Theory QuotRing

theory QuotRing
imports RingHom
```(*  Title:      HOL/Algebra/QuotRing.thy
Author:     Stephan Hohe
Author:     Paulo EmÃ­lio de Vilhena
*)

theory QuotRing
imports RingHom
begin

section ‹Quotient Rings›

subsection ‹Multiplication on Cosets›

definition rcoset_mult :: "[('a, _) ring_scheme, 'a set, 'a set, 'a set] ⇒ 'a set"
("[mod _:] _ ⨂ı _" [81,81,81] 80)
where "rcoset_mult R I A B = (⋃a∈A. ⋃b∈B. I +>⇘R⇙ (a ⊗⇘R⇙ b))"

text ‹@{const "rcoset_mult"} fulfils the properties required by congruences›
assumes "x ∈ carrier R" "y ∈ carrier R"
shows "[mod I:] (I +> x) ⨂ (I +> y) = I +> (x ⊗ y)"
proof -
have 1: "z ∈ I +> x ⊗ y"
if x'rcos: "x' ∈ I +> x" and y'rcos: "y' ∈ I +> y" and zrcos: "z ∈ I +> x' ⊗ y'" for z x' y'
proof -
from that
obtain hx hy hz where hxI: "hx ∈ I" and x': "x' = hx ⊕ x" and hyI: "hy ∈ I" and y': "y' = hy ⊕ y"
and hzI: "hz ∈ I" and z: "z = hz ⊕ (x' ⊗ y')"
by (auto simp: a_r_coset_def r_coset_def)
note carr = assms hxI[THEN a_Hcarr] hyI[THEN a_Hcarr] hzI[THEN a_Hcarr]
from z  x' y' have "z = hz ⊕ ((hx ⊕ x) ⊗ (hy ⊕ y))" by simp
also from carr have "… = (hz ⊕ (hx ⊗ (hy ⊕ y)) ⊕ x ⊗ hy) ⊕ x ⊗ y" by algebra
finally have z2: "z = (hz ⊕ (hx ⊗ (hy ⊕ y)) ⊕ x ⊗ hy) ⊕ x ⊗ y" .
from hxI hyI hzI carr have "hz ⊕ (hx ⊗ (hy ⊕ y)) ⊕ x ⊗ hy ∈ I"
with z2 show ?thesis
by (auto simp add: a_r_coset_def r_coset_def)
qed
have 2: "∃a∈I +> x. ∃b∈I +> y. z ∈ I +> a ⊗ b" if "z ∈ I +> x ⊗ y" for z
using assms a_rcos_self that by blast
show ?thesis
unfolding rcoset_mult_def using assms
by (auto simp: intro!: 1 2)
qed

subsection ‹Quotient Ring Definition›

definition FactRing :: "[('a,'b) ring_scheme, 'a set] ⇒ ('a set) ring"
(infixl "Quot" 65)
where "FactRing R I =
⦇carrier = a_rcosets⇘R⇙ I, mult = rcoset_mult R I,
one = (I +>⇘R⇙ 𝟭⇘R⇙), zero = I, add = set_add R⦈"

lemmas FactRing_simps = FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric]

subsection ‹Factorization over General Ideals›

text ‹The quotient is a ring›
lemma (in ideal) quotient_is_ring: "ring (R Quot I)"
proof (rule ringI)
show "abelian_group (R Quot I)"
apply (rule comm_group_abelian_groupI)
apply (simp add: FactRing_def a_factorgroup_is_comm_group[unfolded A_FactGroup_def'])
done
show "Group.monoid (R Quot I)"
by (rule monoidI)
qed (auto simp: FactRing_simps rcoset_mult_add a_rcos_sum l_distr r_distr)

text ‹This is a ring homomorphism›

lemma (in ideal) rcos_ring_hom: "((+>) I) ∈ ring_hom R (R Quot I)"

lemma (in ideal) rcos_ring_hom_ring: "ring_hom_ring R (R Quot I) ((+>) I)"
by (simp add: local.ring_axioms quotient_is_ring rcos_ring_hom ring_hom_ringI2)

text ‹The quotient of a cring is also commutative›
lemma (in ideal) quotient_is_cring:
assumes "cring R"
shows "cring (R Quot I)"
proof -
interpret cring R by fact
show ?thesis
apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro quotient_is_ring)
apply (rule ring.axioms[OF quotient_is_ring])
done
qed

text ‹Cosets as a ring homomorphism on crings›
lemma (in ideal) rcos_ring_hom_cring:
assumes "cring R"
shows "ring_hom_cring R (R Quot I) ((+>) I)"
proof -
interpret cring R by fact
show ?thesis
apply (rule ring_hom_cringI)
apply (rule rcos_ring_hom_ring)
apply (rule is_cring)
apply (rule quotient_is_cring)
apply (rule is_cring)
done
qed

subsection ‹Factorization over Prime Ideals›

text ‹The quotient ring generated by a prime ideal is a domain›
lemma (in primeideal) quotient_is_domain: "domain (R Quot I)"
proof -
have 1: "I +> 𝟭 = I ⟹ False"
using I_notcarr a_rcos_self one_imp_carrier by blast
have 2: "I +> x = I"
if  carr: "x ∈ carrier R" "y ∈ carrier R"
and a: "I +> x ⊗ y = I"
and b: "I +> y ≠ I" for x y
by (metis I_prime a a_rcos_const a_rcos_self b m_closed that)
show ?thesis
apply (intro domain.intro quotient_is_cring is_cring domain_axioms.intro)
apply (metis "1" FactRing_def monoid.simps(2) ring.simps(1))
qed

text ‹Generating right cosets of a prime ideal is a homomorphism
on commutative rings›
lemma (in primeideal) rcos_ring_hom_cring: "ring_hom_cring R (R Quot I) ((+>) I)"
by (rule rcos_ring_hom_cring) (rule is_cring)

subsection ‹Factorization over Maximal Ideals›

text ‹In a commutative ring, the quotient ring over a maximal ideal is a field.
