Datasets:

yalhessi

/

lemexp

like 0

[]

['Pure.eq', 'AbelCoset.a_r_coset', 'Complete_Lattices.Union', 'Set.image']

lemma a_r_coset_def': fixes G (structure) shows "H +> a \<equiv> \<Union>h\<in>H. {h \<oplus> a}"

[]

['Pure.eq', 'AbelCoset.a_l_coset', 'Complete_Lattices.Union', 'Set.image']

lemma a_l_coset_def': fixes G (structure) shows "a <+ H \<equiv> \<Union>h\<in>H. {a \<oplus> h}"

[]

['Pure.eq', 'AbelCoset.A_RCOSETS', 'Complete_Lattices.Union', 'Set.image', 'Congruence.partial_object.carrier']

lemma A_RCOSETS_def': fixes G (structure) shows "a_rcosets H \<equiv> \<Union>a\<in>carrier G. {H +> a}"

[]

['Pure.eq', 'AbelCoset.set_add', 'Complete_Lattices.Union', 'Set.image']

lemma set_add_def': fixes G (structure) shows "H <+> K \<equiv> \<Union>h\<in>H. \<Union>k\<in>K. {h \<oplus> k}"

[]

['Pure.eq', 'AbelCoset.A_SET_INV', 'Complete_Lattices.Union', 'Set.image']

lemma A_SET_INV_def': fixes G (structure) shows "a_set_inv H \<equiv> \<Union>h\<in>H. {\<ominus> h}"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.subset_eq', 'Congruence.partial_object.carrier', 'Set.member', 'HOL.eq', 'AbelCoset.a_r_coset', 'Ring.ring.add']

lemma (in abelian_group) a_coset_add_assoc: "[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |] ==> (M +> g) +> h = M +> (g \<oplus> h)"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.subset_eq', 'Congruence.partial_object.carrier', 'HOL.eq', 'AbelCoset.a_r_coset', 'Ring.ring.zero']

lemma (in abelian_group) a_coset_add_zero [simp]: "M \<subseteq> carrier G ==> M +> \<zero> = M"

[]

['Pure.imp', 'HOL.Trueprop', 'HOL.eq', 'AbelCoset.a_r_coset', 'Ring.ring.add', 'Ring.a_inv', 'Set.member', 'Congruence.partial_object.carrier', 'Set.subset_eq']

lemma (in abelian_group) a_coset_add_inv1: "[| M +> (x \<oplus> (\<ominus> y)) = M; x \<in> carrier G ; y \<in> carrier G; M \<subseteq> carrier G |] ==> M +> x = M +> y"

[]

['Pure.imp', 'HOL.Trueprop', 'HOL.eq', 'AbelCoset.a_r_coset', 'Set.member', 'Congruence.partial_object.carrier', 'Set.subset_eq', 'Ring.ring.add', 'Ring.a_inv']

lemma (in abelian_group) a_coset_add_inv2: "[| M +> x = M +> y; x \<in> carrier G; y \<in> carrier G; M \<subseteq> carrier G |] ==> M +> (x \<oplus> (\<ominus> y)) = M"

[]

['Pure.imp', 'HOL.Trueprop', 'HOL.eq', 'AbelCoset.a_r_coset', 'Set.member', 'Congruence.partial_object.carrier', 'Group.subgroup', 'Ring.add_monoid']

lemma (in abelian_group) a_coset_join1: "[| H +> x = H; x \<in> carrier G; subgroup H (add_monoid G) |] ==> x \<in> H"

[]

['Pure.imp', 'HOL.Trueprop', 'Group.subgroup', 'Ring.add_monoid', 'Set.member', 'Set.Bex']

lemma (in abelian_group) a_solve_equation: "\<lbrakk>subgroup H (add_monoid G); x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> \<exists>h\<in>H. y = h \<oplus> x"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'AbelCoset.a_r_coset', 'Congruence.partial_object.carrier', 'Group.subgroup', 'Ring.add_monoid', 'HOL.eq']

lemma (in abelian_group) a_repr_independence: "\<lbrakk> y \<in> H +> x; x \<in> carrier G; subgroup H (add_monoid G) \<rbrakk> \<Longrightarrow> H +> x = H +> y"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Congruence.partial_object.carrier', 'Group.subgroup', 'Ring.add_monoid', 'HOL.eq', 'AbelCoset.a_r_coset']

