(* Title: HOL/Algebra/Subrings.thy
Authors: Martin Baillon and Paulo EmÃlio de Vilhena
*)
theory Subrings
imports Ring RingHom QuotRing Multiplicative_Group
begin
section \<open>Subrings\<close>
subsection \<open>Definitions\<close>
locale subring =
subgroup H "add_monoid R" + submonoid H R for H and R (structure)
locale subcring = subring +
assumes sub_m_comm: "\<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<otimes> h2 = h2 \<otimes> h1"
locale subdomain = subcring +
assumes sub_one_not_zero [simp]: "\<one> \<noteq> \<zero>"
assumes subintegral: "\<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<otimes> h2 = \<zero> \<Longrightarrow> h1 = \<zero> \<or> h2 = \<zero>"
locale subfield = subdomain K R for K and R (structure) +
assumes subfield_Units: "Units (R \<lparr> carrier := K \<rparr>) = K - { \<zero> }"
subsection \<open>Basic Properties\<close>
subsubsection \<open>Subrings\<close>
lemma (in ring) subringI:
assumes "H \<subseteq> carrier R"
and "\<one> \<in> H"
and "\<And>h. h \<in> H \<Longrightarrow> \<ominus> h \<in> H"
and "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<otimes> h2 \<in> H"
and "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<oplus> h2 \<in> H"
shows "subring H R"
using add.subgroupI[OF assms(1) _ assms(3, 5)] assms(2)
submonoid.intro[OF assms(1, 4, 2)]
unfolding subring_def by auto
lemma subringE:
assumes "subring H R"
shows "H \<subseteq> carrier R"
and "\<zero>\<^bsub>R\<^esub> \<in> H"
and "\<one>\<^bsub>R\<^esub> \<in> H"
and "H \<noteq> {}"
and "\<And>h. h \<in> H \<Longrightarrow> \<ominus>\<^bsub>R\<^esub> h \<in> H"
and "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<otimes>\<^bsub>R\<^esub> h2 \<in> H"
and "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<oplus>\<^bsub>R\<^esub> h2 \<in> H"
using subring.axioms[OF assms]
unfolding submonoid_def subgroup_def a_inv_def by auto
lemma (in ring) carrier_is_subring: "subring (carrier R) R"
by (simp add: subringI)
lemma (in ring) subring_inter:
assumes "subring I R" and "subring J R"
shows "subring (I \<inter> J) R"
using subringE[OF assms(1)] subringE[OF assms(2)] subringI[of "I \<inter> J"] by auto
lemma (in ring) subring_Inter:
assumes "\<And>I. I \<in> S \<Longrightarrow> subring I R" and "S \<noteq> {}"
shows "subring (\<Inter>S) R"
proof (rule subringI, auto simp add: assms subringE[of _ R])
fix x assume "\<forall>I \<in> S. x \<in> I" thus "x \<in> carrier R"
using assms subringE(1)[of _ R] by blast
qed
lemma (in ring) subring_is_ring:
assumes "subring H R" shows "ring (R \<lparr> carrier := H \<rparr>)"
proof -
interpret group "add_monoid (R \<lparr> carrier := H \<rparr>)" + monoid "R \<lparr> carrier := H \<rparr>"
using subgroup.subgroup_is_group[OF subring.axioms(1) add.is_group] assms
submonoid.submonoid_is_monoid[OF subring.axioms(2) monoid_axioms] by auto
show ?thesis
using subringE(1)[OF assms]
by (unfold_locales, simp_all add: subringE(1)[OF assms] add.m_comm subset_eq l_distr r_distr)
qed
lemma (in ring) ring_incl_imp_subring:
assumes "H \<subseteq> carrier R"
and "ring (R \<lparr> carrier := H \<rparr>)"
shows "subring H R"
using group.group_incl_imp_subgroup[OF add.group_axioms, of H] assms(1)
monoid.monoid_incl_imp_submonoid[OF monoid_axioms assms(1)]