The proof follows ``W. Adkins, S. Weintraub: Algebra -- An Approach via Module Theory''›
proposition (in maximalideal) quotient_is_field:
assumes "cring R"
shows "field (R Quot I)"
proof -
interpret cring R by fact
have 1: "𝟬⇘R Quot I⇙ ≠ 𝟭⇘R Quot I⇙"  ― ‹Quotient is not empty›
proof
assume "𝟬⇘R Quot I⇙ = 𝟭⇘R Quot I⇙"
then have II1: "I = I +> 𝟭" by (simp add: FactRing_def)
then have "I = carrier R"
using a_rcos_self one_imp_carrier by blast
with I_notcarr show False by simp
qed
have 2: "∃y∈carrier R. I +> a ⊗ y = I +> 𝟭" if IanI: "I +> a ≠ I" and acarr: "a ∈ carrier R" for a
― ‹Existence of Inverse›
proof -
― ‹Helper ideal ‹J››
define J :: "'a set" where "J = (carrier R #> a) <+> I"
have idealJ: "ideal J R"
using J_def acarr add_ideals cgenideal_eq_rcos cgenideal_ideal is_ideal by auto
have IinJ: "I ⊆ J"
proof (clarsimp simp: J_def r_coset_def set_add_defs)
fix x
assume xI: "x ∈ I"
have "x = 𝟬 ⊗ a ⊕ x"
with xI show "∃xa∈carrier R. ∃k∈I. x = xa ⊗ a ⊕ k" by fast
qed
have JnI: "J ≠ I"
proof -
have "a ∉ I"
using IanI a_rcos_const by blast
moreover have "a ∈ J"
from acarr
have "a = 𝟭 ⊗ a ⊕ 𝟬" by algebra
show "∃x∈carrier R. ∃k∈I. a = x ⊗ a ⊕ k" by fast
qed
ultimately show ?thesis by blast
qed
then have Jcarr: "J = carrier R"
using I_maximal IinJ additive_subgroup.a_subset idealJ ideal_def by blast

― ‹Calculating an inverse for @{term "a"}›
from one_closed[folded Jcarr]
obtain r i where rcarr: "r ∈ carrier R"
and iI: "i ∈ I" and one: "𝟭 = r ⊗ a ⊕ i"

from one and rcarr and acarr and iI[THEN a_Hcarr]
have rai1: "a ⊗ r = ⊖i ⊕ 𝟭" by algebra

― ‹Lifting to cosets›
from iI have "⊖i ⊕ 𝟭 ∈ I +> 𝟭"
by (intro a_rcosI, simp, intro a_subset, simp)
with rai1 have "a ⊗ r ∈ I +> 𝟭" by simp
then have "I +> 𝟭 = I +> a ⊗ r"
by (rule a_repr_independence, simp) (rule a_subgroup)

from rcarr and this[symmetric]
show "∃r∈carrier R. I +> a ⊗ r = I +> 𝟭" by fast
qed
show ?thesis
apply (intro cring.cring_fieldI2 quotient_is_cring is_cring 1)
done
qed

lemma (in ring_hom_ring) trivial_hom_iff:
"(h ` (carrier R) = { 𝟬⇘S⇙ }) = (a_kernel R S h = carrier R)"
using group_hom.trivial_hom_iff[OF a_group_hom] by (simp add: a_kernel_def)

lemma (in ring_hom_ring) trivial_ker_imp_inj:
assumes "a_kernel R S h = { 𝟬 }"
shows "inj_on h (carrier R)"
using group_hom.trivial_ker_imp_inj[OF a_group_hom] assms a_kernel_def[of R S h] by simp

lemma (in ring_hom_ring) non_trivial_field_hom_imp_inj:
assumes "field R"
shows "h ` (carrier R) ≠ { 𝟬⇘S⇙ } ⟹ inj_on h (carrier R)"
proof -
assume "h ` (carrier R) ≠ { 𝟬⇘S⇙ }"
hence "a_kernel R S h ≠ carrier R"
using trivial_hom_iff by linarith
hence "a_kernel R S h = { 𝟬 }"
using field.all_ideals[OF assms] kernel_is_ideal by blast
thus "inj_on h (carrier R)"
using trivial_ker_imp_inj by blast
qed

shows "subgroup (h ` H) (add_monoid S)"
proof -
have "group ((add_monoid R) ⦇ carrier := H ⦈)"
hence "h ∈ hom ((add_monoid R) ⦇ carrier := H ⦈) (add_monoid S)"
unfolding hom_def by (auto simp add: subsetD)
ultimately have "subgroup (h ` carrier ((add_monoid R) ⦇ carrier := H ⦈)) (add_monoid S)"
using group_hom.img_is_subgroup[of "(add_monoid R) ⦇ carrier := H ⦈" "add_monoid S" h]
using a_group_hom group_hom_axioms.intro group_hom_def by blast
thus "subgroup (h ` H) (add_monoid S)" by simp
qed

lemma (in ring) ring_ideal_imp_quot_ideal:
assumes "ideal I R"
shows "ideal J R ⟹ ideal ((+>) I ` J) (R Quot I)"
proof -
assume A: "ideal J R" show "ideal (((+>) I) ` J) (R Quot I)"
proof (rule idealI)
show "ring (R Quot I)"
next
moreover have "((+>) I) ∈ ring_hom R (R Quot I)"
ultimately show "subgroup ((+>) I ` J) (add_monoid (R Quot I))"
using assms(1) ideal.rcos_ring_hom_ring ring_hom_ring.img_is_add_subgroup by blast
next
fix a x assume "a ∈ (+>) I ` J" "x ∈ carrier (R Quot I)"
then obtain i j where i: "i ∈ carrier R" "x = I +> i"
and j: "j ∈ J" "a = I +> j"
unfolding FactRing_def using A_RCOSETS_def'[of R I] by auto
hence "a ⊗⇘R Quot I⇙ x = [mod I:] (I +> j) ⨂ (I +> i)"
unfolding FactRing_def by simp
hence "a ⊗⇘R Quot I⇙ x = I +> (j ⊗ i)"
using ideal.rcoset_mult_add[OF assms(1), of j i] i(1) j(1) A ideal.Icarr by force
thus "a ⊗⇘R Quot I⇙ x ∈ (+>) I ` J"
using A i(1) j(1) by (simp add: ideal.I_r_closed)

have "x ⊗⇘R Quot I⇙ a = [mod I:] (I +> i) ⨂ (I +> j)"
unfolding FactRing_def i j by simp
hence "x ⊗⇘R Quot I⇙ a = I +> (i ⊗ j)"
using ideal.rcoset_mult_add[OF assms(1), of i j] i(1) j(1) A ideal.Icarr by force
thus "x ⊗⇘R Quot I⇙ a ∈ (+>) I ` J"
using A i(1) j(1) by (simp add: ideal.I_l_closed)
qed
qed

lemma (in ring_hom_ring) ideal_vimage:
assumes "ideal I S"
shows "ideal { r ∈ carrier R. h r ∈ I } R" (* or (carrier R) ∩ (h -` I) *)
proof
show "{ r ∈ carrier R. h r ∈ I } ⊆ carrier (add_monoid R)" by auto
next
show "𝟭⇘add_monoid R⇙ ∈ { r ∈ carrier R. h r ∈ I }"
next
fix a b
assume "a ∈ { r ∈ carrier R. h r ∈ I }"
and "b ∈ { r ∈ carrier R. h r ∈ I }"
hence a: "a ∈ carrier R" "h a ∈ I"
and b: "b ∈ carrier R" "h b ∈ I" by auto
hence "h (a ⊕ b) = (h a) ⊕⇘S⇙ (h b)" using hom_add by blast
moreover have "(h a) ⊕⇘S⇙ (h b) ∈ I" using a b assms
ultimately show "a ⊗⇘add_monoid R⇙ b ∈ { r ∈ carrier R. h r ∈ I }"
