Theorem dfrhm2 20448

Description: The property of a ring homomorphism can be decomposed into separate homomorphic conditions for addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Assertion

Ref	Expression
dfrhm2	⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))

Distinct variable group:   𝑠,𝑟

Proof of Theorem dfrhm2

Dummy variables 𝑣 𝑤 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rhm 20446	. 2 ⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))})
2		ancom 460	. . . . . . 7 ⊢ (((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))) ↔ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠)))
3		r19.26-2 3123	. . . . . . . 8 ⊢ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))) ↔ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))
4	3	anbi1i 625	. . . . . . 7 ⊢ ((∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠)) ↔ ((∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠)))
5		anass 468	. . . . . . 7 ⊢ (((∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠)) ↔ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))))
6	2, 4, 5	3bitri 297	. . . . . 6 ⊢ (((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))) ↔ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))))
7	6	rabbii 3395	. . . . 5 ⊢ {𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))} = {𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠)))}
8		fvex 6848	. . . . . 6 ⊢ (Base‘𝑟) ∈ V
9		fvex 6848	. . . . . 6 ⊢ (Base‘𝑠) ∈ V
10		oveq12 7370	. . . . . . . 8 ⊢ ((𝑤 = (Base‘𝑠) ∧ 𝑣 = (Base‘𝑟)) → (𝑤 ↑m 𝑣) = ((Base‘𝑠) ↑m (Base‘𝑟)))
11	10	ancoms 458	. . . . . . 7 ⊢ ((𝑣 = (Base‘𝑟) ∧ 𝑤 = (Base‘𝑠)) → (𝑤 ↑m 𝑣) = ((Base‘𝑠) ↑m (Base‘𝑟)))
12		raleq 3293	. . . . . . . . . 10 ⊢ (𝑣 = (Base‘𝑟) → (∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))) ↔ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))))
13	12	raleqbi1dv 3306	. . . . . . . . 9 ⊢ (𝑣 = (Base‘𝑟) → (∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))) ↔ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))))
14	13	adantr 480	. . . . . . . 8 ⊢ ((𝑣 = (Base‘𝑟) ∧ 𝑤 = (Base‘𝑠)) → (∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))) ↔ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))))
15	14	anbi2d 631	. . . . . . 7 ⊢ ((𝑣 = (Base‘𝑟) ∧ 𝑤 = (Base‘𝑠)) → (((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))) ↔ ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))))
16	11, 15	rabeqbidv 3408	. . . . . 6 ⊢ ((𝑣 = (Base‘𝑟) ∧ 𝑤 = (Base‘𝑠)) → {𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))} = {𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))})
17	8, 9, 16	csbie2 3877	. . . . 5 ⊢ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))} = {𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))}
18		inrab 4257	. . . . 5 ⊢ ({𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦))} ∩ {𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))}) = {𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠)))}
19	7, 17, 18	3eqtr4i 2770	. . . 4 ⊢ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))} = ({𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦))} ∩ {𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))})
20		ringgrp 20213	. . . . . . . 8 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Grp)
21		ringgrp 20213	. . . . . . . 8 ⊢ (𝑠 ∈ Ring → 𝑠 ∈ Grp)
22		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝑟) = (Base‘𝑟)
23		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝑠) = (Base‘𝑠)
24		eqid 2737	. . . . . . . . 9 ⊢ (+g‘𝑟) = (+g‘𝑟)
25		eqid 2737	. . . . . . . . 9 ⊢ (+g‘𝑠) = (+g‘𝑠)
26	22, 23, 24, 25	isghm3 19186	. . . . . . . 8 ⊢ ((𝑟 ∈ Grp ∧ 𝑠 ∈ Grp) → (𝑓 ∈ (𝑟 GrpHom 𝑠) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)))))
27	20, 21, 26	syl2an 597	. . . . . . 7 ⊢ ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → (𝑓 ∈ (𝑟 GrpHom 𝑠) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)))))
28	27	eqabdv 2870	. . . . . 6 ⊢ ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → (𝑟 GrpHom 𝑠) = {𝑓 ∣ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)))})
29		df-rab 3391	. . . . . . 7 ⊢ {𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦))} = {𝑓 ∣ (𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)))}
30	9, 8	elmap 8813	. . . . . . . . 9 ⊢ (𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ↔ 𝑓:(Base‘𝑟)⟶(Base‘𝑠))