ring.axioms(1, 2)[OF assms(2)] abelian_group.a_group[of "R \<lparr> carrier := H \<rparr>"]
unfolding subring_def by auto
lemma (in ring) subring_iff:
assumes "H \<subseteq> carrier R"
shows "subring H R \<longleftrightarrow> ring (R \<lparr> carrier := H \<rparr>)"
using subring_is_ring ring_incl_imp_subring[OF assms] by auto
subsubsection \<open>Subcrings\<close>
lemma (in ring) subcringI:
assumes "subring H R"
and "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<otimes> h2 = h2 \<otimes> h1"
shows "subcring H R"
unfolding subcring_def subcring_axioms_def using assms by simp+
lemma (in cring) subcringI':
assumes "subring H R"
shows "subcring H R"
using subcringI[OF assms] subringE(1)[OF assms] m_comm by auto
lemma subcringE:
assumes "subcring H R"
shows "H \<subseteq> carrier R"
and "\<zero>\<^bsub>R\<^esub> \<in> H"
and "\<one>\<^bsub>R\<^esub> \<in> H"
and "H \<noteq> {}"
and "\<And>h. h \<in> H \<Longrightarrow> \<ominus>\<^bsub>R\<^esub> h \<in> H"
and "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<otimes>\<^bsub>R\<^esub> h2 \<in> H"
and "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<oplus>\<^bsub>R\<^esub> h2 \<in> H"
and "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<otimes>\<^bsub>R\<^esub> h2 = h2 \<otimes>\<^bsub>R\<^esub> h1"
using subringE[OF subcring.axioms(1)[OF assms]] subcring.sub_m_comm[OF assms] by simp+
lemma (in cring) carrier_is_subcring: "subcring (carrier R) R"
by (simp add: subcringI' carrier_is_subring)
lemma (in ring) subcring_inter:
assumes "subcring I R" and "subcring J R"
shows "subcring (I \<inter> J) R"
using subcringE[OF assms(1)] subcringE[OF assms(2)]
subcringI[of "I \<inter> J"] subringI[of "I \<inter> J"] by auto
lemma (in ring) subcring_Inter:
assumes "\<And>I. I \<in> S \<Longrightarrow> subcring I R" and "S \<noteq> {}"
shows "subcring (\<Inter>S) R"
proof (rule subcringI)
show "subring (\<Inter>S) R"
using subcring.axioms(1)[of _ R] subring_Inter[of S] assms by auto
next
fix h1 h2 assume h1: "h1 \<in> \<Inter>S" and h2: "h2 \<in> \<Inter>S"
obtain S' where S': "S' \<in> S"
using assms(2) by blast
hence "h1 \<in> S'" "h2 \<in> S'"
using h1 h2 by blast+
thus "h1 \<otimes> h2 = h2 \<otimes> h1"
using subcring.sub_m_comm[OF assms(1)[OF S']] by simp
qed
lemma (in ring) subcring_iff:
assumes "H \<subseteq> carrier R"
shows "subcring H R \<longleftrightarrow> cring (R \<lparr> carrier := H \<rparr>)"
proof
assume A: "subcring H R"
hence ring: "ring (R \<lparr> carrier := H \<rparr>)"
using subring_iff[OF assms] subcring.axioms(1)[OF A] by simp
moreover have "comm_monoid (R \<lparr> carrier := H \<rparr>)"
using monoid.monoid_comm_monoidI[OF ring.is_monoid[OF ring]]
subcring.sub_m_comm[OF A] by auto
ultimately show "cring (R \<lparr> carrier := H \<rparr>)"
using cring_def by blast
next
assume A: "cring (R \<lparr> carrier := H \<rparr>)"
hence "subring H R"
using cring.axioms(1) subring_iff[OF assms] by simp
moreover have "comm_monoid (R \<lparr> carrier := H \<rparr>)"
using A unfolding cring_def by simp
hence"\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<otimes> h2 = h2 \<otimes> h1"
using comm_monoid.m_comm[of "R \<lparr> carrier := H \<rparr>"] by auto
ultimately show "subcring H R"