lemma (in abelian_group) a_coset_join2: "\<lbrakk>x \<in> carrier G; subgroup H (add_monoid G); x\<in>H\<rbrakk> \<Longrightarrow> H +> x = H"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.subset_eq', 'Congruence.partial_object.carrier', 'Set.member', 'AbelCoset.a_r_coset']

lemma (in abelian_monoid) a_r_coset_subset_G: "[| H \<subseteq> carrier G; x \<in> carrier G |] ==> H +> x \<subseteq> carrier G"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Set.subset_eq', 'Congruence.partial_object.carrier', 'Ring.ring.add', 'AbelCoset.a_r_coset']

lemma (in abelian_group) a_rcosI: "[| h \<in> H; H \<subseteq> carrier G; x \<in> carrier G|] ==> h \<oplus> x \<in> H +> x"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.subset_eq', 'Congruence.partial_object.carrier', 'Set.member', 'AbelCoset.a_r_coset', 'AbelCoset.A_RCOSETS']

lemma (in abelian_group) a_rcosetsI: "\<lbrakk>H \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> H +> x \<in> a_rcosets H"

[]

['Pure.imp', 'HOL.Trueprop', 'HOL.eq', 'Ring.ring.add', 'Set.member', 'Congruence.partial_object.carrier', 'Ring.a_inv']

lemma (in abelian_group) a_transpose_inv: "[| x \<oplus> y = z; x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> (\<ominus> x) \<oplus> z = y"

[]

['']

lemma (in additive_subgroup) is_additive_subgroup: shows "additive_subgroup H G"

[]

['Group.subgroup', 'Ring.add_monoid']

lemma additive_subgroupI: fixes G (structure) assumes a_subgroup: "subgroup H (add_monoid G)" shows "additive_subgroup H G"

[]

['Set.subset_eq', 'Congruence.partial_object.carrier']

lemma (in additive_subgroup) a_subset: "H \<subseteq> carrier G"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Ring.ring.add']

lemma (in additive_subgroup) a_closed [intro, simp]: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<oplus> y \<in> H"

[]

['Set.member', 'Ring.ring.zero']

lemma (in additive_subgroup) zero_closed [simp]: "\<zero> \<in> H"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Ring.a_inv']

lemma (in additive_subgroup) a_inv_closed [intro,simp]: "x \<in> H \<Longrightarrow> \<ominus> x \<in> H"

[]

['']

lemma (in abelian_subgroup) is_abelian_subgroup: shows "abelian_subgroup H G"

[]

['Coset.normal', 'Ring.add_monoid', 'Pure.all']

lemma abelian_subgroupI: assumes a_normal: "normal H (add_monoid G)" and a_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus>\<^bsub>G\<^esub> y = y \<oplus>\<^bsub>G\<^esub> x" shows "abelian_subgroup H G"

[]

['Group.comm_group', 'Ring.add_monoid', 'Group.subgroup']

lemma abelian_subgroupI2: fixes G (structure) assumes a_comm_group: "comm_group (add_monoid G)" and a_subgroup: "subgroup H (add_monoid G)" shows "abelian_subgroup H G"

[]

['Ring.abelian_group']

lemma abelian_subgroupI3: fixes G (structure) assumes "additive_subgroup H G" and "abelian_group G" shows "abelian_subgroup H G"

[]

['Set.Ball', 'Congruence.partial_object.carrier']

lemma (in abelian_subgroup) a_coset_eq: "(\<forall>x \<in> carrier G. H +> x = x <+ H)"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Congruence.partial_object.carrier', 'Ring.ring.add', 'Ring.a_inv']

lemma (in abelian_subgroup) a_inv_op_closed1: shows "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> (\<ominus> x) \<oplus> h \<oplus> x \<in> H"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Congruence.partial_object.carrier', 'Ring.ring.add', 'Ring.a_inv']

lemma (in abelian_subgroup) a_inv_op_closed2: shows "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> x \<oplus> h \<oplus> (\<ominus> x) \<in> H"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.subset_eq', 'Congruence.partial_object.carrier', 'Set.member', 'HOL.eq', 'AbelCoset.a_l_coset', 'Ring.ring.add']

