Theorem rnghmmul 45798

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 RngHomo 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 45790	. . 3 ⊢ (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))))
5		fvoveq1 7352	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝐴 · 𝑦)))
6		fveq2 6819	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
7	6	oveq1d 7344	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) × (𝐹‘𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦)))
8	5, 7	eqeq12d 2752	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)) ↔ (𝐹‘(𝐴 · 𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦))))
9		oveq2 7337	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
10	9	fveq2d 6823	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴 · 𝑦)) = (𝐹‘(𝐴 · 𝐵)))
11		fveq2 6819	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
12	11	oveq2d 7345	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴) × (𝐹‘𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝐵)))
13	10, 12	eqeq12d 2752	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐹‘(𝐴 · 𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦)) ↔ (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))))
14	8, 13	rspc2va 3580	. . . . 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 414	. . . 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 726	. . 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 216	. 2 ⊢ (𝐹 ∈ (𝑅 RngHomo 𝑆) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))))
18	17	3impib 1115	1 ⊢ ((𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3061  ‘cfv 6473  (class class class)co 7329  Basecbs 17001  ._rcmulr 17052   GrpHom cghm 18919  Rngcrng 45772   RngHomo crngh 45783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-ov 7332  df-oprab 7333  df-mpo 7334  df-map 8680  df-ghm 18920  df-abl 19476  df-rng0 45773  df-rnghomo 45785

This theorem is referenced by: (None)