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Theorem rnghmmul 41685
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 41677 . . 3 (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))))
5 oveq1 6534 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 · 𝑦) = (𝐴 · 𝑦))
65fveq2d 6092 . . . . . . 7 (𝑥 = 𝐴 → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝐴 · 𝑦)))
7 fveq2 6088 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
87oveq1d 6542 . . . . . . 7 (𝑥 = 𝐴 → ((𝐹𝑥) × (𝐹𝑦)) = ((𝐹𝐴) × (𝐹𝑦)))
96, 8eqeq12d 2624 . . . . . 6 (𝑥 = 𝐴 → ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)) ↔ (𝐹‘(𝐴 · 𝑦)) = ((𝐹𝐴) × (𝐹𝑦))))
10 oveq2 6535 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
1110fveq2d 6092 . . . . . . 7 (𝑦 = 𝐵 → (𝐹‘(𝐴 · 𝑦)) = (𝐹‘(𝐴 · 𝐵)))
12 fveq2 6088 . . . . . . . 8 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
1312oveq2d 6543 . . . . . . 7 (𝑦 = 𝐵 → ((𝐹𝐴) × (𝐹𝑦)) = ((𝐹𝐴) × (𝐹𝐵)))
1411, 13eqeq12d 2624 . . . . . 6 (𝑦 = 𝐵 → ((𝐹‘(𝐴 · 𝑦)) = ((𝐹𝐴) × (𝐹𝑦)) ↔ (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
159, 14rspc2va 3293 . . . . 5 (((𝐴𝑋𝐵𝑋) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦))) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵)))
1615expcom 449 . . . 4 (∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)) → ((𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
1716ad2antll 760 . . 3 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))) → ((𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
184, 17sylbi 205 . 2 (𝐹 ∈ (𝑅 RngHomo 𝑆) → ((𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
19183impib 1253 1 ((𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  cfv 5790  (class class class)co 6527  Basecbs 15641  .rcmulr 15715   GrpHom cghm 17426  Rngcrng 41659   RngHomo crngh 41670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-map 7723  df-ghm 17427  df-abl 17965  df-rng0 41660  df-rnghomo 41672
This theorem is referenced by: (None)
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