lemma (in abelian_group) a_lcos_m_assoc: "\<lbrakk> M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G \<rbrakk> \<Longrightarrow> g <+ (h <+ M) = (g \<oplus> h) <+ M"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.subset_eq', 'Congruence.partial_object.carrier', 'HOL.eq', 'AbelCoset.a_l_coset', 'Ring.ring.zero']

lemma (in abelian_group) a_lcos_mult_one: "M \<subseteq> carrier G ==> \<zero> <+ M = M"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.subset_eq', 'Congruence.partial_object.carrier', 'Set.member', 'AbelCoset.a_l_coset']

lemma (in abelian_group) a_l_coset_subset_G: "\<lbrakk> H \<subseteq> carrier G; x \<in> carrier G \<rbrakk> \<Longrightarrow> x <+ H \<subseteq> carrier G"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'AbelCoset.a_l_coset', 'Congruence.partial_object.carrier', 'Group.subgroup', 'Ring.add_monoid']

lemma (in abelian_group) a_l_coset_swap: "\<lbrakk>y \<in> x <+ H; x \<in> carrier G; subgroup H (add_monoid G)\<rbrakk> \<Longrightarrow> x \<in> y <+ H"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'AbelCoset.a_l_coset', 'Congruence.partial_object.carrier', 'Group.subgroup', 'Ring.add_monoid']

lemma (in abelian_group) a_l_coset_carrier: "[| y \<in> x <+ H; x \<in> carrier G; subgroup H (add_monoid G) |] ==> y \<in> carrier G"

[]

['Set.member', 'AbelCoset.a_l_coset', 'Congruence.partial_object.carrier', 'Group.subgroup', 'Ring.add_monoid', 'Set.subset_eq']

lemma (in abelian_group) a_l_repr_imp_subset: assumes "y \<in> x <+ H" "x \<in> carrier G" "subgroup H (add_monoid G)" shows "y <+ H \<subseteq> x <+ H"

[]

['Set.member', 'AbelCoset.a_l_coset', 'Congruence.partial_object.carrier', 'Group.subgroup', 'Ring.add_monoid', 'HOL.eq']

lemma (in abelian_group) a_l_repr_independence: assumes y: "y \<in> x <+ H" and x: "x \<in> carrier G" and sb: "subgroup H (add_monoid G)" shows "x <+ H = y <+ H"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.subset_eq', 'Congruence.partial_object.carrier', 'AbelCoset.set_add']

lemma (in abelian_group) setadd_subset_G: "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G\<rbrakk> \<Longrightarrow> H <+> K \<subseteq> carrier G"

[]

['Pure.imp', 'HOL.Trueprop', 'Group.subgroup', 'Ring.add_monoid', 'HOL.eq', 'AbelCoset.set_add']

lemma (in abelian_group) subgroup_add_id: "subgroup H (add_monoid G) \<Longrightarrow> H <+> H = H"

[]

['Set.member', 'Congruence.partial_object.carrier', 'HOL.eq', 'AbelCoset.A_SET_INV', 'AbelCoset.a_r_coset', 'Ring.a_inv']

lemma (in abelian_subgroup) a_rcos_inv: assumes x: "x \<in> carrier G" shows "a_set_inv (H +> x) = H +> (\<ominus> x)"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.subset_eq', 'Congruence.partial_object.carrier', 'Set.member', 'HOL.eq', 'AbelCoset.set_add', 'AbelCoset.a_r_coset']

lemma (in abelian_group) a_setmult_rcos_assoc: "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> H <+> (K +> x) = (H <+> K) +> x"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.subset_eq', 'Congruence.partial_object.carrier', 'Set.member', 'HOL.eq', 'AbelCoset.set_add', 'AbelCoset.a_r_coset', 'AbelCoset.a_l_coset']

lemma (in abelian_group) a_rcos_assoc_lcos: "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> (H +> x) <+> K = H <+> (x <+ K)"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Congruence.partial_object.carrier', 'HOL.eq', 'AbelCoset.set_add', 'AbelCoset.a_r_coset', 'Ring.ring.add']

lemma (in abelian_subgroup) a_rcos_sum: "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> (H +> x) <+> (H +> y) = H +> (x \<oplus> y)"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'AbelCoset.A_RCOSETS', 'HOL.eq', 'AbelCoset.set_add']

lemma (in abelian_subgroup) rcosets_add_eq: "M \<in> a_rcosets H \<Longrightarrow> H <+> M = M" \<comment> \<open>generalizes \<open>subgroup_mult_id\<close>\<close>

