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Theorem rnghmmul 44366
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 44358 . . 3 (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))))
5 fvoveq1 7161 . . . . . . 7 (𝑥 = 𝐴 → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝐴 · 𝑦)))
6 fveq2 6651 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
76oveq1d 7153 . . . . . . 7 (𝑥 = 𝐴 → ((𝐹𝑥) × (𝐹𝑦)) = ((𝐹𝐴) × (𝐹𝑦)))
85, 7eqeq12d 2840 . . . . . 6 (𝑥 = 𝐴 → ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)) ↔ (𝐹‘(𝐴 · 𝑦)) = ((𝐹𝐴) × (𝐹𝑦))))
9 oveq2 7146 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
109fveq2d 6655 . . . . . . 7 (𝑦 = 𝐵 → (𝐹‘(𝐴 · 𝑦)) = (𝐹‘(𝐴 · 𝐵)))
11 fveq2 6651 . . . . . . . 8 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
1211oveq2d 7154 . . . . . . 7 (𝑦 = 𝐵 → ((𝐹𝐴) × (𝐹𝑦)) = ((𝐹𝐴) × (𝐹𝐵)))
1310, 12eqeq12d 2840 . . . . . 6 (𝑦 = 𝐵 → ((𝐹‘(𝐴 · 𝑦)) = ((𝐹𝐴) × (𝐹𝑦)) ↔ (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
148, 13rspc2va 3619 . . . . 5 (((𝐴𝑋𝐵𝑋) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦))) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵)))
1514expcom 417 . . . 4 (∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)) → ((𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
1615ad2antll 728 . . 3 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))) → ((𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
174, 16sylbi 220 . 2 (𝐹 ∈ (𝑅 RngHomo 𝑆) → ((𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
18173impib 1113 1 ((𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3132  cfv 6336  (class class class)co 7138  Basecbs 16472  .rcmulr 16555   GrpHom cghm 18344  Rngcrng 44340   RngHomo crngh 44351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-map 8391  df-ghm 18345  df-abl 18898  df-rng0 44341  df-rnghomo 44353
This theorem is referenced by: (None)
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