $.
+    $d X v y z $.  $d m ph r u $.  $d n ph s x $.  $d ph u v x y z $.
+    $d ph w $.
+    qsdrng.0 $e |- O = ( oppR ` R ) $.
+    qsdrng.q $e |- Q = ( R /s ( R ~QG M ) ) $.
+    qsdrng.r $e |- ( ph -> R e. NzRing ) $.
+    qsdrngi.1 $e |- ( ph -> M e. ( MaxIdeal ` R ) ) $.
+    qsdrngi.2 $e |- ( ph -> M e. ( MaxIdeal ` O ) ) $.
+    ${
+      qsdrngilem.1 $e |- ( ph -> X e. ( Base ` R ) ) $.
+      qsdrngilem.2 $e |- ( ph -> -. X e. M ) $.
+      $( Lemma for ~ qsdrngi .  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+      qsdrngilem $p |- ( ph -> E. v e. ( Base ` Q )
+                           ( v ( .r ` Q ) [ X ] ( R ~QG M ) ) = ( 1r ` Q ) ) $=
+        ( cfv co wceq wcel eqid ad3antrrr vr vm cur cv cmulr cplusg cqg cec cbs
+        wrex wa cqs simpllr ovex ecelqsi syl cvv cnzr cqus ovexd qusbas eleqtrd
+        a1i wb oveq1 eqeq1d adantl crg nzrring c2idl clidl cin mxidlidl syl2anc
+        opprring elind 2idlval eleqtrrdi qusmul2 wer cnsg csubg lidlnsg nsgsubg
+        cmxidl eqger 3syl wss cminusg lidlss ringcld ringidcl simpr oveq2d cgrp
+        wbr c0g ringgrpd oveq1d grpinvcld simplr sseldd grpassd grplidd 3eqtr3d
+        grplinvd eqtrd eqeltrd w3a eqgval biimpar syl23anc qus1 simprd rspcedvd
+        erthi csn cun crsp snssd unssd rspcl rspssid unssad unssbd snssg eldifd
+        mxidlmaxv eleqtrrd elrspunsn mpbid r19.29vva ) ADUCOZUAUDZGDUEOZPZUBUDZ
+        DUFOZPZQZBUDZGDEUGPZUHZCUEOZPZCUCOZQZBCUIOZUJUAUBDUIOZEAYNUUIRZUKZYQERZ
+        UKZYTUKZUUGYNUUBUHZUUCUUDPZUUFQZBUUOUUHUUNUUOUUIUUBULZUUHUUNUUJUUOUURRA
+        UUJUULYTUMZUUIYNUUBDEUGUNUOUPAUURUUHQUUJUULYTAUUBDCUUIUQURCDUUBUSPQAIVC
+        UUIUUIQAUUISZVCADEUGUTJVATVBUUAUUOQZUUGUUQVDUUNUVAUUEUUPUUFUUAUUOUUCUUD
+        VEVFVGUUNUUPYMUUBUHZUUFUUNUUPYPUUBUHUVBUUNUUICDYOUUDEYNGIUUTYOSZUUDSADV
+        HRZUUJUULYTADURRUVDJDVIUPZTZAEDVJOZRZUUJUULYTAEDVKOZFVKOZVLUVGAUVIUVJEA
+        UVDEDWEOREUVIRZUVEKUUIDEUUTVMVNZAFVHRZEFWEOREUVJRAUVDUVMUVEDFHVOUPLFUIO
+        ZFEUVNSVMVNVPDUVGUVIUVJFUVISZHUVJSUVGSZVQVRTZUUSAGUUIRZUUJUULYTMTZVSUUN
+        YPYMUUBUUIAUUIUUBVTZUUJUULYTAEDWAORZEDWBORUVTAUVDUVKUWAUVEUVLDEWCVNEDWD
+        UUBDUUIEUUTUUBSZWFWGTUUNUVDEUUIWHZYPUUIRZYMUUIRZYPDWIOZOZYMYRPZERZYPYMU
+        UBWPZUVFAUWCUUJUULYTAUVKUWCUVLUUIEUVIDUUTUVOWJUPZTZUUNUUIDYOYNGUUTUVCUV
+        FUUSUVSWKZAUWEUUJUULYTAUVDUWEUVEUUIDYMUUTYMSZWLUPZTUUNUWHYQEUUNUWHUWGYS
+        YRPZYQUUNYMYSUWGYRUUMYTWMWNUUNUWGYPYRPZYQYRPDWQOZYQYRPUWPYQUUNUWQUWRYQY
+        RUUNUUIYRDUWFYPUWRUUTYRSZUWRSZUWFSZADWORUUJUULYTADUVEWRTZUWMXFWSUUNUUIY
+        RDUWGYPYQUUTUWSUXBUUNUUIDUWFYPUUTUXAUXBUWMWTUWMUUNEUUIYQUWLUUKUULYTXAZX
+        BZXCUUNUUIYRDYQUWRUUTUWSUWTUXBUXDXDXEXGUXCXHUVDUWCUKUWJUWDUWEUWIXIYPYMY
+        RUUBEDUWFVHUUIUUTUXAUWSUWBXJXKXLXPXGUUNUVDUVHUVBUUFQZUVFUVQUVDUVHUKCVHR
+        UXEDECYMUVGIUVPUWNXMXNVNXGXOAYMEGXQZXRZDXSOZOZRYTUBEUJUAUUIUJAYMUUIUXIU
+        WOAUUIDUXIEGUUTUVEKAUVDUXGUUIWHZUXIUVIRUVEAEUXFUUIUWKAGUUIMXTYAZUUIDUVI
+        UXGUXHUXHSZUUTUVOYBVNAEUXFUXIAUVDUXJUXGUXIWHUVEUXKUUIDUXGUXHUXLUUTYCVNZ
+        YDAGUXIEAUVRUXFUXIWHZGUXIRZMAEUXFUXIUXMYEUVRUXOUXNGUXIUUIYFXKVNNYGYHYIA
+        YMUUIYRDYOUBEUXHGUWRUAUXLUUTUWTUVCUVEUWSUVLAGUUIEMNYGYJYKYL $.
+    $}
+
+    $( A quotient by a maximal ideal is an division ring.  (Contributed by
path: set.mm
user: login: benjub
body: Thanks. I won't have time to review the rest of the MR, so I removed myself from reviewers.
created_at: 2025-03-09T23:05:38Z
author_association: CONTRIBUTOR
diff_hunk: @@ -500881,6 +501256,228 @@ The sumset operation can be used for both group (additive) operations and
   $}
 