[]

['Equiv_Relations.equiv', 'Congruence.partial_object.carrier', 'AbelCoset.a_r_congruent']

lemma (in abelian_subgroup) a_equiv_rcong: shows "equiv (carrier G) (racong H)"

[]

['Set.member', 'Congruence.partial_object.carrier', 'HOL.eq', 'AbelCoset.a_l_coset', 'Relation.Image', 'AbelCoset.a_r_congruent', 'Set.insert', 'Set.empty']

lemma (in abelian_subgroup) a_l_coset_eq_rcong: assumes a: "a \<in> carrier G" shows "a <+ H = racong H `` {a}"

[]

['Pure.imp', 'HOL.Trueprop', 'HOL.eq', 'Ring.ring.add', 'Set.member', 'Congruence.partial_object.carrier', 'Complete_Lattices.Union', 'Set.image']

lemma (in abelian_subgroup) a_rcos_equation: shows "\<lbrakk>ha \<oplus> a = h \<oplus> b; a \<in> carrier G; b \<in> carrier G; h \<in> H; ha \<in> H; hb \<in> H\<rbrakk> \<Longrightarrow> hb \<oplus> a \<in> (\<Union>h\<in>H. {h \<oplus> b})"

[]

['Set.pairwise', 'Set.disjnt', 'AbelCoset.A_RCOSETS']

lemma (in abelian_subgroup) a_rcos_disjoint: "pairwise disjnt (a_rcosets H)"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Congruence.partial_object.carrier', 'AbelCoset.a_r_coset']

lemma (in abelian_subgroup) a_rcos_self: shows "x \<in> carrier G \<Longrightarrow> x \<in> H +> x"

[]

['HOL.eq', 'Complete_Lattices.Union', 'AbelCoset.A_RCOSETS', 'Congruence.partial_object.carrier']

lemma (in abelian_subgroup) a_rcosets_part_G: shows "\<Union>(a_rcosets H) = carrier G"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'AbelCoset.A_RCOSETS', 'Set.subset_eq', 'Congruence.partial_object.carrier', 'Finite_Set.finite']

lemma (in abelian_subgroup) a_cosets_finite: "\<lbrakk>c \<in> a_rcosets H; H \<subseteq> carrier G; finite (carrier G)\<rbrakk> \<Longrightarrow> finite c"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'AbelCoset.A_RCOSETS', 'Set.subset_eq', 'Congruence.partial_object.carrier', 'Finite_Set.finite', 'HOL.eq', 'Finite_Set.card']

lemma (in abelian_group) a_card_cosets_equal: "\<lbrakk>c \<in> a_rcosets H; H \<subseteq> carrier G; finite(carrier G)\<rbrakk> \<Longrightarrow> card c = card H"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.subset_eq', 'AbelCoset.A_RCOSETS', 'Set.Pow', 'Congruence.partial_object.carrier']

lemma (in abelian_group) rcosets_subset_PowG: "additive_subgroup H G \<Longrightarrow> a_rcosets H \<subseteq> Pow(carrier G)"

[]

['Pure.imp', 'HOL.Trueprop', 'Finite_Set.finite', 'Congruence.partial_object.carrier', 'HOL.eq', 'Groups.times_class.times', 'Finite_Set.card', 'AbelCoset.A_RCOSETS', 'Coset.order']

theorem (in abelian_group) a_lagrange: "\<lbrakk>finite(carrier G); additive_subgroup H G\<rbrakk> \<Longrightarrow> card(a_rcosets H) * card(H) = order(G)"

[]