using a(1) b (1) by auto

have "h (⊖ a) = ⊖⇘S⇙ (h a)" by (simp add: a)
moreover have "⊖⇘S⇙ (h a) ∈ I"
ultimately show "inv⇘add_monoid R⇙ a ∈ { r ∈ carrier R. h r ∈ I }"
using a by (simp add: a_inv_def)
next
fix a r
assume "a ∈ { r ∈ carrier R. h r ∈ I }" and r: "r ∈ carrier R"
hence a: "a ∈ carrier R" "h a ∈ I" by auto

have "h a ⊗⇘S⇙ h r ∈ I"
using assms a r by (simp add: ideal.I_r_closed)
thus "a ⊗ r ∈ { r ∈ carrier R. h r ∈ I }" by (simp add: a(1) r)

have "h r ⊗⇘S⇙ h a ∈ I"
using assms a r by (simp add: ideal.I_l_closed)
thus "r ⊗ a ∈ { r ∈ carrier R. h r ∈ I }" by (simp add: a(1) r)
qed

lemma (in ring) canonical_proj_vimage_in_carrier:
assumes "ideal I R"
shows "J ⊆ carrier (R Quot I) ⟹ ⋃ J ⊆ carrier R"
proof -
assume A: "J ⊆ carrier (R Quot I)" show "⋃ J ⊆ carrier R"
proof
fix j assume j: "j ∈ ⋃ J"
then obtain j' where j': "j' ∈ J" "j ∈ j'" by blast
then obtain r where r: "r ∈ carrier R" "j' = I +> r"
using A j' unfolding FactRing_def using A_RCOSETS_def'[of R I] by auto
thus "j ∈ carrier R" using j' assms
by (meson a_r_coset_subset_G additive_subgroup.a_subset contra_subsetD ideal.axioms(1))
qed
qed

lemma (in ring) canonical_proj_vimage_mem_iff:
assumes "ideal I R" "J ⊆ carrier (R Quot I)"
shows "⋀a. a ∈ carrier R ⟹ (a ∈ (⋃ J)) = (I +> a ∈ J)"
proof -
fix a assume a: "a ∈ carrier R" show "(a ∈ (⋃ J)) = (I +> a ∈ J)"
proof
assume "a ∈ ⋃ J"
then obtain j where j: "j ∈ J" "a ∈ j" by blast
then obtain r where r: "r ∈ carrier R" "j = I +> r"
using assms j unfolding FactRing_def using A_RCOSETS_def'[of R I] by auto
hence "I +> r = I +> a"
using add.repr_independence[of a I r] j r
by (metis a_r_coset_def additive_subgroup.a_subgroup assms(1) ideal.axioms(1))
thus "I +> a ∈ J" using r j by simp
next
assume "I +> a ∈ J"
hence "𝟬 ⊕ a ∈ I +> a"
a_r_coset_def'[of R I a] by blast
thus "a ∈ ⋃ J" using a ‹I +> a ∈ J› by auto
qed
qed

corollary (in ring) quot_ideal_imp_ring_ideal:
assumes "ideal I R"
shows "ideal J (R Quot I) ⟹ ideal (⋃ J) R"
proof -
assume A: "ideal J (R Quot I)"
have "⋃ J = { r ∈ carrier R. I +> r ∈ J }"
using canonical_proj_vimage_in_carrier[OF assms, of J]
canonical_proj_vimage_mem_iff[OF assms, of J]
thus "ideal (⋃ J) R"
using ring_hom_ring.ideal_vimage[OF ideal.rcos_ring_hom_ring[OF assms] A] by simp
qed

lemma (in ring) ideal_incl_iff:
assumes "ideal I R" "ideal J R"
shows "(I ⊆ J) = (J = (⋃ j ∈ J. I +> j))"
proof
assume A: "J = (⋃ j ∈ J. I +> j)" hence "I +> 𝟬 ⊆ J"
using additive_subgroup.zero_closed[OF ideal.axioms(1)[OF assms(2)]] by blast
thus "I ⊆ J" using additive_subgroup.a_subset[OF ideal.axioms(1)[OF assms(1)]] by simp
next
assume A: "I ⊆ J" show "J = (⋃j∈J. I +> j)"
proof
show "J ⊆ (⋃ j ∈ J. I +> j)"
proof
fix j assume j: "j ∈ J"
hence "𝟬 ⊕ j ∈ I +> j"
using a_r_coset_def'[of R I j] by blast
thus "j ∈ (⋃j∈J. I +> j)"
using assms(2) j additive_subgroup.a_Hcarr ideal.axioms(1) by fastforce
qed
next
show "(⋃ j ∈ J. I +> j) ⊆ J"
proof
fix x assume "x ∈ (⋃ j ∈ J. I +> j)"
then obtain j where j: "j ∈ J" "x ∈ I +> j" by blast
then obtain i where i: "i ∈ I" "x = i ⊕ j"
using a_r_coset_def'[of R I j] by blast
thus "x ∈ J"
using assms(2) j A additive_subgroup.a_closed[OF ideal.axioms(1)[OF assms(2)]] by blast
qed
qed
qed

theorem (in ring) quot_ideal_correspondence:
assumes "ideal I R"
shows "bij_betw (λJ. (+>) I ` J) { J. ideal J R ∧ I ⊆ J } { J . ideal J (R Quot I) }"
proof (rule bij_betw_byWitness[where ?f' = "λX. ⋃ X"])
show "∀J ∈ { J. ideal J R ∧ I ⊆ J }. (λX. ⋃ X) ((+>) I ` J) = J"
using assms ideal_incl_iff by blast
next
show "(λJ. (+>) I ` J) ` { J. ideal J R ∧ I ⊆ J } ⊆ { J. ideal J (R Quot I) }"
using assms ring_ideal_imp_quot_ideal by auto
next
show "(λX. ⋃ X) ` { J. ideal J (R Quot I) } ⊆ { J. ideal J R ∧ I ⊆ J }"
proof
fix J assume "J ∈ ((λX. ⋃ X) ` { J. ideal J (R Quot I) })"
then obtain J' where J': "ideal J' (R Quot I)" "J = ⋃ J'" by blast
hence "ideal J R"
using assms quot_ideal_imp_ring_ideal by auto
moreover have "I ∈ J'"
using additive_subgroup.zero_closed[OF ideal.axioms(1)[OF J'(1)]] unfolding FactRing_def by simp
ultimately show "J ∈ { J. ideal J R ∧ I ⊆ J }" using J'(2) by auto
qed
next
show "∀J' ∈ { J. ideal J (R Quot I) }. ((+>) I ` (⋃ J')) = J'"
proof
fix J' assume "J' ∈ { J. ideal J (R Quot I) }"
hence subset: "J' ⊆ carrier (R Quot I) ∧ ideal J' (R Quot I)"
hence "((+>) I ` (⋃ J')) ⊆ J'"
using canonical_proj_vimage_in_carrier canonical_proj_vimage_mem_iff
by (meson assms contra_subsetD image_subsetI)
moreover have "J' ⊆ ((+>) I ` (⋃ J'))"
proof
fix x assume "x ∈ J'"
then obtain r where r: "r ∈ carrier R" "x = I +> r"
using subset unfolding FactRing_def A_RCOSETS_def'[of R I] by auto
hence "r ∈ (⋃ J')"
using ‹x ∈ J'› assms canonical_proj_vimage_mem_iff subset by blast
thus "x ∈ ((+>) I ` (⋃ J'))" using r(2) by blast
qed
ultimately show "((+>) I ` (⋃ J')) = J'" by blast
qed
qed

lemma (in cring) quot_domain_imp_primeideal:
assumes "ideal P R"
shows "domain (R Quot P) ⟹ primeideal P R"
proof -
assume A: "domain (R Quot P)" show "primeideal P R"
proof (rule primeidealI)
show "ideal P R" using assms .
