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Theorem rnghmmul 41690
 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 · = (.r𝑅)
rnghmmul.n × = (.r𝑆)
Assertion
Ref Expression
rnghmmul ((𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵)))

Proof of Theorem rnghmmul
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnghmmul.x . . . 4 𝑋 = (Base‘𝑅)
2 rnghmmul.m . . . 4 · = (.r𝑅)
3 rnghmmul.n . . . 4 × = (.r𝑆)
41, 2, 3isrnghm 41682 . . 3 (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))))
5 oveq1 6556 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 · 𝑦) = (𝐴 · 𝑦))
65fveq2d 6107 . . . . . . 7 (𝑥 = 𝐴 → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝐴 · 𝑦)))
7 fveq2 6103 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
87oveq1d 6564 . . . . . . 7 (𝑥 = 𝐴 → ((𝐹𝑥) × (𝐹𝑦)) = ((𝐹𝐴) × (𝐹𝑦)))
96, 8eqeq12d 2625 . . . . . 6 (𝑥 = 𝐴 → ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)) ↔ (𝐹‘(𝐴 · 𝑦)) = ((𝐹𝐴) × (𝐹𝑦))))
10 oveq2 6557 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
1110fveq2d 6107 . . . . . . 7 (𝑦 = 𝐵 → (𝐹‘(𝐴 · 𝑦)) = (𝐹‘(𝐴 · 𝐵)))
12 fveq2 6103 . . . . . . . 8 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
1312oveq2d 6565 . . . . . . 7 (𝑦 = 𝐵 → ((𝐹𝐴) × (𝐹𝑦)) = ((𝐹𝐴) × (𝐹𝐵)))
1411, 13eqeq12d 2625 . . . . . 6 (𝑦 = 𝐵 → ((𝐹‘(𝐴 · 𝑦)) = ((𝐹𝐴) × (𝐹𝑦)) ↔ (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
159, 14rspc2va 3294 . . . . 5 (((𝐴𝑋𝐵𝑋) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦))) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵)))
1615expcom 450 . . . 4 (∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)) → ((𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
1716ad2antll 761 . . 3 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))) → ((𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
184, 17sylbi 206 . 2 (𝐹 ∈ (𝑅 RngHomo 𝑆) → ((𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
19183impib 1254 1 ((𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  .rcmulr 15769   GrpHom cghm 17480  Rngcrng 41664   RngHomo crngh 41675 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-ghm 17481  df-abl 18019  df-rng0 41665  df-rnghomo 41677 This theorem is referenced by: (None)
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