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Theorem rnghmmul 42225
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 42217 . . 3 (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))))
5 oveq1 6697 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 · 𝑦) = (𝐴 · 𝑦))
65fveq2d 6233 . . . . . . 7 (𝑥 = 𝐴 → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝐴 · 𝑦)))
7 fveq2 6229 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
87oveq1d 6705 . . . . . . 7 (𝑥 = 𝐴 → ((𝐹𝑥) × (𝐹𝑦)) = ((𝐹𝐴) × (𝐹𝑦)))
96, 8eqeq12d 2666 . . . . . 6 (𝑥 = 𝐴 → ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)) ↔ (𝐹‘(𝐴 · 𝑦)) = ((𝐹𝐴) × (𝐹𝑦))))
10 oveq2 6698 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
1110fveq2d 6233 . . . . . . 7 (𝑦 = 𝐵 → (𝐹‘(𝐴 · 𝑦)) = (𝐹‘(𝐴 · 𝐵)))
12 fveq2 6229 . . . . . . . 8 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
1312oveq2d 6706 . . . . . . 7 (𝑦 = 𝐵 → ((𝐹𝐴) × (𝐹𝑦)) = ((𝐹𝐴) × (𝐹𝐵)))
1411, 13eqeq12d 2666 . . . . . 6 (𝑦 = 𝐵 → ((𝐹‘(𝐴 · 𝑦)) = ((𝐹𝐴) × (𝐹𝑦)) ↔ (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
159, 14rspc2va 3354 . . . . 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 765 . . 3 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))) → ((𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
184, 17sylbi 207 . 2 (𝐹 ∈ (𝑅 RngHomo 𝑆) → ((𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
19183impib 1281 1 ((𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cfv 5926  (class class class)co 6690  Basecbs 15904  .rcmulr 15989   GrpHom cghm 17704  Rngcrng 42199   RngHomo crngh 42210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-ghm 17705  df-abl 18242  df-rng0 42200  df-rnghomo 42212
This theorem is referenced by: (None)
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