show "cring R" using is_cring .
next
show "carrier R ≠ P"
proof (rule ccontr)
assume "¬ carrier R ≠ P" hence "carrier R = P" by simp
hence "⋀I. I ∈ carrier (R Quot P) ⟹ I = P"
unfolding FactRing_def A_RCOSETS_def' apply simp
using a_coset_join2 additive_subgroup.a_subgroup assms ideal.axioms(1) by blast
hence "𝟭⇘(R Quot P)⇙ = 𝟬⇘(R Quot P)⇙"
by (metis assms ideal.quotient_is_ring ring.ring_simprules(2) ring.ring_simprules(6))
thus False using domain.one_not_zero[OF A] by simp
qed
next
fix a b assume a: "a ∈ carrier R" and b: "b ∈ carrier R" and ab: "a ⊗ b ∈ P"
hence "P +> (a ⊗ b) = 𝟬⇘(R Quot P)⇙" unfolding FactRing_def
moreover have "(P +> a) ⊗⇘(R Quot P)⇙ (P +> b) = P +> (a ⊗ b)" unfolding FactRing_def
moreover have "P +> a ∈ carrier (R Quot P) ∧ P +> b ∈ carrier (R Quot P)"
ultimately have "P +> a = 𝟬⇘(R Quot P)⇙ ∨ P +> b = 𝟬⇘(R Quot P)⇙"
using domain.integral[OF A, of "P +> a" "P +> b"] by auto
thus "a ∈ P ∨ b ∈ P" unfolding FactRing_def apply simp
using a b assms a_coset_join1 additive_subgroup.a_subgroup ideal.axioms(1) by blast
qed
qed

lemma (in cring) quot_domain_iff_primeideal:
assumes "ideal P R"
shows "domain (R Quot P) = primeideal P R"
using quot_domain_imp_primeideal[OF assms] primeideal.quotient_is_domain[of P R] by auto

subsection ‹Isomorphism›

definition
ring_iso :: "_ ⇒ _ ⇒ ('a ⇒ 'b) set"
where "ring_iso R S = { h. h ∈ ring_hom R S ∧ bij_betw h (carrier R) (carrier S) }"

definition
is_ring_iso :: "_ ⇒ _ ⇒ bool" (infixr "≃" 60)
where "R ≃ S = (ring_iso R S ≠ {})"

definition
morphic_prop :: "_ ⇒ ('a ⇒ bool) ⇒ bool"
where "morphic_prop R P =
((P 𝟭⇘R⇙) ∧
(∀r ∈ carrier R. P r) ∧
(∀r1 ∈ carrier R. ∀r2 ∈ carrier R. P (r1 ⊗⇘R⇙ r2)) ∧
(∀r1 ∈ carrier R. ∀r2 ∈ carrier R. P (r1 ⊕⇘R⇙ r2)))"

lemma ring_iso_memI:
fixes R (structure) and S (structure)
assumes "⋀x. x ∈ carrier R ⟹ h x ∈ carrier S"
and "⋀x y. ⟦ x ∈ carrier R; y ∈ carrier R ⟧ ⟹ h (x ⊗ y) = h x ⊗⇘S⇙ h y"
and "⋀x y. ⟦ x ∈ carrier R; y ∈ carrier R ⟧ ⟹ h (x ⊕ y) = h x ⊕⇘S⇙ h y"
and "h 𝟭 = 𝟭⇘S⇙"
and "bij_betw h (carrier R) (carrier S)"
shows "h ∈ ring_iso R S"
by (auto simp add: ring_hom_memI assms ring_iso_def)

lemma ring_iso_memE:
fixes R (structure) and S (structure)
assumes "h ∈ ring_iso R S"
shows "⋀x. x ∈ carrier R ⟹ h x ∈ carrier S"
and "⋀x y. ⟦ x ∈ carrier R; y ∈ carrier R ⟧ ⟹ h (x ⊗ y) = h x ⊗⇘S⇙ h y"
and "⋀x y. ⟦ x ∈ carrier R; y ∈ carrier R ⟧ ⟹ h (x ⊕ y) = h x ⊕⇘S⇙ h y"
and "h 𝟭 = 𝟭⇘S⇙"
and "bij_betw h (carrier R) (carrier S)"
using assms unfolding ring_iso_def ring_hom_def by auto

lemma morphic_propI:
fixes R (structure)
assumes "P 𝟭"
and "⋀r. r ∈ carrier R ⟹ P r"
and "⋀r1 r2. ⟦ r1 ∈ carrier R; r2 ∈ carrier R ⟧ ⟹ P (r1 ⊗ r2)"
and "⋀r1 r2. ⟦ r1 ∈ carrier R; r2 ∈ carrier R ⟧ ⟹ P (r1 ⊕ r2)"
shows "morphic_prop R P"
unfolding morphic_prop_def using assms by auto

lemma morphic_propE:
fixes R (structure)
assumes "morphic_prop R P"
shows "P 𝟭"
and "⋀r. r ∈ carrier R ⟹ P r"
and "⋀r1 r2. ⟦ r1 ∈ carrier R; r2 ∈ carrier R ⟧ ⟹ P (r1 ⊗ r2)"
and "⋀r1 r2. ⟦ r1 ∈ carrier R; r2 ∈ carrier R ⟧ ⟹ P (r1 ⊕ r2)"
using assms unfolding morphic_prop_def by auto

lemma ring_iso_restrict:
assumes "f ∈ ring_iso R S"
and "⋀r. r ∈ carrier R ⟹ f r = g r"
and "ring R"
shows "g ∈ ring_iso R S"
proof (rule ring_iso_memI)
show "bij_betw g (carrier R) (carrier S)"
using assms(1-2) bij_betw_cong ring_iso_memE(5) by blast
show "g 𝟭⇘R⇙ = 𝟭⇘S⇙"
using assms ring.ring_simprules(6) ring_iso_memE(4) by force
next
fix x y assume x: "x ∈ carrier R" and y: "y ∈ carrier R"
show "g x ∈ carrier S"
using assms(1-2) ring_iso_memE(1) x by fastforce
show "g (x ⊗⇘R⇙ y) = g x ⊗⇘S⇙ g y"
by (metis assms ring.ring_simprules(5) ring_iso_memE(2) x y)
show "g (x ⊕⇘R⇙ y) = g x ⊕⇘S⇙ g y"
by (metis assms ring.ring_simprules(1) ring_iso_memE(3) x y)
qed

lemma ring_iso_morphic_prop:
assumes "f ∈ ring_iso R S"