 
+  ${
+    $d B x y $.  $d I x y $.  $d O x y $.  $d R x y $.
+    oppreqg.o $e |- O = ( oppR ` R ) $.
+    ${
+      oppreqg.b $e |- B = ( Base ` R ) $.
+      $( Group coset equivalence relation for the opposite ring.  (Contributed
+         by Thierry Arnoux, 9-Mar-2025.) $)
+      oppreqg $p |- ( ( R e. V /\ I C_ B ) -> ( R ~QG I ) = ( O ~QG I ) ) $=
+        ( vx vy wcel wss wa cqg co cv cpr cfv eqid eqgfval cvv copab wceq coppr
+        cminusg cplusg fvexi opprbas opprneg oppradd mpan adantl eqtr4d ) BEJZC
+        AKZLBCMNZHOZIOZPAKUPBUDQZQUQBUEQZNCJLHIUAZDCMNZHIUSUOCBUREAGURRZUSRZUOR
+        SUNVAUTUBZUMDTJUNVDDBUCFUFHIUSVACDURTAABDFGUGBURDFVBUHUSBDFVCUIVARSUJUK
+        UL $.
+    $}
+
+    $d O a b i x $.  $d R a b i x $.  $d a b i ph x $.
+    oppr2idl.2 $e |- ( ph -> R e. Ring ) $.
+    $( The ideals of the opposite ring's opposite ring are the ideals of the
+       original ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+    opprlidlabs $p |- ( ph -> ( LIdeal ` R ) = ( LIdeal ` ( oppR ` O ) ) ) $=
+      ( vi vx va vb clidl cfv cv cbs cmulr co wcel wral w3a wa eqid coppr c0 wb
+      wss wne cplusg opprmul eqtr2i a1i oveq1d eleq1d ralbidva anasss 2ralbidva
+      wceq 3anbi3d islidl opprbas oppradd 3bitr4g eqrdv ) AFBJKZCUAKZJKZAFLZBMK
+      ZUDZVEUBUEZGLZHLZBNKZOZILZBUFKZOZVEPZIVEQZHVEQGVFQZRVGVHVIVJVCNKZOZVMVNOZ
+      VEPZIVEQZHVEQGVFQZRVEVBPVEVDPAVRWDVGVHAVQWCGHVFVEAVIVFPZVJVEPZVQWCUCAWESW
+      FSZVPWBIVEWGVMVEPSZVOWAVEWHVLVTVMVNVLVTUOWHVTVJVICNKZOVLCMKZCVSWIVCVIVJWJ
+      TWITZVCTZVSTZUGVFBWIVKCVJVIVFTZVKTZDWKUGUHUIUJUKULUMUNUPGVFVNBVKVBVEHIVBT
+      WNVNTZWOUQGVFVNVCVSVDVEHIVDTVFCVCWLVFBCDWNURURVNCVCWLVNBCDWPUSUSWMUQUTVA
+      $.
+
+    $( Two sided ideal of the opposite ring.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    oppr2idl $p |- ( ph -> ( 2Ideal ` R ) = ( 2Ideal ` O ) ) $=
+      ( clidl cfv cin coppr c2idl incom opprlidlabs ineq2d eqid 2idlval 3eqtr4g
+      eqtrid ) ABFGZCFGZHZSCIGZFGZHZBJGZCJGZATSRHUCRSKARUBSABCDELMQBUDRSCRNDSNZ
+      UDNOCUESUBUAUFUANUBNUENOP $.
+
+    $d M j $.  $d O j $.  $d j ph $.
+    opprmxidl.3 $e |- ( ph -> M e. ( MaxIdeal ` R ) ) $.
+    $( The maximal ideal of the opposite ring's opposite ring.  (Contributed by
+       Thierry Arnoux, 9-Mar-2025.) $)
+    opprmxidlabs $p |- ( ph -> M e. ( MaxIdeal ` ( oppR ` O ) ) ) $=
+      ( vj cfv wcel clidl wceq cmxidl opprring eqid syl2anc wa ad2antrr opprbas
+      crg coppr cbs wne cv wo wi wral 3syl mxidlidl opprlidlabs eleqtrd mxidlnr
+      wss simplr eleqtrrd simpr mxidlmax syl22anc ralrimiva w3a ismxidl biimpar
+      ex syl13anc ) ADUAIZTJZCVEKIZJZCBUBIZUCZCHUDZUMZVKCLVKVILUEZUFZHVGUGZCVEM
+      IJZABTJZDTJVFFBDENDVEVEOZNUHACBKIZVGAVQCBMIJZCVSJFGVIBCVIOZUIPABDEFUJZUKA
+      VQVTVJFGVIBCWAULPAVNHVGAVKVGJZQZVLVMWDVLQZVQVTVKVSJVLVMAVQWCVLFRAVTWCVLGR
+      WEVKVGVSAWCVLUNAVSVGLWCVLWBRUOWDVLUPVIBVKCWAUQURVCUSVFVPVHVJVOUTVIVEHCVID
+      VEVRVIBDEWASSVAVBVD $.
+  $}
+
+  ${
+    $d B p q $.  $d I p q x y $.  $d O p q x y $.  $d Q p q x y $.  $d R p q $.
+    $d X p q $.  $d Y p q $.  $d p ph q x y $.
+    opprqus.b $e |- B = ( Base ` R ) $.
+    opprqus.o $e |- O = ( oppR ` R ) $.
+    opprqus.q $e |- Q = ( R /s ( R ~QG I ) ) $.
+    ${
+      opprqusbas.r $e |- ( ph -> R e. V ) $.
+      opprqusbas.i $e |- ( ph -> I C_ B ) $.
+      $( The base of the quotient of the opposite ring is the same as the base
+         of the opposite of the quotient ring.  (Contributed by Thierry Arnoux,
