Theorem rnghmval 20461

Description: The set of the non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 22-Feb-2020.)

Hypotheses

Ref	Expression
isrnghm.b	⊢ 𝐵 = (Base‘𝑅)
isrnghm.t	⊢ · = (.r‘𝑅)
isrnghm.m	⊢ ∗ = (.r‘𝑆)
rnghmval.c	⊢ 𝐶 = (Base‘𝑆)
rnghmval.p	⊢ + = (+g‘𝑅)
rnghmval.a	⊢ ✚ = (+g‘𝑆)

Assertion

Ref	Expression
rnghmval	⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHom 𝑆) = {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))})

Distinct variable groups:   𝐵,𝑓,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦   𝑆,𝑓,𝑥,𝑦   𝐶,𝑓

Allowed substitution hints:   𝐶(𝑥,𝑦)   + (𝑥,𝑦,𝑓)   ✚ (𝑥,𝑦,𝑓)   · (𝑥,𝑦,𝑓)   ∗ (𝑥,𝑦,𝑓)

Proof of Theorem rnghmval

Dummy variables 𝑟 𝑠 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rnghm 20457	. . 3 ⊢ RngHom = (𝑟 ∈ Rng, 𝑠 ∈ Rng ↦ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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	1	a1i 11	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → RngHom = (𝑟 ∈ Rng, 𝑠 ∈ Rng ↦ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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)))}))
3		fveq2 6856	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4		isrnghm.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
5	3, 4	eqtr4di 2809	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6	5	csbeq1d 3851	. . . . 5 ⊢ (𝑟 = 𝑅 → ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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)))} = ⦋𝐵 / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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)))})
7		fveq2 6856	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
8		rnghmval.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑆)
9	7, 8	eqtr4di 2809	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝐶)
10	9	csbeq1d 3851	. . . . . 6 ⊢ (𝑠 = 𝑆 → ⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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)))} = ⦋𝐶 / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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)))})
11	10	csbeq2dv 3854	. . . . 5 ⊢ (𝑠 = 𝑆 → ⦋𝐵 / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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)))} = ⦋𝐵 / 𝑣⦌⦋𝐶 / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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)))})
12	6, 11	sylan9eq 2811	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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)))} = ⦋𝐵 / 𝑣⦌⦋𝐶 / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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)))})
13	12	adantl 484	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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)))} = ⦋𝐵 / 𝑣⦌⦋𝐶 / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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)))})
14	4	fvexi 6870	. . . . . 6 ⊢ 𝐵 ∈ V
15	8	fvexi 6870	. . . . . 6 ⊢ 𝐶 ∈ V
16		oveq12 7394	. . . . . . . 8 ⊢ ((𝑤 = 𝐶 ∧ 𝑣 = 𝐵) → (𝑤 ↑m 𝑣) = (𝐶 ↑m 𝐵))
17	16	ancoms 461	. . . . . . 7 ⊢ ((𝑣 = 𝐵 ∧ 𝑤 = 𝐶) → (𝑤 ↑m 𝑣) = (𝐶 ↑m 𝐵))
18		raleq 3311	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → (∀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 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))))
19	18	raleqbi1dv 3324	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → (∀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 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))))
20	19	adantr 483	. . . . . . 7 ⊢ ((𝑣 = 𝐵 ∧ 𝑤 = 𝐶) → (∀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 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))))
21	17, 20	rabeqbidv 3426	. . . . . 6 ⊢ ((𝑣 = 𝐵 ∧ 𝑤 = 𝐶) → {𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))})
22	14, 15, 21	csbie2 3886	. . . . 5 ⊢ ⦋𝐵 / 𝑣⦌⦋𝐶 / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))}
23		fveq2 6856	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = (+g‘𝑅))
24		rnghmval.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑅)
25	23, 24	eqtr4di 2809	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = + )
26	25	oveqdr 7413	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥(+g‘𝑟)𝑦) = (𝑥 + 𝑦))
27	26	fveq2d 6860	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑓‘(𝑥(+g‘𝑟)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
28		fveq2 6856	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
29		rnghmval.a	. . . . . . . . . . . 12 ⊢ ✚ = (+g‘𝑆)
30	28, 29	eqtr4di 2809	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = ✚ )
31	30	adantl 484	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (+g‘𝑠) = ✚ )
32	31	oveqd 7402	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)))
33	27, 32	eqeq12d 2772	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦))))
34		fveq2 6856	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
35		isrnghm.t	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑅)
36	34, 35	eqtr4di 2809	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
37	36	oveqdr 7413	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
38	37	fveq2d 6860	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑓‘(𝑥(.r‘𝑟)𝑦)) = (𝑓‘(𝑥 · 𝑦)))
39		fveq2 6856	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (.r‘𝑠) = (.r‘𝑆))
40		isrnghm.m	. . . . . . . . . . . 12 ⊢ ∗ = (.r‘𝑆)
41	39, 40	eqtr4di 2809	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (.r‘𝑠) = ∗ )
42	41	adantl 484	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (.r‘𝑠) = ∗ )
43	42	oveqd 7402	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))
44	38, 43	eqeq12d 2772	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦))))
45	33, 44	anbi12d 640	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))) ↔ ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))))
46	45	2ralbidv 3220	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))))
47	46	rabbidv 3415	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))} = {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))})
48	22, 47	eqtrid 2803	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⦋𝐵 / 𝑣⦌⦋𝐶 / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))})
49	48	adantl 484	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → ⦋𝐵 / 𝑣⦌⦋𝐶 / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))})
50	13, 49	eqtrd 2791	. 2 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))})
51		simpl 485	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → 𝑅 ∈ Rng)
52		simpr 487	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → 𝑆 ∈ Rng)
53		ovex 7418	. . . 4 ⊢ (𝐶 ↑m 𝐵) ∈ V
54	53	rabex 5289	. . 3 ⊢ {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))} ∈ V
55	54	a1i 11	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))} ∈ V)
56	2, 50, 51, 52, 55	ovmpod 7537	1 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHom 𝑆) = {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1554   ∈ wcel 2136  ∀wral 3070  {crab 3408  Vcvv 3448  ⦋csb 3847  ‘cfv 6510  (class class class)co 7385   ∈ cmpo 7387   ↑_m cmap 8796  Basecbs 17221  +_gcplusg 17262  ._rcmulr 17263  Rngcrng 20174   RngHom crnghm 20455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6512  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-rnghm 20457

This theorem is referenced by:  isrnghm  20462