['Pure.eq', 'AbelCoset.A_FactGroup', 'Congruence.partial_object.partial_object_ext', 'AbelCoset.A_RCOSETS', 'Group.monoid.monoid_ext', 'AbelCoset.set_add', 'Product_Type.Unity']

lemma A_FactGroup_def': fixes G (structure) shows "G A_Mod H \<equiv> \<lparr>carrier = a_rcosets\<^bsub>G\<^esub> H, mult = set_add G, one = H\<rparr>"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'AbelCoset.A_RCOSETS', 'AbelCoset.set_add']

lemma (in abelian_subgroup) a_setmult_closed: "\<lbrakk>K1 \<in> a_rcosets H; K2 \<in> a_rcosets H\<rbrakk> \<Longrightarrow> K1 <+> K2 \<in> a_rcosets H"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'AbelCoset.A_RCOSETS', 'AbelCoset.A_SET_INV']

lemma (in abelian_subgroup) a_setinv_closed: "K \<in> a_rcosets H \<Longrightarrow> a_set_inv K \<in> a_rcosets H"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'AbelCoset.A_RCOSETS', 'HOL.eq', 'AbelCoset.set_add']

lemma (in abelian_subgroup) a_rcosets_assoc: "\<lbrakk>M1 \<in> a_rcosets H; M2 \<in> a_rcosets H; M3 \<in> a_rcosets H\<rbrakk> \<Longrightarrow> M1 <+> M2 <+> M3 = M1 <+> (M2 <+> M3)"

[]

['Set.member', 'AbelCoset.A_RCOSETS']

lemma (in abelian_subgroup) a_subgroup_in_rcosets: "H \<in> a_rcosets H"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'AbelCoset.A_RCOSETS', 'HOL.eq', 'AbelCoset.set_add', 'AbelCoset.A_SET_INV']

lemma (in abelian_subgroup) a_rcosets_inv_mult_group_eq: "M \<in> a_rcosets H \<Longrightarrow> a_set_inv M <+> M = H"

[]

['Group.group', 'AbelCoset.A_FactGroup']

theorem (in abelian_subgroup) a_factorgroup_is_group: "group (G A_Mod H)"

[]

['Group.comm_group', 'AbelCoset.A_FactGroup']

theorem (in abelian_subgroup) a_factorgroup_is_comm_group: "comm_group (G A_Mod H)"

[]

['HOL.eq', 'Group.monoid.mult', 'AbelCoset.A_FactGroup', 'AbelCoset.set_add']

lemma add_A_FactGroup [simp]: "X \<otimes>\<^bsub>(G A_Mod H)\<^esub> X' = X <+>\<^bsub>G\<^esub> X'"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Congruence.partial_object.carrier', 'AbelCoset.A_FactGroup', 'HOL.eq', 'Group.m_inv', 'AbelCoset.A_SET_INV']

lemma (in abelian_subgroup) a_inv_FactGroup: "X \<in> carrier (G A_Mod H) \<Longrightarrow> inv\<^bsub>G A_Mod H\<^esub> X = a_set_inv X"

[]

['Set.member', 'Group.hom', 'Ring.add_monoid', 'AbelCoset.A_FactGroup']

lemma (in abelian_subgroup) a_r_coset_hom_A_Mod: "(\<lambda>a. H +> a) \<in> hom (add_monoid G) (G A_Mod H)"

[]

['HOL.eq', 'AbelCoset.a_kernel', 'Set.Collect']

lemma a_kernel_def': "a_kernel R S h = {x \<in> carrier R. h x = \<zero>\<^bsub>S\<^esub>}"

[]

['Ring.abelian_group', 'Group.group_hom', 'Ring.add_monoid', 'AbelCoset.abelian_group_hom']

lemma abelian_group_homI: assumes "abelian_group G" assumes "abelian_group H" assumes a_group_hom: "group_hom (add_monoid G) (add_monoid H) h" shows "abelian_group_hom G H h"

[]

['AbelCoset.abelian_group_hom']

lemma (in abelian_group_hom) is_abelian_group_hom: "abelian_group_hom G H h"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Congruence.partial_object.carrier', 'HOL.eq', 'Ring.ring.add']

lemma (in abelian_group_hom) hom_add [simp]: "[| x \<in> carrier G; y \<in> carrier G |] ==> h (x \<oplus>\<^bsub>G\<^esub> y) = h x \<oplus>\<^bsub>H\<^esub> h y"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Congruence.partial_object.carrier']

lemma (in abelian_group_hom) hom_closed [simp]: "x \<in> carrier G \<Longrightarrow> h x \<in> carrier H"

[]

['Set.member', 'Ring.ring.zero', 'Congruence.partial_object.carrier']

lemma (in abelian_group_hom) zero_closed [simp]: "h \<zero> \<in> carrier H"