+         9-Mar-2025.) $)
+      opprqusbas $p |- ( ph -> ( Base ` ( oppR ` Q ) )
+                               = ( Base ` ( O /s ( O ~QG I ) ) ) ) $=
+        ( coppr cfv cbs cqg co wceq cvv a1i cqus opprbas cqs wss oppreqg qseq2d
+        eqid wcel syl2anc ovexd qusbas eqidd fvexi 3eqtr3d eqtr3id ) ACMNZONCON
+        ZFFEPQZUAQZONZUQCUPUPUGUQUGUBABDEPQZUCBURUCUQUTAVAURBADGUHEBUDVAURRKLBD
+        EFGIHUEUIUFAVADCBSGCDVAUAQRAJTBDONRAHTADEPUJKUKAURFUSBSSAUSULBFONRABDFI
+        HUBTAFEPUJFSUHAFDMIUMTUKUNUO $.
+    $}
+
+    opprqus1r.r $e |- ( ph -> R e. Ring ) $.
+    opprqus1r.i $e |- ( ph -> I e. ( 2Ideal ` R ) ) $.
+    ${
+      opprqusmulr.e $e |- E = ( Base ` Q ) $.
+      opprqusmulr.x $e |- ( ph -> X e. E ) $.
+      opprqusmulr.y $e |- ( ph -> Y e. E ) $.
+      $( The multiplication operation of the quotient of the opposite ring is
+         the same as the multiplication operation of the opposite of the
+         quotient ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+      opprqusmulr $p |- ( ph -> ( X ( .r ` ( oppR ` Q ) ) Y )
+                                = ( X ( .r ` ( O /s ( O ~QG I ) ) ) Y ) ) $=
+        ( cfv co wcel vp vq coppr cmulr cqg cqus eqid opprmul cv cec wa ad4antr
+        crg c2idl simplr simp-4r qusmul2 simpr simpllr oveq12d opprbas opprring
+        wceq syl oppr2idl eleqtrd clidl 2idllidld lidlss oppreqg syl2anc eceq2d
+        wss eqtrd a1i eceq1d eqtr3d 3eqtr4d cqs cbs eleqtrdi ad2antrr cvv ovexd
+        wrex qusbas eqtr3i eqtr2di elqsi r19.29a eqtrid ) AHICUCRZUDRZSIHCUDRZS
+        ZHIGGFUESZUFSZUDRZSZECWMWNWLHIOWNUGZWLUGZWMUGUHAHUAUIZDFUESZUJZVCZWOWSV
+        CZUABAXBBTZUKZXEUKZIUBUIZXCUJZVCZXFUBBXIXJBTZUKZXLUKZXKXDWNSXJXBDUDRZSZ
+        XCUJZWOWSXOBCDXPWNFXJXBLJXPUGZWTADUMTZXGXEXMXLMULAFDUNRZTXGXEXMXLNULXIX
+        MXLUOZAXGXEXMXLUPZUQXOIXKHXDWNXNXLURZXHXEXMXLUSZUTXOXBWPUJZXJWPUJZWRSXB
+        XJGUDRZSZWPUJZWSXRXOBWQGYHWRFXBXJWQUGBDGKJVAYHUGZWRUGAGUMTZXGXEXMXLAXTY
+        LMDGKVBVDULAFGUNRZTXGXEXMXLAFYAYMNADGKMVEVFULYCYBUQXOHYFIYGWRXOHXDYFYEX
+        OXCWPXBAXCWPVCZXGXEXMXLAXTFBVMZYNMAFDVGRZTYOADFNVHBFYPDJYPUGVIVDBDFGUMK
+        JVJVKULZVLVNXOIXKYGYDXOXCWPXJYQVLVNUTXOYIXCUJXRYJXOYIXQXCYIXQVCXOBDYHXP
+        GXBXJJXSKYKUHVOVPXOXCWPYIYQVLVQVRVRXIIBXCVSZTXLUBBWEXIIWLVTRZYRAIYSTXGX
+        EAIEYSQECWLXAOVAZWAWBAYSYRVCXGXEAYRCVTRZYSAXCDCBWCUMCDXCUFSVCALVOBDVTRV
+        CAJVOADFUEWDMWFEUUAYSOYTWGWHZWBVFUBBIXCWIVDWJAHYRTXEUABWEAHYSYRAHEYSPYT
+        WAUUBVFUABHXCWIVDWJWK $.
+    $}
+
+    $( The ring unity of the quotient of the opposite ring is the same as the
+       ring unity of the opposite of the quotient ring.  (Contributed by
+       Thierry Arnoux, 9-Mar-2025.) $)
+    opprqus1r $p |- ( ph -> ( 1r ` ( oppR ` Q ) ) = ( 1r ` ( O /s ( O ~QG I ) )
+       ) ) $=
+      ( vx vy coppr cfv cbs co eqid wcel wa cqg cvv fvexd ovexd clidl 2idllidld
+      cqus crg wss lidlss syl opprqusbas ad2antrr c2idl simpr opprbas eleqtrrdi
+      cv adantr adantlr opprqusmulr urpropd ) ALMCNOZPOZVCFFEUAQZUGQUBUBVDRACNU
+      CAFVEUGUDABCDEFUHGHIJAEDUEOZSEBUIADEKUFBEVFDGVFRUJUKULALURZVDSZTZMURZVDSZ
+      TBCDCPOZEFVGVJGHIADUHSVHVKJUMAEDUNOSVHVKKUMVLRZVIVGVLSVKVIVGVDVLAVHUOVLCV
+      CVCRVMUPZUQUSAVKVJVLSVHAVKTVJVDVLAVKUOVNUQUTVAVB $.
+  $}
+
+  ${
+    $d M m r u v x $.  $d M n s $.  $d M u v x y z $.  $d M w $.  $d O r s w $.
+    $d Q m r u $.  $d Q n s $.  $d Q u v x y z $.  $d Q w $.  $d R m r u v x $.
+    $d R n u x $.  $d R s u v w x $.  $d R u v x y z $.  $d X m r $.