unfolding subcring_def subcring_axioms_def by auto
qed
subsubsection \<open>Subdomains\<close>
lemma (in ring) subdomainI:
assumes "subcring H R"
and "\<one> \<noteq> \<zero>"
and "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<otimes> h2 = \<zero> \<Longrightarrow> h1 = \<zero> \<or> h2 = \<zero>"
shows "subdomain H R"
unfolding subdomain_def subdomain_axioms_def using assms by simp+
lemma (in domain) subdomainI':
assumes "subring H R"
shows "subdomain H R"
proof (rule subdomainI[OF subcringI[OF assms]], simp_all)
show "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<otimes> h2 = h2 \<otimes> h1"
using m_comm subringE(1)[OF assms] by auto
show "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H; h1 \<otimes> h2 = \<zero> \<rbrakk> \<Longrightarrow> (h1 = \<zero>) \<or> (h2 = \<zero>)"
using integral subringE(1)[OF assms] by auto
qed
lemma subdomainE:
assumes "subdomain H R"
shows "H \<subseteq> carrier R"
and "\<zero>\<^bsub>R\<^esub> \<in> H"
and "\<one>\<^bsub>R\<^esub> \<in> H"
and "H \<noteq> {}"
and "\<And>h. h \<in> H \<Longrightarrow> \<ominus>\<^bsub>R\<^esub> h \<in> H"
and "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<otimes>\<^bsub>R\<^esub> h2 \<in> H"
and "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<oplus>\<^bsub>R\<^esub> h2 \<in> H"
and "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<otimes>\<^bsub>R\<^esub> h2 = h2 \<otimes>\<^bsub>R\<^esub> h1"
and "\<And>h1 h2. \<lbrakk> h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<otimes>\<^bsub>R\<^esub> h2 = \<zero>\<^bsub>R\<^esub> \<Longrightarrow> h1 = \<zero>\<^bsub>R\<^esub> \<or> h2 = \<zero>\<^bsub>R\<^esub>"
and "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>"
using subcringE[OF subdomain.axioms(1)[OF assms]] assms
unfolding subdomain_def subdomain_axioms_def by auto
lemma (in ring) subdomain_iff:
assumes "H \<subseteq> carrier R"
shows "subdomain H R \<longleftrightarrow> domain (R \<lparr> carrier := H \<rparr>)"
proof
assume A: "subdomain H R"
hence cring: "cring (R \<lparr> carrier := H \<rparr>)"
using subcring_iff[OF assms] subdomain.axioms(1)[OF A] by simp
thus "domain (R \<lparr> carrier := H \<rparr>)"
using domain.intro[OF cring] subdomain.subintegral[OF A] subdomain.sub_one_not_zero[OF A]
unfolding domain_axioms_def by auto
next
assume A: "domain (R \<lparr> carrier := H \<rparr>)"
hence subcring: "subcring H R"
using subcring_iff[OF assms] unfolding domain_def by simp
thus "subdomain H R"
using subdomain.intro[OF subcring] domain.integral[OF A] domain.one_not_zero[OF A]
unfolding subdomain_axioms_def by auto
qed
lemma (in domain) subring_is_domain:
assumes "subring H R" shows "domain (R \<lparr> carrier := H \<rparr>)"
using subdomainI'[OF assms] unfolding subdomain_iff[OF subringE(1)[OF assms]] .
(* NEW ====================== *)
lemma (in ring) subdomain_is_domain:
assumes "subdomain H R" shows "domain (R \<lparr> carrier := H \<rparr>)"
using assms unfolding subdomain_iff[OF subdomainE(1)[OF assms]] .
subsubsection \<open>Subfields\<close>
lemma (in ring) subfieldI:
assumes "subcring K R" and "Units (R \<lparr> carrier := K \<rparr>) = K - { \<zero> }"
shows "subfield K R"
proof (rule subfield.intro)
show "subfield_axioms K R"
using assms(2) unfolding subfield_axioms_def .