and "morphic_prop R P"
and "⋀r. P r ⟹ f r = g r"
shows "g ∈ ring_iso R S"
proof -
have eq0: "⋀r. r ∈ carrier R ⟹ f r = g r"
and eq1: "f 𝟭⇘R⇙ = g 𝟭⇘R⇙"
and eq2: "⋀r1 r2. ⟦ r1 ∈ carrier R; r2 ∈ carrier R ⟧ ⟹ f (r1 ⊗⇘R⇙ r2) = g (r1 ⊗⇘R⇙ r2)"
and eq3: "⋀r1 r2. ⟦ r1 ∈ carrier R; r2 ∈ carrier R ⟧ ⟹ f (r1 ⊕⇘R⇙ r2) = g (r1 ⊕⇘R⇙ r2)"
using assms(2-3) unfolding morphic_prop_def by auto
show ?thesis
apply (rule ring_iso_memI)
using assms(1) eq0 ring_iso_memE(1) apply fastforce
apply (metis assms(1) eq0 eq2 ring_iso_memE(2))
apply (metis assms(1) eq0 eq3 ring_iso_memE(3))
using assms(1) eq1 ring_iso_memE(4) apply fastforce
using assms(1) bij_betw_cong eq0 ring_iso_memE(5) by blast
qed

lemma (in ring) ring_hom_imp_img_ring:
assumes "h ∈ ring_hom R S"
shows "ring (S ⦇ carrier := h ` (carrier R), one := h 𝟭, zero := h 𝟬 ⦈)" (is "ring ?h_img")
proof -
using assms unfolding hom_def ring_hom_def by auto
hence "comm_group ((add_monoid S) ⦇  carrier := h ` (carrier R), one := h 𝟬 ⦈)"
by (auto intro: comm_monoidI simp add: monoid.defs)

moreover have "h ∈ hom R S"
using assms unfolding ring_hom_def hom_def by auto
hence "monoid (S ⦇  carrier := h ` (carrier R), one := h 𝟭 ⦈)"
using hom_imp_img_monoid[of h S] by simp
hence monoid: "monoid ?h_img"
unfolding monoid_def by (simp add: monoid.defs)

show ?thesis
proof (rule ringI, simp_all add: comm_group_abelian_groupI[OF comm_group] monoid)
fix x y z assume "x ∈ h ` carrier R" "y ∈ h ` carrier R" "z ∈ h ` carrier R"
then obtain r1 r2 r3
where r1: "r1 ∈ carrier R" "x = h r1"
and r2: "r2 ∈ carrier R" "y = h r2"
and r3: "r3 ∈ carrier R" "z = h r3" by blast
hence "(x ⊕⇘S⇙ y) ⊗⇘S⇙ z = h ((r1 ⊕ r2) ⊗ r3)"
using ring_hom_memE[OF assms] by auto
also have " ... = h ((r1 ⊗ r3) ⊕ (r2 ⊗ r3))"
using l_distr[OF r1(1) r2(1) r3(1)] by simp
also have " ... = (x ⊗⇘S⇙ z) ⊕⇘S⇙ (y ⊗⇘S⇙ z)"
using ring_hom_memE[OF assms] r1 r2 r3 by auto
finally show "(x ⊕⇘S⇙ y) ⊗⇘S⇙ z = (x ⊗⇘S⇙ z) ⊕⇘S⇙ (y ⊗⇘S⇙ z)" .

have "z ⊗⇘S⇙ (x ⊕⇘S⇙ y) = h (r3 ⊗ (r1 ⊕ r2))"
using ring_hom_memE[OF assms] r1 r2 r3 by auto
also have " ... =  h ((r3 ⊗ r1) ⊕ (r3 ⊗ r2))"
using r_distr[OF r1(1) r2(1) r3(1)] by simp
also have " ... = (z ⊗⇘S⇙ x) ⊕⇘S⇙ (z ⊗⇘S⇙ y)"
using ring_hom_memE[OF assms] r1 r2 r3 by auto
finally show "z ⊗⇘S⇙ (x ⊕⇘S⇙ y) = (z ⊗⇘S⇙ x) ⊕⇘S⇙ (z ⊗⇘S⇙ y)" .
qed
qed

lemma (in ring) ring_iso_imp_img_ring:
assumes "h ∈ ring_iso R S"
shows "ring (S ⦇ one := h 𝟭, zero := h 𝟬 ⦈)"
proof -
have "ring (S ⦇ carrier := h ` (carrier R), one := h 𝟭, zero := h 𝟬 ⦈)"
using ring_hom_imp_img_ring[of h S] assms unfolding ring_iso_def by auto
moreover have "h ` (carrier R) = carrier S"
using assms unfolding ring_iso_def bij_betw_def by auto
ultimately show ?thesis by simp
qed

lemma (in cring) ring_iso_imp_img_cring:
assumes "h ∈ ring_iso R S"
shows "cring (S ⦇ one := h 𝟭, zero := h 𝟬 ⦈)" (is "cring ?h_img")
proof -
note m_comm
interpret h_img?: ring ?h_img
using ring_iso_imp_img_ring[OF assms] .
show ?thesis
proof (unfold_locales)
fix x y assume "x ∈ carrier ?h_img" "y ∈ carrier ?h_img"
then obtain r1 r2
where r1: "r1 ∈ carrier R" "x = h r1"
and r2: "r2 ∈ carrier R" "y = h r2"
using assms image_iff[where ?f = h and ?A = "carrier R"]
unfolding ring_iso_def bij_betw_def by auto
have "x ⊗⇘(?h_img)⇙ y = h (r1 ⊗ r2)"
using assms r1 r2 unfolding ring_iso_def ring_hom_def by auto
also have " ... = h (r2 ⊗ r1)"
using m_comm[OF r1(1) r2(1)] by simp
also have " ... = y ⊗⇘(?h_img)⇙ x"
using assms r1 r2 unfolding ring_iso_def ring_hom_def by auto
finally show "x ⊗⇘(?h_img)⇙ y = y ⊗⇘(?h_img)⇙ x" .
qed
qed

lemma (in domain) ring_iso_imp_img_domain:
assumes "h ∈ ring_iso R S"
shows "domain (S ⦇ one := h 𝟭, zero := h 𝟬 ⦈)" (is "domain ?h_img")
proof -
note aux = m_closed integral one_not_zero one_closed zero_closed
interpret h_img?: cring ?h_img
using ring_iso_imp_img_cring[OF assms] .
