Theorem rnghmmul 20466

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 20458	. . 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 7454	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝐴 · 𝑦)))
6		fveq2 6907	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
7	6	oveq1d 7446	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) × (𝐹‘𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦)))
8	5, 7	eqeq12d 2751	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)) ↔ (𝐹‘(𝐴 · 𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦))))
9		oveq2 7439	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
10	9	fveq2d 6911	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴 · 𝑦)) = (𝐹‘(𝐴 · 𝐵)))
11		fveq2 6907	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
12	11	oveq2d 7447	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴) × (𝐹‘𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝐵)))
13	10, 12	eqeq12d 2751	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐹‘(𝐴 · 𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦)) ↔ (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))))
14	8, 13	rspc2va 3634	. . . . 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 413	. . . 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 729	. . 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 217	. 2 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))))
18	17	3impib 1115	1 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  ._rcmulr 17299   GrpHom cghm 19243  Rngcrng 20170   RngHom crnghm 20451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-ghm 19244  df-abl 19816  df-rng 20171  df-rnghm 20453

This theorem is referenced by:  rngisom1  20483