+    $d X v y z $.  $d m ph r u $.  $d n ph s x $.  $d ph u v x y z $.
+    $d ph w $.
+    qsdrng.0 $e |- O = ( oppR ` R ) $.
+    qsdrng.q $e |- Q = ( R /s ( R ~QG M ) ) $.
+    qsdrng.r $e |- ( ph -> R e. NzRing ) $.
+    qsdrngi.1 $e |- ( ph -> M e. ( MaxIdeal ` R ) ) $.
+    qsdrngi.2 $e |- ( ph -> M e. ( MaxIdeal ` O ) ) $.
+    ${
+      qsdrngilem.1 $e |- ( ph -> X e. ( Base ` R ) ) $.
+      qsdrngilem.2 $e |- ( ph -> -. X e. M ) $.
+      $( Lemma for ~ qsdrngi .  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
+      qsdrngilem $p |- ( ph -> E. v e. ( Base ` Q )
+                           ( v ( .r ` Q ) [ X ] ( R ~QG M ) ) = ( 1r ` Q ) ) $=
+        ( cfv co wceq wcel eqid ad3antrrr vr vm cur cv cmulr cplusg cqg cec cbs
+        wrex wa cqs simpllr ovex ecelqsi syl cvv cnzr cqus ovexd qusbas eleqtrd
+        a1i wb oveq1 eqeq1d adantl crg nzrring c2idl clidl cin mxidlidl syl2anc
+        opprring elind 2idlval eleqtrrdi qusmul2 wer cnsg csubg lidlnsg nsgsubg
+        cmxidl eqger 3syl wss cminusg lidlss ringcld ringidcl simpr oveq2d cgrp
+        wbr c0g ringgrpd oveq1d grpinvcld simplr sseldd grpassd grplidd 3eqtr3d
+        grplinvd eqtrd eqeltrd w3a eqgval biimpar syl23anc qus1 simprd rspcedvd
+        erthi csn cun crsp snssd unssd rspcl rspssid unssad unssbd snssg eldifd
+        mxidlmaxv eleqtrrd elrspunsn mpbid r19.29vva ) ADUCOZUAUDZGDUEOZPZUBUDZ
+        DUFOZPZQZBUDZGDEUGPZUHZCUEOZPZCUCOZQZBCUIOZUJUAUBDUIOZEAYNUUIRZUKZYQERZ
+        UKZYTUKZUUGYNUUBUHZUUCUUDPZUUFQZBUUOUUHUUNUUOUUIUUBULZUUHUUNUUJUUOUURRA
+        UUJUULYTUMZUUIYNUUBDEUGUNUOUPAUURUUHQUUJUULYTAUUBDCUUIUQURCDUUBUSPQAIVC
+        UUIUUIQAUUISZVCADEUGUTJVATVBUUAUUOQZUUGUUQVDUUNUVAUUEUUPUUFUUAUUOUUCUUD
+        VEVFVGUUNUUPYMUUBUHZUUFUUNUUPYPUUBUHUVBUUNUUICDYOUUDEYNGIUUTYOSZUUDSADV
+        HRZUUJUULYTADURRUVDJDVIUPZTZAEDVJOZRZUUJUULYTAEDVKOZFVKOZVLUVGAUVIUVJEA
+        UVDEDWEOREUVIRZUVEKUUIDEUUTVMVNZAFVHRZEFWEOREUVJRAUVDUVMUVEDFHVOUPLFUIO
+        ZFEUVNSVMVNVPDUVGUVIUVJFUVISZHUVJSUVGSZVQVRTZUUSAGUUIRZUUJUULYTMTZVSUUN
+        YPYMUUBUUIAUUIUUBVTZUUJUULYTAEDWAORZEDWBORUVTAUVDUVKUWAUVEUVLDEWCVNEDWD
+        UUBDUUIEUUTUUBSZWFWGTUUNUVDEUUIWHZYPUUIRZYMUUIRZYPDWIOZOZYMYRPZERZYPYMU
+        UBWPZUVFAUWCUUJUULYTAUVKUWCUVLUUIEUVIDUUTUVOWJUPZTZUUNUUIDYOYNGUUTUVCUV
+        FUUSUVSWKZAUWEUUJUULYTAUVDUWEUVEUUIDYMUUTYMSZWLUPZTUUNUWHYQEUUNUWHUWGYS
+        YRPZYQUUNYMYSUWGYRUUMYTWMWNUUNUWGYPYRPZYQYRPDWQOZYQYRPUWPYQUUNUWQUWRYQY
+        RUUNUUIYRDUWFYPUWRUUTYRSZUWRSZUWFSZADWORUUJUULYTADUVEWRTZUWMXFWSUUNUUIY
+        RDUWGYPYQUUTUWSUXBUUNUUIDUWFYPUUTUXAUXBUWMWTUWMUUNEUUIYQUWLUUKUULYTXAZX
+        BZXCUUNUUIYRDYQUWRUUTUWSUWTUXBUXDXDXEXGUXCXHUVDUWCUKUWJUWDUWEUWIXIYPYMY
+        RUUBEDUWFVHUUIUUTUXAUWSUWBXJXKXLXPXGUUNUVDUVHUVBUUFQZUVFUVQUVDUVHUKCVHR
+        UXEDECYMUVGIUVPUWNXMXNVNXGXOAYMEGXQZXRZDXSOZOZRYTUBEUJUAUUIUJAYMUUIUXIU
+        WOAUUIDUXIEGUUTUVEKAUVDUXGUUIWHZUXIUVIRUVEAEUXFUUIUWKAGUUIMXTYAZUUIDUVI
+        UXGUXHUXHSZUUTUVOYBVNAEUXFUXIAUVDUXJUXGUXIWHUVEUXKUUIDUXGUXHUXLUUTYCVNZ
+        YDAGUXIEAUVRUXFUXIWHZGUXIRZMAEUXFUXIUXMYEUVRUXOUXNGUXIUUIYFXKVNNYGYHYIA
+        YMUUIYRDYOUBEUXHGUWRUAUXLUUTUWTUVCUVEUWSUVLAGUUIEMNYGYJYKYL $.
+    $}
+
+    $( A quotient by a maximal ideal is an division ring.  (Contributed by
path: set.mm
user: login: tirix
body: OK, thanks for your time anyway!
created_at: 2025-03-09T23:41:55Z
author_association: CONTRIBUTOR
diff_hunk: @@ -498243,6 +498422,29 @@ then this is also a division ring (see ~ fldgensdrg ).  If the base
   $}
 