show ?thesis
proof (unfold_locales)
show "𝟭⇘?h_img⇙ ≠ 𝟬⇘?h_img⇙"
using ring_iso_memE(5)[OF assms] aux(3-4)
unfolding bij_betw_def inj_on_def by force
next
fix a b
assume A: "a ⊗⇘?h_img⇙ b = 𝟬⇘?h_img⇙" "a ∈ carrier ?h_img" "b ∈ carrier ?h_img"
then obtain r1 r2
where r1: "r1 ∈ carrier R" "a = h r1"
and r2: "r2 ∈ carrier R" "b = h r2"
using assms image_iff[where ?f = h and ?A = "carrier R"]
unfolding ring_iso_def bij_betw_def by auto
hence "a ⊗⇘?h_img⇙ b = h (r1 ⊗ r2)"
using assms r1 r2 unfolding ring_iso_def ring_hom_def by auto
hence "h (r1 ⊗ r2) = h 𝟬"
using A(1) by simp
hence "r1 ⊗ r2 = 𝟬"
using ring_iso_memE(5)[OF assms] aux(1)[OF r1(1) r2(1)] aux(5)
unfolding bij_betw_def inj_on_def by force
hence "r1 = 𝟬 ∨ r2 = 𝟬"
using aux(2)[OF _ r1(1) r2(1)] by simp
thus "a = 𝟬⇘?h_img⇙ ∨ b = 𝟬⇘?h_img⇙"
unfolding r1 r2 by auto
qed
qed

lemma (in field) ring_iso_imp_img_field:
assumes "h ∈ ring_iso R S"
shows "field (S ⦇ one := h 𝟭, zero := h 𝟬 ⦈)" (is "field ?h_img")
proof -
interpret h_img?: domain ?h_img
using ring_iso_imp_img_domain[OF assms] .
show ?thesis
proof (unfold_locales, auto simp add: Units_def)
interpret field R using field_axioms .
fix a assume a: "a ∈ carrier S" "a ⊗⇘S⇙ h 𝟬 = h 𝟭"
then obtain r where r: "r ∈ carrier R" "a = h r"
using assms image_iff[where ?f = h and ?A = "carrier R"]
unfolding ring_iso_def bij_betw_def by auto
have "a ⊗⇘S⇙ h 𝟬 = h (r ⊗ 𝟬)" unfolding r(2)
using ring_iso_memE(2)[OF assms r(1)] by simp
hence "h 𝟭 = h 𝟬"
using r(1) a(2) by simp
thus False
using ring_iso_memE(5)[OF assms]
unfolding bij_betw_def inj_on_def by force
next
interpret field R using field_axioms .
fix s assume s: "s ∈ carrier S" "s ≠ h 𝟬"
then obtain r where r: "r ∈ carrier R" "s = h r"
using assms image_iff[where ?f = h and ?A = "carrier R"]
unfolding ring_iso_def bij_betw_def by auto
hence "r ≠ 𝟬" using s(2) by auto
hence inv_r: "inv r ∈ carrier R" "inv r ≠ 𝟬" "r ⊗ inv r = 𝟭" "inv r ⊗ r = 𝟭"
using field_Units r(1) by auto
have "h (inv r) ⊗⇘S⇙ h r = h 𝟭" and "h r ⊗⇘S⇙ h (inv r) = h 𝟭"
using ring_iso_memE(2)[OF assms inv_r(1) r(1)] inv_r(3-4)
ring_iso_memE(2)[OF assms r(1) inv_r(1)] by auto
thus "∃s' ∈ carrier S. s' ⊗⇘S⇙ s = h 𝟭 ∧ s ⊗⇘S⇙ s' = h 𝟭"
using ring_iso_memE(1)[OF assms inv_r(1)] r(2) by auto
qed
qed

lemma ring_iso_same_card: "R ≃ S ⟹ card (carrier R) = card (carrier S)"
proof -
assume "R ≃ S"
then obtain h where "bij_betw h (carrier R) (carrier S)"
unfolding is_ring_iso_def ring_iso_def by auto
thus "card (carrier R) = card (carrier S)"
using bij_betw_same_card[of h "carrier R" "carrier S"] by simp
qed

lemma ring_iso_set_refl: "id ∈ ring_iso R R"
by (rule ring_iso_memI) (auto)

corollary ring_iso_refl: "R ≃ R"
using is_ring_iso_def ring_iso_set_refl by auto

lemma ring_iso_set_trans:
"⟦ f ∈ ring_iso R S; g ∈ ring_iso S Q ⟧ ⟹ (g ∘ f) ∈ ring_iso R Q"
unfolding ring_iso_def using bij_betw_trans ring_hom_trans by fastforce

corollary ring_iso_trans: "⟦ R ≃ S; S ≃ Q ⟧ ⟹ R ≃ Q"
using ring_iso_set_trans unfolding is_ring_iso_def by blast

lemma ring_iso_set_sym:
assumes "ring R" and h: "h ∈ ring_iso R S"
shows "(inv_into (carrier R) h) ∈ ring_iso S R"
proof -
have h_hom: "h ∈ ring_hom R S"
and h_surj: "h ` (carrier R) = (carrier S)"
and h_inj:  "⋀ x1 x2. ⟦ x1 ∈ carrier R; x2 ∈ carrier R ⟧ ⟹  h x1 = h x2 ⟹ x1 = x2"
using h unfolding ring_iso_def bij_betw_def inj_on_def by auto

have h_inv_bij: "bij_betw (inv_into (carrier R) h) (carrier S) (carrier R)"
using bij_betw_inv_into h ring_iso_def by fastforce

show "inv_into (carrier R) h ∈ ring_iso S R"
apply (rule ring_iso_memI)
using ring_iso_memE [OF h] bij_betwE [OF h_inv_bij]
apply (simp_all add: ‹ring R› bij_betw_def bij_betw_inv_into_right inv_into_f_eq ring.ring_simprules(5))
using ring_iso_memE [OF h] bij_betw_inv_into_right [of h "carrier R" "carrier S"]
apply (simp add: ‹ring R› inv_into_f_eq ring.ring_simprules(1))
by (simp add: ‹ring R› inv_into_f_eq ring.ring_simprules(6))
qed

corollary ring_iso_sym:
assumes "ring R"
shows "R ≃ S ⟹ S ≃ R"
using assms ring_iso_set_sym unfolding is_ring_iso_def by auto

lemma (in ring_hom_ring) the_elem_simp [simp]:
"⋀x. x ∈ carrier R ⟹ the_elem (h ` ((a_kernel R S h) +> x)) = h x"
proof -
fix x assume x: "x ∈ carrier R"
hence "h x ∈ h ` ((a_kernel R S h) +> x)"
using homeq_imp_rcos by blast
thus "the_elem (h ` ((a_kernel R S h) +> x)) = h x"
by (metis (no_types, lifting) x empty_iff homeq_imp_rcos rcos_imp_homeq the_elem_image_unique)
qed

lemma (in ring_hom_ring) the_elem_inj:
"⋀X Y. ⟦ X ∈ carrier (R Quot (a_kernel R S h)); Y ∈ carrier (R Quot (a_kernel R S h)) ⟧ ⟹