31	30	anbi1i 625	. . . . . . . 8 ⊢ ((𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦))) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦))))
32	31	abbii 2804	. . . . . . 7 ⊢ {𝑓 ∣ (𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)))} = {𝑓 ∣ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)))}
33	29, 32	eqtri 2760	. . . . . 6 ⊢ {𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦))} = {𝑓 ∣ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)))}
34	28, 33	eqtr4di 2790	. . . . 5 ⊢ ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → (𝑟 GrpHom 𝑠) = {𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦))})
35		eqid 2737	. . . . . . . . 9 ⊢ (mulGrp‘𝑟) = (mulGrp‘𝑟)
36	35	ringmgp 20214	. . . . . . . 8 ⊢ (𝑟 ∈ Ring → (mulGrp‘𝑟) ∈ Mnd)
37		eqid 2737	. . . . . . . . 9 ⊢ (mulGrp‘𝑠) = (mulGrp‘𝑠)
38	37	ringmgp 20214	. . . . . . . 8 ⊢ (𝑠 ∈ Ring → (mulGrp‘𝑠) ∈ Mnd)
39	35, 22	mgpbas 20120	. . . . . . . . . 10 ⊢ (Base‘𝑟) = (Base‘(mulGrp‘𝑟))
40	37, 23	mgpbas 20120	. . . . . . . . . 10 ⊢ (Base‘𝑠) = (Base‘(mulGrp‘𝑠))
41		eqid 2737	. . . . . . . . . . 11 ⊢ (.r‘𝑟) = (.r‘𝑟)
42	35, 41	mgpplusg 20119	. . . . . . . . . 10 ⊢ (.r‘𝑟) = (+g‘(mulGrp‘𝑟))
43		eqid 2737	. . . . . . . . . . 11 ⊢ (.r‘𝑠) = (.r‘𝑠)
44	37, 43	mgpplusg 20119	. . . . . . . . . 10 ⊢ (.r‘𝑠) = (+g‘(mulGrp‘𝑠))
45		eqid 2737	. . . . . . . . . . 11 ⊢ (1r‘𝑟) = (1r‘𝑟)
46	35, 45	ringidval 20158	. . . . . . . . . 10 ⊢ (1r‘𝑟) = (0g‘(mulGrp‘𝑟))
47		eqid 2737	. . . . . . . . . . 11 ⊢ (1r‘𝑠) = (1r‘𝑠)
48	37, 47	ringidval 20158	. . . . . . . . . 10 ⊢ (1r‘𝑠) = (0g‘(mulGrp‘𝑠))
49	39, 40, 42, 44, 46, 48	ismhm 18747	. . . . . . . . 9 ⊢ (𝑓 ∈ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)) ↔ (((mulGrp‘𝑟) ∈ Mnd ∧ (mulGrp‘𝑠) ∈ Mnd) ∧ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))))
50	49	baib 535	. . . . . . . 8 ⊢ (((mulGrp‘𝑟) ∈ Mnd ∧ (mulGrp‘𝑠) ∈ Mnd) → (𝑓 ∈ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))))
51	36, 38, 50	syl2an 597	. . . . . . 7 ⊢ ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → (𝑓 ∈ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))))
52	51	eqabdv 2870	. . . . . 6 ⊢ ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)) = {𝑓 ∣ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))})
53		df-rab 3391	. . . . . . 7 ⊢ {𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))} = {𝑓 ∣ (𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠)))}
54	30	anbi1i 625	. . . . . . . . 9 ⊢ ((𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))))
55		3anass 1095	. . . . . . . . 9 ⊢ ((𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠)) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))))
56	54, 55	bitr4i 278	. . . . . . . 8 ⊢ ((𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠)))
57	56	abbii 2804	. . . . . . 7 ⊢ {𝑓 ∣ (𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠)))} = {𝑓 ∣ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))}
58	53, 57	eqtri 2760	. . . . . 6 ⊢ {𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))} = {𝑓 ∣ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))}
59	52, 58	eqtr4di 2790	. . . . 5 ⊢ ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)) = {𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))})
60	34, 59	ineq12d 4162	. . . 4 ⊢ ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))) = ({𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦))} ∩ {𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(1r‘𝑟)) = (1r‘𝑠))}))
61	19, 60	eqtr4id 2791	. . 3 ⊢ ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))} = ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))
62	61	mpoeq3ia 7439	. 2 ⊢ (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))}) = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))
63	1, 62	eqtri 2760	1 ⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  {crab 3390  ⦋csb 3838   ∩ cin 3889  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363   ↑_m cmap 8767  Basecbs 17173  +_gcplusg 17214  ._rcmulr 17215  Mndcmnd 18696   MndHom cmhm 18743  Grpcgrp 18903   GrpHom cghm 19181  mulGrpcmgp 20115  1_rcur 20156  Ringcrg 20208   RingHom crh 20443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-plusg 17227  df-0g 17398  df-mhm 18745  df-ghm 19182  df-mgp 20116  df-ur 20157  df-ring 20210  df-rhm 20446

This theorem is referenced by:  rhmrcl1  20450  rhmrcl2  20451  isrhm  20452  rhmfn  20470  rhmval  20471  rhmsubclem1  20656  zrhval  21500  rhmsubcALTVlem1  48772