 
+$(
+-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
+  Ideals
+-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
+$)
+
+  ${
+    2idllidld.1 $e |- ( ph -> I e. ( 2Ideal ` R ) ) $.
+    $( A two-sided ideal is a left ideal.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    2idllidld $p |- ( ph -> I e. ( LIdeal ` R ) ) $=
+      ( clidl cfv coppr c2idl cin eqid 2idlval eleqtrdi elin1d ) ABEFZBGFZEFZCA
+      CBHFZNPIDBQNPONJOJPJQJKLM $.
+
+    2idllidld.o $e |- O = ( oppR ` R ) $.
+    $( A two-sided ideal is a right ideal.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    2idlridld $p |- ( ph -> I e. ( LIdeal ` O ) ) $=
+      ( clidl cfv c2idl cin eqid 2idlval eleqtrdi elin2d ) ABGHZDGHZCACBIHZOPJE
+      BQOPDOKFPKQKLMN $.
+  $}
+
+
path: set.mm
user: login: icecream17
body: They seem useful enough to be in main
created_at: 2025-03-10T01:15:28Z
author_association: CONTRIBUTOR
diff_hunk: @@ -498243,6 +498422,29 @@ then this is also a division ring (see ~ fldgensdrg ).  If the base
   $}
 
 
+$(
+-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
+  Ideals
+-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
+$)
+
+  ${
+    2idllidld.1 $e |- ( ph -> I e. ( 2Ideal ` R ) ) $.
+    $( A two-sided ideal is a left ideal.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    2idllidld $p |- ( ph -> I e. ( LIdeal ` R ) ) $=
+      ( clidl cfv coppr c2idl cin eqid 2idlval eleqtrdi elin1d ) ABEFZBGFZEFZCA
+      CBHFZNPIDBQNPONJOJPJQJKLM $.
+
+    2idllidld.o $e |- O = ( oppR ` R ) $.
+    $( A two-sided ideal is a right ideal.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    2idlridld $p |- ( ph -> I e. ( LIdeal ` O ) ) $=
+      ( clidl cfv c2idl cin eqid 2idlval eleqtrdi elin2d ) ABGHZDGHZCACBIHZOPJE
+      BQOPDOKFPKQKLMN $.
+  $}
+
+
path: set.mm
user: login: avekens
body: > Usually we move them to main in the same PR where we use them, but if you prefer I could move them in this PR?

