Theorem rnghmmul 20596

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 20588	. . 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 7440	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝐴 · 𝑦)))
6		fveq2 6882	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
7	6	oveq1d 7432	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) × (𝐹‘𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦)))
8	5, 7	eqeq12d 2778	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)) ↔ (𝐹‘(𝐴 · 𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦))))
9		oveq2 7425	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
10	9	fveq2d 6886	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴 · 𝑦)) = (𝐹‘(𝐴 · 𝐵)))
11		fveq2 6882	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
12	11	oveq2d 7433	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴) × (𝐹‘𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝐵)))
13	10, 12	eqeq12d 2778	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐹‘(𝐴 · 𝑦)) = ((𝐹‘𝐴) × (𝐹‘𝑦)) ↔ (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))))
14	8, 13	rspc2va 3591	. . . . 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 1570   ∈ wcel 2145  ∀wral 3078  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  ._rcmulr 17349   GrpHom cghm 19346  Rngcrng 20293   RngHom crnghm 20581

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-map 8832  df-ghm 19347  df-abl 19916  df-rng 20294  df-rnghm 20583

This theorem is used by:  rngisom1  20613