Theorem rnghmmul 20684

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 20676	. . 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 7443	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝐴 · 𝑦)))
6		fveq2 6885	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
7	6	oveq1d 7435	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) × (𝐹‘𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦)))
8	5, 7	eqeq12d 2776	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)) ↔ (𝐹‘(𝐴 · 𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦))))
9		oveq2 7428	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
10	9	fveq2d 6889	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴 · 𝑦)) = (𝐹‘(𝐴 · 𝐵)))
11		fveq2 6885	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
12	11	oveq2d 7436	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴) × (𝐹‘𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝐵)))
13	10, 12	eqeq12d 2776	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐹‘(𝐴 · 𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦)) ↔ (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))))
14	8, 13	rspc2va 3587	. . . . 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 419	. . . 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 742	. . 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)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  ._rcmulr 17434   GrpHom cghm 19432  Rngcrng 20379   RngHom crnghm 20669

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8852  df-ghm 19433  df-abl 20002  df-rng 20380  df-rnghm 20671

This theorem is used by:  rngisom1  20701