Theorem dfrhm2 14232

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 14230	. 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 266	. . . . . . 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 2663	. . . . . . . 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 458	. . . . . . 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 401	. . . . . . 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 206	. . . . . 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 2790	. . . . 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		basfn 13204	. . . . . . 7 ⊢ Base Fn V
9		vex 2806	. . . . . . 7 ⊢ 𝑟 ∈ V
10		funfvex 5665	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑟 ∈ dom Base) → (Base‘𝑟) ∈ V)
11	10	funfni 5439	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑟 ∈ V) → (Base‘𝑟) ∈ V)
12	8, 9, 11	mp2an 426	. . . . . 6 ⊢ (Base‘𝑟) ∈ V
13		vex 2806	. . . . . . 7 ⊢ 𝑠 ∈ V
14		funfvex 5665	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑠 ∈ dom Base) → (Base‘𝑠) ∈ V)
15	14	funfni 5439	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑠 ∈ V) → (Base‘𝑠) ∈ V)
16	8, 13, 15	mp2an 426	. . . . . 6 ⊢ (Base‘𝑠) ∈ V
17		oveq12 6037	. . . . . . . 8 ⊢ ((𝑤 = (Base‘𝑠) ∧ 𝑣 = (Base‘𝑟)) → (𝑤 ↑𝑚 𝑣) = ((Base‘𝑠) ↑𝑚 (Base‘𝑟)))
18	17	ancoms 268	. . . . . . 7 ⊢ ((𝑣 = (Base‘𝑟) ∧ 𝑤 = (Base‘𝑠)) → (𝑤 ↑𝑚 𝑣) = ((Base‘𝑠) ↑𝑚 (Base‘𝑟)))
19		raleq 2731	. . . . . . . . . 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)))))
20	19	raleqbi1dv 2743	. . . . . . . . 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)))))
21	20	adantr 276	. . . . . . . 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)))))
22	21	anbi2d 464	. . . . . . 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))))))
23	18, 22	rabeqbidv 2798	. . . . . 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))))})
24	12, 16, 23	csbie2 3178	. . . . 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))))}
25		inrab 3481	. . . . 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)))}
26	7, 24, 25	3eqtr4i 2262	. . . 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))})
27		ringgrp 14078	. . . . . . . 8 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Grp)
28		ringgrp 14078	. . . . . . . 8 ⊢ (𝑠 ∈ Ring → 𝑠 ∈ Grp)
29		eqid 2231	. . . . . . . . 9 ⊢ (Base‘𝑟) = (Base‘𝑟)
30		eqid 2231	. . . . . . . . 9 ⊢ (Base‘𝑠) = (Base‘𝑠)
31		eqid 2231	. . . . . . . . 9 ⊢ (+g‘𝑟) = (+g‘𝑟)
32		eqid 2231	. . . . . . . . 9 ⊢ (+g‘𝑠) = (+g‘𝑠)
33	29, 30, 31, 32	isghm3 13894	. . . . . . . 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)))))
34	27, 28, 33	syl2an 289	. . . . . . 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)))))
35	34	eqabdv 2361	. . . . . 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)))})
36		df-rab 2520	. . . . . . 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)))}
37	16, 12	elmap 6889	. . . . . . . . 9 ⊢ (𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ↔ 𝑓:(Base‘𝑟)⟶(Base‘𝑠))
38	37	anbi1i 458	. . . . . . . 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))))
39	38	abbii 2347	. . . . . . 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)))}
40	36, 39	eqtri 2252	. . . . . 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)))}
41	35, 40	eqtr4di 2282	. . . . 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))})
42		eqid 2231	. . . . . . . . 9 ⊢ (mulGrp‘𝑟) = (mulGrp‘𝑟)
