Theorem rhmval0 20698

Description: The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by AV, 24-Jul-2026.)

Hypotheses

Ref	Expression
rhmval0.b	⊢ 𝐵 = (Base‘𝑅)
rhmval0.c	⊢ 𝐶 = (Base‘𝑆)
rhmval0.1	⊢ 1 = (1r‘𝑅)
rhmval0.i	⊢ 𝑁 = (1r‘𝑆)
rhmval0.m	⊢ · = (.r‘𝑅)
rhmval0.n	⊢ × = (.r‘𝑆)
rhmval0.p	⊢ + = (+g‘𝑅)
rhmval0.q	⊢ ⨣ = (+g‘𝑆)

Assertion

Ref	Expression
rhmval0	⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝑅 RingHom 𝑆) = {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀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   𝐶,𝑓   𝑓,𝑁   𝑅,𝑓,𝑥,𝑦   𝑆,𝑓,𝑥,𝑦   1 ,𝑓   + ,𝑓   ⨣ ,𝑓   · ,𝑓   × ,𝑓

Allowed substitution hints:   𝐶(𝑥, 𝑦)   + (𝑥, 𝑦)   ⨣ (𝑥, 𝑦)   · (𝑥, 𝑦)   × (𝑥, 𝑦)   1 (𝑥, 𝑦)   𝑁(𝑥, 𝑦)

Proof of Theorem rhmval0

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

Step	Hyp	Ref	Expression
1		fveq2 6883	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
2		rhmval0.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
3	1, 2	eqtr4di 2814	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
4	3	adantr 486	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Base‘𝑟) = 𝐵)
5		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
6		rhmval0.1	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑅)
7	5, 6	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
8	7	fveq2d 6887	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑓‘(1r‘𝑟)) = (𝑓‘ 1 ))
9		fveq2 6883	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (1r‘𝑠) = (1r‘𝑆))
10		rhmval0.i	. . . . . . . . 9 ⊢ 𝑁 = (1r‘𝑆)
11	9, 10	eqtr4di 2814	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1r‘𝑠) = 𝑁)
12	8, 11	eqeqan12d 2775	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ↔ (𝑓‘ 1 ) = 𝑁))
13		fveq2 6883	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = (+g‘𝑅))
14		rhmval0.p	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝑅)
15	13, 14	eqtr4di 2814	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = + )
16	15	oveqd 7435	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑥(+g‘𝑟)𝑦) = (𝑥 + 𝑦))
17	16	fveq2d 6887	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑓‘(𝑥(+g‘𝑟)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
18		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
19		rhmval0.q	. . . . . . . . . . . 12 ⊢ ⨣ = (+g‘𝑆)
20	18, 19	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = ⨣ )
21	20	oveqd 7435	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)))
22	17, 21	eqeqan12d 2775	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦))))
23		fveq2 6883	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
24		rhmval0.m	. . . . . . . . . . . . 13 ⊢ · = (.r‘𝑅)
25	23, 24	eqtr4di 2814	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
26	25	oveqd 7435	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
27	26	fveq2d 6887	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑓‘(𝑥(.r‘𝑟)𝑦)) = (𝑓‘(𝑥 · 𝑦)))
28		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (.r‘𝑠) = (.r‘𝑆))
29		rhmval0.n	. . . . . . . . . . . 12 ⊢ × = (.r‘𝑆)
30	28, 29	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (.r‘𝑠) = × )
31	30	oveqd 7435	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦)))
32	27, 31	eqeqan12d 2775	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))
33	22, 32	anbi12d 644	. . . . . . . 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)))))
34	33	2ralbidv 3227	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 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 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦)))))
35	12, 34	anbi12d 644	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 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‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))))
36	35	rabbidv 3420	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 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 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))})
37	36	csbeq2dv 3854	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⦋(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))))} = ⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))})
38	4, 37	csbeq12dv 3856	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⦋(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))))} = ⦋𝐵 / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))})
39		fveq2 6883	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
40		rhmval0.c	. . . . . . 7 ⊢ 𝐶 = (Base‘𝑆)
41	39, 40	eqtr4di 2814	. . . . . 6 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝐶)
42	41	adantl 487	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Base‘𝑠) = 𝐶)
43	42	csbeq1d 3851	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))} = ⦋𝐶 / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))})
44	43	csbeq2dv 3854	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⦋𝐵 / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))} = ⦋𝐵 / 𝑣⦌⦋𝐶 / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))})
45	2	fvexi 6897	. . . . 5 ⊢ 𝐵 ∈ V
46	40	fvexi 6897	. . . . 5 ⊢ 𝐶 ∈ V
47		oveq12 7427	. . . . . . 7 ⊢ ((𝑤 = 𝐶 ∧ 𝑣 = 𝐵) → (𝑤 ↑m 𝑣) = (𝐶 ↑m 𝐵))
48	47	ancoms 464	. . . . . 6 ⊢ ((𝑣 = 𝐵 ∧ 𝑤 = 𝐶) → (𝑤 ↑m 𝑣) = (𝐶 ↑m 𝐵))
49		raleq 3317	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → (∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦)))))
50		raleq 3317	. . . . . . . . . 10 ⊢ (𝑣 = 𝐵 → (∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))) ↔ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦)))))
51	50	ralbidv 3186	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦)))))
52	49, 51	bitrd 282	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → (∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦)))))
53	52	adantr 486	. . . . . . 7 ⊢ ((𝑣 = 𝐵 ∧ 𝑤 = 𝐶) → (∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦)))))
54	53	anbi2d 642	. . . . . 6 ⊢ ((𝑣 = 𝐵 ∧ 𝑤 = 𝐶) → (((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦)))) ↔ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))))
55	48, 54	rabeqbidv 3430	. . . . 5 ⊢ ((𝑣 = 𝐵 ∧ 𝑤 = 𝐶) → {𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))} = {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))})
56	45, 46, 55	csbie2 3886	. . . 4 ⊢ ⦋𝐵 / 𝑣⦌⦋𝐶 / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))} = {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))}
57	56	a1i 11	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⦋𝐵 / 𝑣⦌⦋𝐶 / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))} = {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))})
58	38, 44, 57	3eqtrd 2800	. 2 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⦋(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 ∈ (𝐶 ↑m 𝐵) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))})
59		df-rhm 20695	. 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))))})
60		ovex 7451	. . 3 ⊢ (𝐶 ↑m 𝐵) ∈ V
61	60	rabex 5300	. 2 ⊢ {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))} ∈ V
62	58, 59, 61	ovmpoa 7573	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝑅 RingHom 𝑆) = {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413  ⦋csb 3847  ‘cfv 6537  (class class class)co 7418   ↑_m cmap 8840  Basecbs 17380  +_gcplusg 17421  ._rcmulr 17422  1_rcur 20400  Ringcrg 20452   RingHom crh 20692

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-rhm 20695

This theorem is used by:  isrhm0  20699