show "subdomain K R"
proof (rule subdomainI[OF assms(1)], auto)
have subM: "submonoid K R"
using subring.axioms(2)[OF subcring.axioms(1)[OF assms(1)]] .
show contr: "\<one> = \<zero> \<Longrightarrow> False"
proof -
assume one_eq_zero: "\<one> = \<zero>"
have "\<one> \<in> K" and "\<one> \<otimes> \<one> = \<one>"
using submonoid.one_closed[OF subM] by simp+
hence "\<one> \<in> Units (R \<lparr> carrier := K \<rparr>)"
unfolding Units_def by (simp, blast)
hence "\<one> \<noteq> \<zero>"
using assms(2) by simp
thus False
using one_eq_zero by simp
qed
fix k1 k2 assume k1: "k1 \<in> K" and k2: "k2 \<in> K" "k2 \<noteq> \<zero>" and k12: "k1 \<otimes> k2 = \<zero>"
obtain k2' where k2': "k2' \<in> K" "k2' \<otimes> k2 = \<one>" "k2 \<otimes> k2' = \<one>"
using assms(2) k2 unfolding Units_def by auto
have "\<zero> = (k1 \<otimes> k2) \<otimes> k2'"
using k12 k2'(1) submonoid.mem_carrier[OF subM] by fastforce
also have "... = k1"
using k1 k2(1) k2'(1,3) submonoid.mem_carrier[OF subM] by (simp add: m_assoc)
finally have "\<zero> = k1" .
thus "k1 = \<zero>" by simp
qed
qed
lemma (in field) subfieldI':
assumes "subring K R" and "\<And>k. k \<in> K - { \<zero> } \<Longrightarrow> inv k \<in> K"
shows "subfield K R"
proof (rule subfieldI)
show "subcring K R"
using subcringI[OF assms(1)] m_comm subringE(1)[OF assms(1)] by auto
show "Units (R \<lparr> carrier := K \<rparr>) = K - { \<zero> }"
proof
show "K - { \<zero> } \<subseteq> Units (R \<lparr> carrier := K \<rparr>)"
proof
fix k assume k: "k \<in> K - { \<zero> }"
hence inv_k: "inv k \<in> K"
using assms(2) by simp
moreover have "k \<in> carrier R - { \<zero> }"
using subringE(1)[OF assms(1)] k by auto
ultimately have "k \<otimes> inv k = \<one>" "inv k \<otimes> k = \<one>"
by (simp add: field_Units)+
thus "k \<in> Units (R \<lparr> carrier := K \<rparr>)"
unfolding Units_def using k inv_k by auto
qed
next
show "Units (R \<lparr> carrier := K \<rparr>) \<subseteq> K - { \<zero> }"
proof
fix k assume k: "k \<in> Units (R \<lparr> carrier := K \<rparr>)"
then obtain k' where k': "k' \<in> K" "k \<otimes> k' = \<one>"
unfolding Units_def by auto
hence "k \<in> carrier R" and "k' \<in> carrier R"
using k subringE(1)[OF assms(1)] unfolding Units_def by auto
hence "\<zero> = \<one>" if "k = \<zero>"
using that k'(2) by auto
thus "k \<in> K - { \<zero> }"
using k unfolding Units_def by auto
qed
qed
qed
lemma (in field) carrier_is_subfield: "subfield (carrier R) R"
by (auto intro: subfieldI[OF carrier_is_subcring] simp add: field_Units)
lemma subfieldE:
assumes "subfield K R"
shows "subring K R" and "subcring K R"
and "K \<subseteq> carrier R"
and "\<And>k1 k2. \<lbrakk> k1 \<in> K; k2 \<in> K \<rbrakk> \<Longrightarrow> k1 \<otimes>\<^bsub>R\<^esub> k2 = k2 \<otimes>\<^bsub>R\<^esub> k1"
and "\<And>k1 k2. \<lbrakk> k1 \<in> K; k2 \<in> K \<rbrakk> \<Longrightarrow> k1 \<otimes>\<^bsub>R\<^esub> k2 = \<zero>\<^bsub>R\<^esub> \<Longrightarrow> k1 = \<zero>\<^bsub>R\<^esub> \<or> k2 = \<zero>\<^bsub>R\<^esub>"
and "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>"
using subdomain.axioms(1)[OF subfield.axioms(1)[OF assms]] subcring_def
subdomainE(1, 8, 9, 10)[OF subfield.axioms(1)[OF assms]] by auto
lemma (in ring) subfield_m_inv:
assumes "subfield K R" and "k \<in> K - { \<zero> }"
shows "inv k \<in> K - { \<zero> }" and "k \<otimes> inv k = \<one>" and "inv k \<otimes> k = \<one>"
proof -
have K: "subring K R" "submonoid K R"
using subfieldE(1)[OF assms(1)] subring.axioms(2) by auto
have monoid: "monoid (R \<lparr> carrier := K \<rparr>)"
using submonoid.submonoid_is_monoid[OF subring.axioms(2)[OF K(1)] is_monoid] .