Yes, please.
created_at: 2025-03-10T04:56:19Z
author_association: CONTRIBUTOR
diff_hunk: @@ -500881,6 +501256,228 @@ The sumset operation can be used for both group (additive) operations and
   $}
 
 
+  ${
+    $d B x y $.  $d I x y $.  $d O x y $.  $d R x y $.
+    oppreqg.o $e |- O = ( oppR ` R ) $.
+    ${
+      oppreqg.b $e |- B = ( Base ` R ) $.
+      $( Group coset equivalence relation for the opposite ring.  (Contributed
+         by Thierry Arnoux, 9-Mar-2025.) $)
+      oppreqg $p |- ( ( R e. V /\ I C_ B ) -> ( R ~QG I ) = ( O ~QG I ) ) $=
+        ( vx vy wcel wss wa cqg co cv cpr cfv eqid eqgfval cvv copab wceq coppr
+        cminusg cplusg fvexi opprbas opprneg oppradd mpan adantl eqtr4d ) BEJZC
+        AKZLBCMNZHOZIOZPAKUPBUDQZQUQBUEQZNCJLHIUAZDCMNZHIUSUOCBUREAGURRZUSRZUOR
+        SUNVAUTUBZUMDTJUNVDDBUCFUFHIUSVACDURTAABDFGUGBURDFVBUHUSBDFVCUIVARSUJUK
+        UL $.
+    $}
+
+    $d O a b i x $.  $d R a b i x $.  $d a b i ph x $.
+    oppr2idl.2 $e |- ( ph -> R e. Ring ) $.
+    $( The ideals of the opposite ring's opposite ring are the ideals of the
+       original ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
path: set.mm
user: login: avekens
body: OK, I see , it is not that easy as it would be expected. Nevertheless, I would like to have ~opprabs in set.mm to visualize these difficulties.
created_at: 2025-03-10T05:00:22Z
author_association: CONTRIBUTOR
diff_hunk: @@ -498243,6 +498422,29 @@ then this is also a division ring (see ~ fldgensdrg ).  If the base
   $}
 