[]

['HOL.eq', 'Ring.ring.zero']

lemma (in abelian_group_hom) hom_zero [simp]: "h \<zero> = \<zero>\<^bsub>H\<^esub>"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Congruence.partial_object.carrier', 'Ring.a_inv']

lemma (in abelian_group_hom) a_inv_closed [simp]: "x \<in> carrier G ==> h (\<ominus>x) \<in> carrier H"

[]

['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Congruence.partial_object.carrier', 'HOL.eq', 'Ring.a_inv']

lemma (in abelian_group_hom) hom_a_inv [simp]: "x \<in> carrier G ==> h (\<ominus>x) = \<ominus>\<^bsub>H\<^esub> (h x)"

[]

['AbelCoset.a_kernel']

lemma (in abelian_group_hom) additive_subgroup_a_kernel: "additive_subgroup (a_kernel G H h) G"

[]

['AbelCoset.a_kernel']

lemma (in abelian_group_hom) abelian_subgroup_a_kernel: "abelian_subgroup (a_kernel G H h) G"

[]

['Set.member', 'Congruence.partial_object.carrier', 'AbelCoset.A_FactGroup', 'AbelCoset.a_kernel', 'HOL.not_equal', 'Set.empty']

lemma (in abelian_group_hom) A_FactGroup_nonempty: assumes X: "X \<in> carrier (G A_Mod a_kernel G H h)" shows "X \<noteq> {}"

[]

['Set.member', 'Congruence.partial_object.carrier', 'AbelCoset.A_FactGroup', 'AbelCoset.a_kernel', 'Set.the_elem', 'Set.image']

lemma (in abelian_group_hom) FactGroup_the_elem_mem: assumes X: "X \<in> carrier (G A_Mod (a_kernel G H h))" shows "the_elem (h`X) \<in> carrier H"

[]

['Set.member', 'Group.hom', 'AbelCoset.A_FactGroup', 'AbelCoset.a_kernel', 'Ring.add_monoid']

lemma (in abelian_group_hom) A_FactGroup_hom: "(\<lambda>X. the_elem (h`X)) \<in> hom (G A_Mod (a_kernel G H h)) (add_monoid H)"

[]

['Fun.inj_on', 'Congruence.partial_object.carrier', 'AbelCoset.A_FactGroup', 'AbelCoset.a_kernel']

lemma (in abelian_group_hom) A_FactGroup_inj_on: "inj_on (\<lambda>X. the_elem (h ` X)) (carrier (G A_Mod a_kernel G H h))"

[]

['HOL.eq', 'Set.image', 'Congruence.partial_object.carrier', 'AbelCoset.A_FactGroup', 'AbelCoset.a_kernel']

lemma (in abelian_group_hom) A_FactGroup_onto: assumes h: "h ` carrier G = carrier H" shows "(\<lambda>X. the_elem (h ` X)) ` carrier (G A_Mod a_kernel G H h) = carrier H"

[]

['Pure.imp', 'HOL.Trueprop', 'HOL.eq', 'Set.image', 'Congruence.partial_object.carrier', 'Set.member', 'Group.iso', 'AbelCoset.A_FactGroup', 'AbelCoset.a_kernel', 'Ring.add_monoid']

theorem (in abelian_group_hom) A_FactGroup_iso_set: "h ` carrier G = carrier H \<Longrightarrow> (\<lambda>X. the_elem (h`X)) \<in> iso (G A_Mod (a_kernel G H h)) (add_monoid H)"

[]

['Set.member', 'Congruence.partial_object.carrier']

lemma (in additive_subgroup) a_Hcarr [simp]: assumes hH: "h \<in> H" shows "h \<in> carrier G"

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['Set.member', 'Congruence.partial_object.carrier', 'AbelCoset.a_r_coset']

lemma (in abelian_subgroup) a_elemrcos_carrier: assumes acarr: "a \<in> carrier G" and a': "a' \<in> H +> a" shows "a' \<in> carrier G"

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['Set.member', 'HOL.eq', 'AbelCoset.a_r_coset']

lemma (in abelian_subgroup) a_rcos_const: assumes hH: "h \<in> H" shows "H +> h = H"