have "monoid R"
by (simp add: monoid_axioms)
hence k: "k \<in> Units (R \<lparr> carrier := K \<rparr>)"
using subfield.subfield_Units[OF assms(1)] assms(2) by blast
hence unit_of_R: "k \<in> Units R"
using assms(2) subringE(1)[OF subfieldE(1)[OF assms(1)]] unfolding Units_def by auto
have "inv\<^bsub>(R \<lparr> carrier := K \<rparr>)\<^esub> k \<in> Units (R \<lparr> carrier := K \<rparr>)"
by (simp add: k monoid monoid.Units_inv_Units)
hence "inv\<^bsub>(R \<lparr> carrier := K \<rparr>)\<^esub> k \<in> K - { \<zero> }"
using subfield.subfield_Units[OF assms(1)] by blast
thus "inv k \<in> K - { \<zero> }" and "k \<otimes> inv k = \<one>" and "inv k \<otimes> k = \<one>"
using Units_l_inv[OF unit_of_R] Units_r_inv[OF unit_of_R]
using monoid.m_inv_monoid_consistent[OF monoid_axioms k K(2)] by auto
qed
lemma (in ring) subfield_m_inv_simprule:
assumes "subfield K R"
shows "\<lbrakk> k \<in> K - { \<zero> }; a \<in> carrier R \<rbrakk> \<Longrightarrow> k \<otimes> a \<in> K \<Longrightarrow> a \<in> K"
proof -
note subring_props = subringE[OF subfieldE(1)[OF assms]]
assume A: "k \<in> K - { \<zero> }" "a \<in> carrier R" "k \<otimes> a \<in> K"
then obtain k' where k': "k' \<in> K" "k \<otimes> a = k'" by blast
have inv_k: "inv k \<in> K" "inv k \<otimes> k = \<one>"
using subfield_m_inv[OF assms A(1)] by auto
hence "inv k \<otimes> (k \<otimes> a) \<in> K"
using k' A(3) subring_props(6) by auto
thus "a \<in> K"
using m_assoc[of "inv k" k a] A(2) inv_k subring_props(1)
by (metis (no_types, opaque_lifting) A(1) Diff_iff l_one subsetCE)
qed
lemma (in ring) subfield_iff:
shows "\<lbrakk> field (R \<lparr> carrier := K \<rparr>); K \<subseteq> carrier R \<rbrakk> \<Longrightarrow> subfield K R"
and "subfield K R \<Longrightarrow> field (R \<lparr> carrier := K \<rparr>)"
proof-
assume A: "field (R \<lparr> carrier := K \<rparr>)" "K \<subseteq> carrier R"
have "\<And>k1 k2. \<lbrakk> k1 \<in> K; k2 \<in> K \<rbrakk> \<Longrightarrow> k1 \<otimes> k2 = k2 \<otimes> k1"
using comm_monoid.m_comm[OF cring.axioms(2)[OF fieldE(1)[OF A(1)]]] by simp
moreover have "subring K R"
using ring_incl_imp_subring[OF A(2) cring.axioms(1)[OF fieldE(1)[OF A(1)]]] .