43	42	ringmgp 14079	. . . . . . . 8 ⊢ (𝑟 ∈ Ring → (mulGrp‘𝑟) ∈ Mnd)
44		eqid 2231	. . . . . . . . 9 ⊢ (mulGrp‘𝑠) = (mulGrp‘𝑠)
45	44	ringmgp 14079	. . . . . . . 8 ⊢ (𝑠 ∈ Ring → (mulGrp‘𝑠) ∈ Mnd)
46	42, 29	mgpbasg 14003	. . . . . . . . . . 11 ⊢ (𝑟 ∈ V → (Base‘𝑟) = (Base‘(mulGrp‘𝑟)))
47	46	elv 2807	. . . . . . . . . 10 ⊢ (Base‘𝑟) = (Base‘(mulGrp‘𝑟))
48	44, 30	mgpbasg 14003	. . . . . . . . . . 11 ⊢ (𝑠 ∈ V → (Base‘𝑠) = (Base‘(mulGrp‘𝑠)))
49	48	elv 2807	. . . . . . . . . 10 ⊢ (Base‘𝑠) = (Base‘(mulGrp‘𝑠))
50		eqid 2231	. . . . . . . . . . . 12 ⊢ (.r‘𝑟) = (.r‘𝑟)
51	42, 50	mgpplusgg 14001	. . . . . . . . . . 11 ⊢ (𝑟 ∈ V → (.r‘𝑟) = (+g‘(mulGrp‘𝑟)))
52	51	elv 2807	. . . . . . . . . 10 ⊢ (.r‘𝑟) = (+g‘(mulGrp‘𝑟))
53		eqid 2231	. . . . . . . . . . . 12 ⊢ (.r‘𝑠) = (.r‘𝑠)
54	44, 53	mgpplusgg 14001	. . . . . . . . . . 11 ⊢ (𝑠 ∈ V → (.r‘𝑠) = (+g‘(mulGrp‘𝑠)))
55	54	elv 2807	. . . . . . . . . 10 ⊢ (.r‘𝑠) = (+g‘(mulGrp‘𝑠))
56		eqid 2231	. . . . . . . . . . . 12 ⊢ (1r‘𝑟) = (1r‘𝑟)
57	42, 56	ringidvalg 14038	. . . . . . . . . . 11 ⊢ (𝑟 ∈ V → (1r‘𝑟) = (0g‘(mulGrp‘𝑟)))
58	57	elv 2807	. . . . . . . . . 10 ⊢ (1r‘𝑟) = (0g‘(mulGrp‘𝑟))
59		eqid 2231	. . . . . . . . . . . 12 ⊢ (1r‘𝑠) = (1r‘𝑠)
60	44, 59	ringidvalg 14038	. . . . . . . . . . 11 ⊢ (𝑠 ∈ V → (1r‘𝑠) = (0g‘(mulGrp‘𝑠)))
61	60	elv 2807	. . . . . . . . . 10 ⊢ (1r‘𝑠) = (0g‘(mulGrp‘𝑠))
62	47, 49, 52, 55, 58, 61	ismhm 13607	. . . . . . . . 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))))
63	62	baib 927	. . . . . . . 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))))
64	43, 45, 63	syl2an 289	. . . . . . 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))))
65	64	eqabdv 2361	. . . . . 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))})
66		df-rab 2520	. . . . . . 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)))}
67	37	anbi1i 458	. . . . . . . . 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))))
68		3anass 1009	. . . . . . . . 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))))
69	67, 68	bitr4i 187	. . . . . . . 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)))
70	69	abbii 2347	. . . . . . 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))}
71	66, 70	eqtri 2252	. . . . . 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))}
72	65, 71	eqtr4di 2282	. . . . 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))})
73	41, 72	ineq12d 3411	. . . 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))}))
74	26, 73	eqtr4id 2283	. . 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))))
75	74	mpoeq3ia 6096	. 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))))
76	1, 75	eqtri 2252	1 ⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  {cab 2217  ∀wral 2511  {crab 2515  Vcvv 2803  ⦋csb 3128   ∩ cin 3200   Fn wfn 5328  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028   ∈ cmpo 6030   ↑_𝑚 cmap 6860  Basecbs 13145  +_gcplusg 13223  ._rcmulr 13224  0_gc0g 13402  Mndcmnd 13562   MndHom cmhm 13603  Grpcgrp 13646   GrpHom cghm 13890  mulGrpcmgp 13997  1_rcur 14036  Ringcrg 14073   RingHom crh 14228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-3 9245  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-plusg 13236  df-mulr 13237  df-mhm 13605  df-ghm 13891  df-mgp 13998  df-ur 14037  df-ring 14075  df-rhm 14230

This theorem is referenced by:  rhmrcl1  14233  rhmrcl2  14234  isrhm  14236  rhmfn  14250  rhmval  14251