Theorem isrhm0 20757

Description: The predicate "is a ring homomorphism from 𝑅 to 𝑆". (Contributed by Jeff Madsen, 19-Jun-2010.) (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
isrhm0	⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ (𝐹:𝐵⟶𝐶 ∧ (𝐹‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))))))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

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

Proof of Theorem isrhm0

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rhmval0.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
2		rhmval0.c	. . . 4 ⊢ 𝐶 = (Base‘𝑆)
3		rhmval0.1	. . . 4 ⊢ 1 = (1r‘𝑅)
4		rhmval0.i	. . . 4 ⊢ 𝑁 = (1r‘𝑆)
5		rhmval0.m	. . . 4 ⊢ · = (.r‘𝑅)
6		rhmval0.n	. . . 4 ⊢ × = (.r‘𝑆)
7		rhmval0.p	. . . 4 ⊢ + = (+g‘𝑅)
8		rhmval0.q	. . . 4 ⊢ ⨣ = (+g‘𝑆)
9	1, 2, 3, 4, 5, 6, 7, 8	rhmval0 20756	. . 3 ⊢ ((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))))})
10	9	eleq2d 2850	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ 𝐹 ∈ {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))))}))
11	2	fvexi 6930	. . . . 5 ⊢ 𝐶 ∈ V
12	1	fvexi 6930	. . . . 5 ⊢ 𝐵 ∈ V
13	11, 12	elmap 8937	. . . 4 ⊢ (𝐹 ∈ (𝐶 ↑m 𝐵) ↔ 𝐹:𝐵⟶𝐶)
14	13	anbi1i 636	. . 3 ⊢ ((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))))) ↔ (𝐹:𝐵⟶𝐶 ∧ ((𝐹‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))))))
15		fveq1 6915	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘ 1 ) = (𝐹‘ 1 ))
16	15	eqeq1d 2765	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘ 1 ) = 𝑁 ↔ (𝐹‘ 1 ) = 𝑁))
17		fveq1 6915	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
18		fveq1 6915	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
19		fveq1 6915	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
20	18, 19	oveq12d 7469	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
21	17, 20	eqeq12d 2779	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))
22		fveq1 6915	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
23	18, 19	oveq12d 7469	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) × (𝑓‘𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))
24	22, 23	eqeq12d 2779	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))))
25	21, 24	anbi12d 644	. . . . . 6 ⊢ (𝑓 = 𝐹 → (((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))) ↔ ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))))
26	25	2ralbidv 3230	. . . . 5 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((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)))))
27	16, 26	anbi12d 644	. . . 4 ⊢ (𝑓 = 𝐹 → (((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((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))))))
28	27	elrab 3648	. . 3 ⊢ (𝐹 ∈ {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))))} ↔ (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))))))
29		3anass 1111	. . 3 ⊢ ((𝐹:𝐵⟶𝐶 ∧ (𝐹‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))) ↔ (𝐹:𝐵⟶𝐶 ∧ ((𝐹‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))))))
30	14, 28, 29	3bitr4i 306	. 2 ⊢ (𝐹 ∈ {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))))} ↔ (𝐹:𝐵⟶𝐶 ∧ (𝐹‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))))
31	10, 30	bitrdi 290	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ (𝐹:𝐵⟶𝐶 ∧ (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   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ↑_m cmap 8885  Basecbs 17431  +_gcplusg 17472  ._rcmulr 17473  1_rcur 20458  Ringcrg 20510   RingHom crh 20750

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-map 8887  df-rhm 20753

This theorem is used by: (None)