ultimately have "subcring K R"
using subcringI by simp
thus "subfield K R"
using field.field_Units[OF A(1)] subfieldI by auto
next
assume A: "subfield K R"
have cring: "cring (R \<lparr> carrier := K \<rparr>)"
using subcring_iff[OF subringE(1)[OF subfieldE(1)[OF A]]] subfieldE(2)[OF A] by simp
thus "field (R \<lparr> carrier := K \<rparr>)"
using cring.cring_fieldI[OF cring] subfield.subfield_Units[OF A] by simp
qed
lemma (in field) subgroup_mult_of :
assumes "subfield K R"
shows "subgroup (K - {\<zero>}) (mult_of R)"
proof (intro group.group_incl_imp_subgroup[OF field_mult_group])
show "K - {\<zero>} \<subseteq> carrier (mult_of R)"
by (simp add: Diff_mono assms carrier_mult_of subfieldE(3))
show "group ((mult_of R) \<lparr> carrier := K - {\<zero>} \<rparr>)"
using field.field_mult_group[OF subfield_iff(2)[OF assms]]
unfolding mult_of_def by simp
qed
subsection \<open>Subring Homomorphisms\<close>
lemma (in ring) hom_imp_img_subring:
assumes "h \<in> ring_hom R S" and "subring K R"
shows "ring (S \<lparr> carrier := h ` K, one := h \<one>, zero := h \<zero> \<rparr>)"
proof -
have [simp]: "h \<one> = \<one>\<^bsub>S\<^esub>"
using assms ring_hom_one by blast
have "ring (R \<lparr> carrier := K \<rparr>)"
by (simp add: assms(2) subring_is_ring)
moreover have "h \<in> ring_hom (R \<lparr> carrier := K \<rparr>) S"
using assms subringE(1)[OF assms (2)] unfolding ring_hom_def
apply simp
apply blast
done
ultimately show ?thesis
using ring.ring_hom_imp_img_ring[of "R \<lparr> carrier := K \<rparr>" h S] by simp
qed
lemma (in ring_hom_ring) img_is_subring:
assumes "subring K R" shows "subring (h ` K) S"
proof -
have "ring (S \<lparr> carrier := h ` K \<rparr>)"
using R.hom_imp_img_subring[OF homh assms] hom_zero hom_one by simp
moreover have "h ` K \<subseteq> carrier S"
using ring_hom_memE(1)[OF homh] subringE(1)[OF assms] by auto
ultimately show ?thesis
using ring_incl_imp_subring by simp
qed
lemma (in ring_hom_ring) img_is_subfield:
assumes "subfield K R" and "\<one>\<^bsub>S\<^esub> \<noteq> \<zero>\<^bsub>S\<^esub>"
shows "inj_on h K" and "subfield (h ` K) S"
proof -
have K: "K \<subseteq> carrier R" "subring K R" "subring (h ` K) S"
using subfieldE(1)[OF assms(1)] subringE(1) img_is_subring by auto
have field: "field (R \<lparr> carrier := K \<rparr>)"
using R.subfield_iff(2) \<open>subfield K R\<close> by blast
moreover have ring: "ring (R \<lparr> carrier := K \<rparr>)"
using K R.ring_axioms R.subring_is_ring by blast
moreover have ringS: "ring (S \<lparr> carrier := h ` K \<rparr>)"
using subring_is_ring K by simp
ultimately have h: "h \<in> ring_hom (R \<lparr> carrier := K \<rparr>) (S \<lparr> carrier := h ` K \<rparr>)"
unfolding ring_hom_def apply auto
using ring_hom_memE[OF homh] K
by (meson contra_subsetD)+
hence ring_hom: "ring_hom_ring (R \<lparr> carrier := K \<rparr>) (S \<lparr> carrier := h ` K \<rparr>) h"