 
+$(
+-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
+  Ideals
+-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
+$)
+
+  ${
+    2idllidld.1 $e |- ( ph -> I e. ( 2Ideal ` R ) ) $.
+    $( A two-sided ideal is a left ideal.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    2idllidld $p |- ( ph -> I e. ( LIdeal ` R ) ) $=
+      ( clidl cfv coppr c2idl cin eqid 2idlval eleqtrdi elin1d ) ABEFZBGFZEFZCA
+      CBHFZNPIDBQNPONJOJPJQJKLM $.
+
+    2idllidld.o $e |- O = ( oppR ` R ) $.
+    $( A two-sided ideal is a right ideal.  (Contributed by Thierry Arnoux,
+       9-Mar-2025.) $)
+    2idlridld $p |- ( ph -> I e. ( LIdeal ` O ) ) $=
+      ( clidl cfv c2idl cin eqid 2idlval eleqtrdi elin2d ) ABGHZDGHZCACBIHZOPJE
+      BQOPDOKFPKQKLMN $.
+  $}
+
+
path: set.mm
user: login: tirix
body: OK! Done in https://github.com/metamath/set.mm/pull/4702/commits/9d63a4d7a9982b3829be673dac2a3babfa94ed23
I did not mention it in `change-set.txt` because those theorems were new in my mathbox anyway.
created_at: 2025-03-10T09:40:58Z
author_association: CONTRIBUTOR
diff_hunk: @@ -500881,6 +501256,228 @@ The sumset operation can be used for both group (additive) operations and
   $}
 
 
+  ${
+    $d B x y $.  $d I x y $.  $d O x y $.  $d R x y $.
+    oppreqg.o $e |- O = ( oppR ` R ) $.
+    ${
+      oppreqg.b $e |- B = ( Base ` R ) $.
+      $( Group coset equivalence relation for the opposite ring.  (Contributed
+         by Thierry Arnoux, 9-Mar-2025.) $)
+      oppreqg $p |- ( ( R e. V /\ I C_ B ) -> ( R ~QG I ) = ( O ~QG I ) ) $=
+        ( vx vy wcel wss wa cqg co cv cpr cfv eqid eqgfval cvv copab wceq coppr
+        cminusg cplusg fvexi opprbas opprneg oppradd mpan adantl eqtr4d ) BEJZC
+        AKZLBCMNZHOZIOZPAKUPBUDQZQUQBUEQZNCJLHIUAZDCMNZHIUSUOCBUREAGURRZUSRZUOR
+        SUNVAUTUBZUMDTJUNVDDBUCFUFHIUSVACDURTAABDFGUGBURDFVBUHUSBDFVCUIVARSUJUK
+        UL $.
+    $}
+
+    $d O a b i x $.  $d R a b i x $.  $d a b i ph x $.
+    oppr2idl.2 $e |- ( ph -> R e. Ring ) $.
+    $( The ideals of the opposite ring's opposite ring are the ideals of the
+       original ring.  (Contributed by Thierry Arnoux, 9-Mar-2025.) $)
path: set.mm
user: login: tirix
body: Done in https://github.com/metamath/set.mm/pull/4702/commits/9d63a4d7a9982b3829be673dac2a3babfa94ed23
I included a comment on the difficulties to use the theorem.
created_at: 2025-03-10T09:42:08Z
author_association: CONTRIBUTOR
diff_hunk: @@ -84,6 +84,17 @@ make a github issue.)
 
 DONE:
 Date      Old       New         Notes
+10-Mar-25 brrici    [same]     Moved from SN's mathbox to main set.mm
path: changes-set.txt
user: login: avekens
body: indentation of notes
created_at: 2025-03-10T10:57:03Z
author_association: CONTRIBUTOR
diff_hunk: @@ -84,6 +84,17 @@ make a github issue.)
 
