Theorem rnghmval 41891

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 RngHomo 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-rnghomo 41887	. . 3 ⊢ RngHomo = (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) → RngHomo = (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 6191	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4		isrnghm.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
5	3, 4	syl6eqr 2674	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6	5	csbeq1d 3540	. . . . 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 6191	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
8		rnghmval.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑆)
9	7, 8	syl6eqr 2674	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝐶)
10	9	csbeq1d 3540	. . . . . 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 3992	. . . . 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 2676	. . . 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 482	. . 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		fvex 6201	. . . . . . . 8 ⊢ (Base‘𝑅) ∈ V
15	4, 14	eqeltri 2697	. . . . . . 7 ⊢ 𝐵 ∈ V
16		fvex 6201	. . . . . . . 8 ⊢ (Base‘𝑆) ∈ V
17	8, 16	eqeltri 2697	. . . . . . 7 ⊢ 𝐶 ∈ V
18		oveq12 6659	. . . . . . . . 9 ⊢ ((𝑤 = 𝐶 ∧ 𝑣 = 𝐵) → (𝑤 ↑𝑚 𝑣) = (𝐶 ↑𝑚 𝐵))
19	18	ancoms 469	. . . . . . . 8 ⊢ ((𝑣 = 𝐵 ∧ 𝑤 = 𝐶) → (𝑤 ↑𝑚 𝑣) = (𝐶 ↑𝑚 𝐵))
20		raleq 3138	. . . . . . . . . 10 ⊢ (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)))))
21	20	raleqbi1dv 3146	. . . . . . . . 9 ⊢ (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)))))
22	21	adantr 481	. . . . . . . 8 ⊢ ((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)))))
23	19, 22	rabeqbidv 3195	. . . . . . 7 ⊢ ((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)))})
24	15, 17, 23	csbie2 3563	. . . . . 6 ⊢ ⦋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)))}
25	24	a1i 11	. . . . 5 ⊢ ((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(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))})
26		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = (+g‘𝑅))
27		rnghmval.p	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝑅)
28	26, 27	syl6eqr 2674	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = + )
29	28	oveqdr 6674	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥(+g‘𝑟)𝑦) = (𝑥 + 𝑦))
30	29	fveq2d 6195	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑓‘(𝑥(+g‘𝑟)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
31		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
32		rnghmval.a	. . . . . . . . . . . . 13 ⊢ ✚ = (+g‘𝑆)
33	31, 32	syl6eqr 2674	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = ✚ )
34	33	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (+g‘𝑠) = ✚ )
35	34	oveqd 6667	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)))
36	30, 35	eqeq12d 2637	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦))))
37		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
38		isrnghm.t	. . . . . . . . . . . . 13 ⊢ · = (.r‘𝑅)
39	37, 38	syl6eqr 2674	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
40	39	oveqdr 6674	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
41	40	fveq2d 6195	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑓‘(𝑥(.r‘𝑟)𝑦)) = (𝑓‘(𝑥 · 𝑦)))
42		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (.r‘𝑠) = (.r‘𝑆))
43		isrnghm.m	. . . . . . . . . . . . 13 ⊢ ∗ = (.r‘𝑆)
44	42, 43	syl6eqr 2674	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (.r‘𝑠) = ∗ )
45	44	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (.r‘𝑠) = ∗ )
46	45	oveqd 6667	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))
47	41, 46	eqeq12d 2637	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦))))
48	36, 47	anbi12d 747	. . . . . . . 8 ⊢ ((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)))))
49	48	ralbidv 2986	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (∀𝑦 ∈ 𝐵 ((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 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))))
50	49	ralbidv 2986	. . . . . 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)))))
51	50	rabbidv 3189	. . . . 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)))})
52	25, 51	eqtrd 2656	. . . 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)))})
53	52	adantl 482	. . 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)))})
54	13, 53	eqtrd 2656	. 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)))})
55		simpl 473	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → 𝑅 ∈ Rng)
56		simpr 477	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → 𝑆 ∈ Rng)
57		ovex 6678	. . . 4 ⊢ (𝐶 ↑𝑚 𝐵) ∈ V
58	57	rabex 4813	. . 3 ⊢ {𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))} ∈ V
59	58	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)
60	2, 54, 55, 56, 59	ovmpt2d 6788	1 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHomo 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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200  ⦋csb 3533  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ↑_𝑚 cmap 7857  Basecbs 15857  +_gcplusg 15941  ._rcmulr 15942  Rngcrng 41874   RngHomo crngh 41885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rnghomo 41887

This theorem is referenced by:  isrnghm  41892