using ring_axioms ring ringS ring_hom_ringI2 by blast
have "h ` K \<noteq> { \<zero>\<^bsub>S\<^esub> }"
using subfieldE(1, 5)[OF assms(1)] subringE(3) assms(2)
by (metis hom_one image_eqI singletonD)
thus "inj_on h K"
using ring_hom_ring.non_trivial_field_hom_imp_inj[OF ring_hom field] by auto
hence "h \<in> ring_iso (R \<lparr> carrier := K \<rparr>) (S \<lparr> carrier := h ` K \<rparr>)"
using h unfolding ring_iso_def bij_betw_def by auto
hence "field (S \<lparr> carrier := h ` K \<rparr>)"
using field.ring_iso_imp_img_field[OF field, of h "S \<lparr> carrier := h ` K \<rparr>"] by auto
thus "subfield (h ` K) S"
using S.subfield_iff[of "h ` K"] K(1) ring_hom_memE(1)[OF homh] by blast
qed
(* NEW ========================================================================== *)
lemma (in ring_hom_ring) induced_ring_hom:
assumes "subring K R" shows "ring_hom_ring (R \<lparr> carrier := K \<rparr>) S h"
proof -
have "h \<in> ring_hom (R \<lparr> carrier := K \<rparr>) S"
using homh subringE(1)[OF assms] unfolding ring_hom_def
by (auto, meson hom_mult hom_add subsetCE)+
thus ?thesis
using R.subring_is_ring[OF assms] ring_axioms
unfolding ring_hom_ring_def ring_hom_ring_axioms_def by auto
qed
(* NEW ========================================================================== *)
lemma (in ring_hom_ring) inj_on_subgroup_iff_trivial_ker:
assumes "subring K R"
shows "inj_on h K \<longleftrightarrow> a_kernel (R \<lparr> carrier := K \<rparr>) S h = { \<zero> }"
using ring_hom_ring.inj_iff_trivial_ker[OF induced_ring_hom[OF assms]] by simp
lemma (in ring_hom_ring) inv_ring_hom:
assumes "inj_on h K" and "subring K R"
shows "ring_hom_ring (S \<lparr> carrier := h ` K \<rparr>) R (inv_into K h)"
proof (intro ring_hom_ringI[OF _ R.ring_axioms], auto)
show "ring (S \<lparr> carrier := h ` K \<rparr>)"
using subring_is_ring[OF img_is_subring[OF assms(2)]] .
next
show "inv_into K h \<one>\<^bsub>S\<^esub> = \<one>\<^bsub>R\<^esub>"
using assms(1) subringE(3)[OF assms(2)] hom_one by (simp add: inv_into_f_eq)
next
fix k1 k2
assume k1: "k1 \<in> K" and k2: "k2 \<in> K"
with \<open>k1 \<in> K\<close> show "inv_into K h (h k1) \<in> carrier R"
using assms(1) subringE(1)[OF assms(2)] by (simp add: subset_iff)
from \<open>k1 \<in> K\<close> and \<open>k2 \<in> K\<close>
have "h k1 \<oplus>\<^bsub>S\<^esub> h k2 = h (k1 \<oplus>\<^bsub>R\<^esub> k2)" and "k1 \<oplus>\<^bsub>R\<^esub> k2 \<in> K"
and "h k1 \<otimes>\<^bsub>S\<^esub> h k2 = h (k1 \<otimes>\<^bsub>R\<^esub> k2)" and "k1 \<otimes>\<^bsub>R\<^esub> k2 \<in> K"
using subringE(1,6,7)[OF assms(2)] by (simp add: subset_iff)+
thus "inv_into K h (h k1 \<oplus>\<^bsub>S\<^esub> h k2) = inv_into K h (h k1) \<oplus>\<^bsub>R\<^esub> inv_into K h (h k2)"
and "inv_into K h (h k1 \<otimes>\<^bsub>S\<^esub> h k2) = inv_into K h (h k1) \<otimes>\<^bsub>R\<^esub> inv_into K h (h k2)"
using assms(1) k1 k2 by simp+
qed
end