 DONE:
 Date      Old       New         Notes
+10-Mar-25 brrici    [same]     Moved from SN's mathbox to main set.mm
path: changes-set.txt
user: login: tirix
body: Fixed in https://github.com/metamath/set.mm/pull/4702/commits/5fbe092e361a3293b804e9f4a7baba377f5c7875
(I feel a bit bad to use GitHub server time to rerun all tests because of a single space character!)
created_at: 2025-03-10T13:03:45Z
author_association: CONTRIBUTOR

PR comments:
user: login: savask
created_at: 2025-03-09T15:33:21Z
body: Amazing work as always! I'm baffled by the size of some of these theorems, I would separate them into several steps, but probably not worth worrying about that now.

Shouldn't it be possible to use `elrspunidl` to prove `elrnspunsn`? If `x` is an element of a ring `R`, then `Rx` is a left ideal and the situation seems to reduce to the span of two ideals. Also I think the hypothesis `elrspunsn.x` is unnecessary.
author_association: CONTRIBUTOR
user: login: tirix
created_at: 2025-03-09T18:20:28Z
body: > Shouldn't it be possible to use `elrspunidl` to prove `elrnspunsn`? If `x` is an element of a ring `R`, then `Rx` is a left ideal and the situation seems to reduce to the span of two ideals. 

Yes - I think I've initially considered it and probably ruled it out too quickly!

> Also I think the hypothesis `elrspunsn.x` is unnecessary.

Yes, that is unnecessary, but the way I built the proof it was easier to make that assumption (I'm using [`gsumsplit2`](https://us.metamath.org/mpeuni/gsumsplit2.html) which requires disjoint summation domains). I thought about removing the condition by handling two cases, `X e. I` and `-. X e. I`, but in my case I have `-. X e. I`. A proof using `elrspunidl` would not require that.
author_association: CONTRIBUTOR
user: login: avekens
created_at: 2025-03-09T18:38:02Z
body: > I've renamed [`grpridd`](https://us.metamath.org/mpeuni/grpridd.html) to `grprida` and chose to use @icecream17's version [`sn-grpidd`](https://us.metamath.org/mpeuni/sn-grpridd.html) instead. Was that the correct way? Should I have revised the old `grpridd` instead? It has only one use.

I am also working on rings and ideals, and I came over `grpridd`, too. I called it `grplinvlidridd`, not bothering that this label is quite long, because it is not used often. To rename the existing `grpridd` and to remove the prefix "sn-" from `sn-grpidd` is a good idea.

author_association: CONTRIBUTOR
user: login: tirix
created_at: 2025-03-09T19:10:50Z
body: > > I've renamed [`grpridd`](https://us.metamath.org/mpeuni/grpridd.html) to `grprida` and chose to use @icecream17's version [`sn-grpidd`](https://us.metamath.org/mpeuni/sn-grpridd.html) instead. Was that the correct way? Should I have revised the old `grpridd` instead? It has only one use.
> 
> I am also working on rings and ideals, and I came over `grpridd`, too. I called it `grplinvlidridd`, not bothering that this label is quite long, because it is not used often. To rename the existing `grpridd` and to remove the prefix "sn-" from `sn-grpidd` is a good idea.

Alternatively we could completely remove `grplinvlidridd` / `grplinvda` and use `grplinvdd` instead.
author_association: CONTRIBUTOR
user: login: avekens
created_at: 2025-03-10T05:04:17Z
body: 


> > > I've renamed [`grpridd`](https://us.metamath.org/mpeuni/grpridd.html) to `grprida` and chose to use @icecream17's version [`sn-grpidd`](https://us.metamath.org/mpeuni/sn-grpridd.html) instead. Was that the correct way? Should I have revised the old `grpridd` instead? It has only one use.
> > 
> > 
> > I am also working on rings and ideals, and I came over `grpridd`, too. I called it `grplinvlidridd`, not bothering that this label is quite long, because it is not used often. To rename the existing `grpridd` and to remove the prefix "sn-" from `sn-grpidd` is a good idea.
> 
> Alternatively we could completely remove `grplinvlidridd` / `grplinvda` and use `grplinvdd` instead.

I don't think so -  the theorems are quite different and have different intentions. 
author_association: CONTRIBUTOR
user: login: tirix
created_at: 2025-03-10T09:45:44Z
body: **Note:** I decided last minute to add 3 small theorems to this PR: `gicqusker`, `ricqusker` and `lmicqusker`, which are just restatements of `ghmqusker` and co. using the `~=g` isomorphic relations. I'm not usually doing this, sorry for the late addition.
author_association: CONTRIBUTOR
user: login: tirix
created_at: 2025-03-10T13:07:44Z
body: > I am also working on rings and ideals

I am thrilled to see what you're posting next! I want to first prove the reverse direction of `qsdrngi`, I don't expect any big issues.

Since I have necessary approvals and you seem to be expecting some theorems in main for your own needs, I will not wait longer and merge this PR.
author_association: CONTRIBUTOR