the_elem (h ` X) = the_elem (h ` Y) ⟹ X = Y"
proof -
fix X Y
assume "X ∈ carrier (R Quot (a_kernel R S h))"
and "Y ∈ carrier (R Quot (a_kernel R S h))"
and Eq: "the_elem (h ` X) = the_elem (h ` Y)"
then obtain x y where x: "x ∈ carrier R" "X = (a_kernel R S h) +> x"
and y: "y ∈ carrier R" "Y = (a_kernel R S h) +> y"
unfolding FactRing_def A_RCOSETS_def' by auto
hence "h x = h y" using Eq by simp
hence "x ⊖ y ∈ (a_kernel R S h)"
abelian_subgroup_a_kernel homeq_imp_rcos x(1) y(1))
thus "X = Y"
qed

lemma (in ring_hom_ring) quot_mem:
"⋀X. X ∈ carrier (R Quot (a_kernel R S h)) ⟹ ∃x ∈ carrier R. X = (a_kernel R S h) +> x"
proof -
fix X assume "X ∈ carrier (R Quot (a_kernel R S h))"
thus "∃x ∈ carrier R. X = (a_kernel R S h) +> x"
unfolding FactRing_simps by (simp add: a_r_coset_def)
qed

lemma (in ring_hom_ring) the_elem_wf:
"⋀X. X ∈ carrier (R Quot (a_kernel R S h)) ⟹ ∃y ∈ carrier S. (h ` X) = { y }"
proof -
fix X assume "X ∈ carrier (R Quot (a_kernel R S h))"
then obtain x where x: "x ∈ carrier R" and X: "X = (a_kernel R S h) +> x"
using quot_mem by blast
hence "⋀x'. x' ∈ X ⟹ h x' = h x"
proof -
fix x' assume "x' ∈ X" hence "x' ∈ (a_kernel R S h) +> x" using X by simp
then obtain k where k: "k ∈ a_kernel R S h" "x' = k ⊕ x"
abelian_subgroup.a_elemrcos_carrier
abelian_subgroup.a_rcos_module_imp abelian_subgroup_a_kernel x)
hence "h x' = h k ⊕⇘S⇙ h x"
also have " ... =  h x"
using k by (auto simp add: x)
finally show "h x' = h x" .
qed
moreover have "h x ∈ h ` X"
by (simp add: X homeq_imp_rcos x)
ultimately have "(h ` X) = { h x }"
by blast
thus "∃y ∈ carrier S. (h ` X) = { y }" using x by simp
qed

corollary (in ring_hom_ring) the_elem_wf':
"⋀X. X ∈ carrier (R Quot (a_kernel R S h)) ⟹ ∃r ∈ carrier R. (h ` X) = { h r }"
using the_elem_wf by (metis quot_mem the_elem_eq the_elem_simp)

lemma (in ring_hom_ring) the_elem_hom:
"(λX. the_elem (h ` X)) ∈ ring_hom (R Quot (a_kernel R S h)) S"
proof (rule ring_hom_memI)
show "⋀x. x ∈ carrier (R Quot a_kernel R S h) ⟹ the_elem (h ` x) ∈ carrier S"
using the_elem_wf by fastforce

show "the_elem (h ` 𝟭⇘R Quot a_kernel R S h⇙) = 𝟭⇘S⇙"
unfolding FactRing_def  using the_elem_simp[of "𝟭⇘R⇙"] by simp

fix X Y
assume "X ∈ carrier (R Quot a_kernel R S h)"
and "Y ∈ carrier (R Quot a_kernel R S h)"
then obtain x y where x: "x ∈ carrier R" "X = (a_kernel R S h) +> x"
and y: "y ∈ carrier R" "Y = (a_kernel R S h) +> y"
using quot_mem by blast

have "X ⊗⇘R Quot a_kernel R S h⇙ Y = (a_kernel R S h) +> (x ⊗ y)"
thus "the_elem (h ` (X ⊗⇘R Quot a_kernel R S h⇙ Y)) = the_elem (h ` X) ⊗⇘S⇙ the_elem (h ` Y)"

have "X ⊕⇘R Quot a_kernel R S h⇙ Y = (a_kernel R S h) +> (x ⊕ y)"
using ideal.rcos_ring_hom kernel_is_ideal ring_hom_add x y by fastforce
thus "the_elem (h ` (X ⊕⇘R Quot a_kernel R S h⇙ Y)) = the_elem (h ` X) ⊕⇘S⇙ the_elem (h ` Y)"
qed

lemma (in ring_hom_ring) the_elem_surj:
"(λX. (the_elem (h ` X))) ` carrier (R Quot (a_kernel R S h)) = (h ` (carrier R))"
proof
show "(λX. the_elem (h ` X)) ` carrier (R Quot a_kernel R S h) ⊆ h ` carrier R"
using the_elem_wf' by fastforce
next
show "h ` carrier R ⊆ (λX. the_elem (h ` X)) ` carrier (R Quot a_kernel R S h)"
proof
fix y assume "y ∈ h ` carrier R"
then obtain x where x: "x ∈ carrier R" "h x = y"
by (metis image_iff)
hence "the_elem (h ` ((a_kernel R S h) +> x)) = y" by simp
moreover have "(a_kernel R S h) +> x ∈ carrier (R Quot (a_kernel R S h))"
unfolding FactRing_simps by (auto simp add: x a_r_coset_def)
ultimately show "y ∈ (λX. (the_elem (h ` X))) ` carrier (R Quot (a_kernel R S h))" by blast
qed
qed

proposition (in ring_hom_ring) FactRing_iso_set_aux:
"(λX. the_elem (h ` X)) ∈ ring_iso (R Quot (a_kernel R S h)) (S ⦇ carrier := h ` (carrier R) ⦈)"
proof -
have "bij_betw (λX. the_elem (h ` X)) (carrier (R Quot a_kernel R S h)) (h ` (carrier R))"
unfolding bij_betw_def inj_on_def using the_elem_surj the_elem_inj by simp

moreover
have "(λX. the_elem (h ` X)): carrier (R Quot (a_kernel R S h)) → h ` (carrier R)"
using the_elem_wf' by fastforce
hence "(λX. the_elem (h ` X)) ∈ ring_hom (R Quot (a_kernel R S h)) (S ⦇ carrier := h ` (carrier R) ⦈)"
using the_elem_hom the_elem_wf' unfolding ring_hom_def by simp

ultimately show ?thesis unfolding ring_iso_def using the_elem_hom by simp
qed

theorem (in ring_hom_ring) FactRing_iso_set:
assumes "h ` carrier R = carrier S"
shows "(λX. the_elem (h ` X)) ∈ ring_iso (R Quot (a_kernel R S h)) S"
using FactRing_iso_set_aux assms by auto

corollary (in ring_hom_ring) FactRing_iso:
