Theorem rnghmmul 20532

Description: A homomorphism of non-unital rings preserves multiplication. (Contributed by AV, 23-Feb-2020.)

Hypotheses

Ref	Expression
rnghmmul.x	⊢ 𝑋 = (Base‘𝑅)
rnghmmul.m	⊢ · = (.r‘𝑅)
rnghmmul.n	⊢ × = (.r‘𝑆)

Assertion

Ref	Expression
rnghmmul	⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵)))

Proof of Theorem rnghmmul

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnghmmul.x	. . . 4 ⊢ 𝑋 = (Base‘𝑅)
2		rnghmmul.m	. . . 4 ⊢ · = (.r‘𝑅)
3		rnghmmul.n	. . . 4 ⊢ × = (.r‘𝑆)
4	1, 2, 3	isrnghm 20524	. . 3 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))))
5		fvoveq1 7435	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝐴 · 𝑦)))
6		fveq2 6883	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
7	6	oveq1d 7427	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) × (𝐹‘𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦)))
8	5, 7	eqeq12d 2779	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)) ↔ (𝐹‘(𝐴 · 𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦))))
9		oveq2 7420	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
10	9	fveq2d 6887	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴 · 𝑦)) = (𝐹‘(𝐴 · 𝐵)))
11		fveq2 6883	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
12	11	oveq2d 7428	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴) × (𝐹‘𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝐵)))
13	10, 12	eqeq12d 2779	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐹‘(𝐴 · 𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦)) ↔ (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))))
14	8, 13	rspc2va 3594	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵)))
15	14	expcom 418	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))))
16	15	ad2antll 741	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))))
17	4, 16	sylbi 220	. 2 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))))
18	17	3impib 1134	1 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  ._rcmulr 17312   GrpHom cghm 19284  Rngcrng 20231   RngHom crnghm 20517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-map 8827  df-ghm 19285  df-abl 19854  df-rng 20232  df-rnghm 20519

This theorem is referenced by:  rngisom1  20549