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['Set.member', 'Congruence.partial_object.carrier', 'AbelCoset.a_r_coset', 'Ring.ring.add', 'Ring.a_inv']

lemma (in abelian_subgroup) a_rcos_module_imp: assumes xcarr: "x \<in> carrier G" and x'cos: "x' \<in> H +> x" shows "(x' \<oplus> \<ominus>x) \<in> H"

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['Set.member', 'Congruence.partial_object.carrier', 'Ring.ring.add', 'Ring.a_inv', 'AbelCoset.a_r_coset']

lemma (in abelian_subgroup) a_rcos_module_rev: assumes "x \<in> carrier G" "x' \<in> carrier G" and "(x' \<oplus> \<ominus>x) \<in> H" shows "x' \<in> H +> x"

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['Set.member', 'Congruence.partial_object.carrier', 'HOL.eq', 'AbelCoset.a_r_coset', 'Ring.ring.add', 'Ring.a_inv']

lemma (in abelian_subgroup) a_rcos_module: assumes "x \<in> carrier G" "x' \<in> carrier G" shows "(x' \<in> H +> x) = (x' \<oplus> \<ominus>x \<in> H)"

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['Ring.ring', 'Set.member', 'Congruence.partial_object.carrier', 'HOL.eq', 'AbelCoset.a_r_coset', 'Ring.a_minus']

lemma (in abelian_subgroup) a_rcos_module_minus: assumes "ring G" assumes carr: "x \<in> carrier G" "x' \<in> carrier G" shows "(x' \<in> H +> x) = (x' \<ominus> x \<in> H)"

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['Set.member', 'AbelCoset.a_r_coset', 'Congruence.partial_object.carrier', 'HOL.eq']

lemma (in abelian_subgroup) a_repr_independence': assumes "y \<in> H +> x" "x \<in> carrier G" shows "H +> x = H +> y"

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['Set.member', 'Congruence.partial_object.carrier', 'HOL.eq', 'AbelCoset.a_r_coset']

lemma (in abelian_subgroup) a_repr_independenceD: assumes "y \<in> carrier G" "H +> x = H +> y" shows "y \<in> H +> x"

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['Pure.imp', 'HOL.Trueprop', 'Set.member', 'AbelCoset.A_RCOSETS', 'Set.subset_eq', 'Congruence.partial_object.carrier']

lemma (in abelian_subgroup) a_rcosets_carrier: "X \<in> a_rcosets H \<Longrightarrow> X \<subseteq> carrier G"

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['Set.subset_eq', 'Congruence.partial_object.carrier', 'AbelCoset.set_add']

lemma (in abelian_monoid) set_add_closed: assumes "A \<subseteq> carrier G" "B \<subseteq> carrier G" shows "A <+> B \<subseteq> carrier G"

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['AbelCoset.set_add']

lemma (in abelian_group) add_additive_subgroups: assumes subH: "additive_subgroup H G" and subK: "additive_subgroup K G" shows "additive_subgroup (H <+> K) G"

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['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Bij.Bij', 'FuncSet.extensional']

lemma Bij_imp_extensional: "f \<in> Bij S \<Longrightarrow> f \<in> extensional S"

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['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Bij.Bij', 'FuncSet.funcset']

lemma Bij_imp_funcset: "f \<in> Bij S \<Longrightarrow> f \<in> S \<rightarrow> S"

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['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Bij.Bij', 'FuncSet.restrict']

lemma restrict_inv_into_Bij: "f \<in> Bij S \<Longrightarrow> (\<lambda>x \<in> S. (inv_into S f) x) \<in> Bij S"

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['Set.member', 'FuncSet.restrict', 'Bij.Bij']

lemma id_Bij: "(\<lambda>x\<in>S. x) \<in> Bij S "

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['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Bij.Bij', 'FuncSet.compose']

lemma compose_Bij: "\<lbrakk>x \<in> Bij S; y \<in> Bij S\<rbrakk> \<Longrightarrow> compose S x y \<in> Bij S"

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['Pure.imp', 'HOL.Trueprop', 'Set.member', 'Bij.Bij', 'HOL.eq', 'FuncSet.compose', 'FuncSet.restrict', 'Hilbert_Choice.inv_into']

lemma Bij_compose_restrict_eq: "f \<in> Bij S \<Longrightarrow> compose S (restrict (inv_into S f) S) f = (\<lambda>x\<in>S. x)"