assumes "h ` carrier R = carrier S"
shows "R Quot (a_kernel R S h) ≃ S"
using FactRing_iso_set assms is_ring_iso_def by auto

corollary (in ring) FactRing_zeroideal:
shows "R Quot { 𝟬 } ≃ R" and "R ≃ R Quot { 𝟬 }"
proof -
have "ring_hom_ring R R id"
using ring_axioms by (auto intro: ring_hom_ringI)
moreover have "a_kernel R R id = { 𝟬 }"
unfolding a_kernel_def' by auto
ultimately show "R Quot { 𝟬 } ≃ R" and "R ≃ R Quot { 𝟬 }"
using ring_hom_ring.FactRing_iso[of R R id]
ring_iso_sym[OF ideal.quotient_is_ring[OF zeroideal], of R] by auto
qed

lemma (in ring_hom_ring) img_is_ring: "ring (S ⦇ carrier := h ` (carrier R) ⦈)"
proof -
let ?the_elem = "λX. the_elem (h ` X)"
have FactRing_is_ring: "ring (R Quot (a_kernel R S h))"
have "ring ((S ⦇ carrier := ?the_elem ` (carrier (R Quot (a_kernel R S h))) ⦈)
⦇     one := ?the_elem 𝟭⇘(R Quot (a_kernel R S h))⇙,
zero := ?the_elem 𝟬⇘(R Quot (a_kernel R S h))⇙ ⦈)"
using ring.ring_iso_imp_img_ring[OF FactRing_is_ring, of ?the_elem
"S ⦇ carrier := ?the_elem ` (carrier (R Quot (a_kernel R S h))) ⦈"]
FactRing_iso_set_aux the_elem_surj by auto

moreover
have "𝟬 ∈ (a_kernel R S h)"
using a_kernel_def'[of R S h] by auto
hence "𝟭 ∈ (a_kernel R S h) +> 𝟭"
using a_r_coset_def'[of R "a_kernel R S h" 𝟭] by force
hence "𝟭⇘S⇙ ∈ (h ` ((a_kernel R S h) +> 𝟭))"
using hom_one by force
hence "?the_elem 𝟭⇘(R Quot (a_kernel R S h))⇙ = 𝟭⇘S⇙"
using the_elem_wf[of "(a_kernel R S h) +> 𝟭"] by (simp add: FactRing_def)

moreover
have "𝟬⇘S⇙ ∈ (h ` (a_kernel R S h))"
using a_kernel_def'[of R S h] hom_zero by force
hence "𝟬⇘S⇙ ∈ (h ` 𝟬⇘(R Quot (a_kernel R S h))⇙)"
hence "?the_elem 𝟬⇘(R Quot (a_kernel R S h))⇙ = 𝟬⇘S⇙"
using the_elem_wf[OF ring.ring_simprules(2)[OF FactRing_is_ring]]
by (metis singletonD the_elem_eq)

ultimately
have "ring ((S ⦇ carrier := h ` (carrier R) ⦈) ⦇ one := 𝟭⇘S⇙, zero := 𝟬⇘S⇙ ⦈)"
using the_elem_surj by simp
thus ?thesis
by auto
qed

lemma (in ring_hom_ring) img_is_cring:
assumes "cring S"
shows "cring (S ⦇ carrier := h ` (carrier R) ⦈)"
proof -
interpret ring "S ⦇ carrier := h ` (carrier R) ⦈"
using img_is_ring .
show ?thesis
apply unfold_locales
using assms unfolding cring_def comm_monoid_def comm_monoid_axioms_def by auto
qed

lemma (in ring_hom_ring) img_is_domain:
assumes "domain S"
shows "domain (S ⦇ carrier := h ` (carrier R) ⦈)"
proof -
interpret cring "S ⦇ carrier := h ` (carrier R) ⦈"
using img_is_cring assms unfolding domain_def by simp
show ?thesis
apply unfold_locales
using assms unfolding domain_def domain_axioms_def apply auto
using hom_closed by blast
qed

proposition (in ring_hom_ring) primeideal_vimage:
assumes "cring R"
shows "primeideal P S ⟹ primeideal { r ∈ carrier R. h r ∈ P } R"
proof -
assume A: "primeideal P S"
hence is_ideal: "ideal P S" unfolding primeideal_def by simp
have "ring_hom_ring R (S Quot P) (((+>⇘S⇙) P) ∘ h)" (is "ring_hom_ring ?A ?B ?h")
using ring_hom_trans[OF homh, of "(+>⇘S⇙) P" "S Quot P"]
ideal.rcos_ring_hom_ring[OF is_ideal] assms
unfolding ring_hom_ring_def ring_hom_ring_axioms_def cring_def by simp
then interpret hom: ring_hom_ring R "S Quot P" "((+>⇘S⇙) P) ∘ h" by simp

have "inj_on (λX. the_elem (?h ` X)) (carrier (R Quot (a_kernel R (S Quot P) ?h)))"
using hom.the_elem_inj unfolding inj_on_def by simp
moreover
have "ideal (a_kernel R (S Quot P) ?h) R"
using hom.kernel_is_ideal by auto
have hom': "ring_hom_ring (R Quot (a_kernel R (S Quot P) ?h)) (S Quot P) (λX. the_elem (?h ` X))"
using hom.the_elem_hom hom.kernel_is_ideal
by (meson hom.ring_hom_ring_axioms ideal.rcos_ring_hom_ring ring_hom_ring_axioms_def ring_hom_ring_def)

ultimately
have "primeideal (a_kernel R (S Quot P) ?h) R"
using ring_hom_ring.inj_on_domain[OF hom'] primeideal.quotient_is_domain[OF A]
cring.quot_domain_imp_primeideal[OF assms hom.kernel_is_ideal] by simp

moreover have "a_kernel R (S Quot P) ?h = { r ∈ carrier R. h r ∈ P }"
proof
show "a_kernel R (S Quot P) ?h ⊆ { r ∈ carrier R. h r ∈ P }"
proof
fix r assume "r ∈ a_kernel R (S Quot P) ?h"
hence r: "r ∈ carrier R" "P +>⇘S⇙ (h r) = P"
unfolding a_kernel_def kernel_def FactRing_def by auto
hence "h r ∈ P"
using S.a_rcosI R.l_zero S.l_zero additive_subgroup.a_subset[OF ideal.axioms(1)[OF is_ideal]]
additive_subgroup.zero_closed[OF ideal.axioms(1)[OF is_ideal]] hom_closed by metis
thus "r ∈ { r ∈ carrier R. h r ∈ P }" using r by simp
qed
next
show "{ r ∈ carrier R. h r ∈ P } ⊆ a_kernel R (S Quot P) ?h"
proof
fix r assume "r ∈ { r ∈ carrier R. h r ∈ P }"
hence r: "r ∈ carrier R" "h r ∈ P" by simp_all
hence "?h r = P"