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Theorem rnghmmul 44099
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 44091 . . 3 (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))))
5 fvoveq1 7168 . . . . . . 7 (𝑥 = 𝐴 → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝐴 · 𝑦)))
6 fveq2 6663 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
76oveq1d 7160 . . . . . . 7 (𝑥 = 𝐴 → ((𝐹𝑥) × (𝐹𝑦)) = ((𝐹𝐴) × (𝐹𝑦)))
85, 7eqeq12d 2834 . . . . . 6 (𝑥 = 𝐴 → ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)) ↔ (𝐹‘(𝐴 · 𝑦)) = ((𝐹𝐴) × (𝐹𝑦))))
9 oveq2 7153 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
109fveq2d 6667 . . . . . . 7 (𝑦 = 𝐵 → (𝐹‘(𝐴 · 𝑦)) = (𝐹‘(𝐴 · 𝐵)))
11 fveq2 6663 . . . . . . . 8 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
1211oveq2d 7161 . . . . . . 7 (𝑦 = 𝐵 → ((𝐹𝐴) × (𝐹𝑦)) = ((𝐹𝐴) × (𝐹𝐵)))
1310, 12eqeq12d 2834 . . . . . 6 (𝑦 = 𝐵 → ((𝐹‘(𝐴 · 𝑦)) = ((𝐹𝐴) × (𝐹𝑦)) ↔ (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
148, 13rspc2va 3631 . . . . 5 (((𝐴𝑋𝐵𝑋) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦))) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵)))
1514expcom 414 . . . 4 (∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)) → ((𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
1615ad2antll 725 . . 3 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))) → ((𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
174, 16sylbi 218 . 2 (𝐹 ∈ (𝑅 RngHomo 𝑆) → ((𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵))))
18173impib 1108 1 ((𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145  Basecbs 16471  .rcmulr 16554   GrpHom cghm 18293  Rngcrng 44073   RngHomo crngh 44084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-ghm 18294  df-abl 18838  df-rng0 44074  df-rnghomo 44086
This theorem is referenced by: (None)
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