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Theorem rnghmghm 41690
Description: A non-unital ring homomorphism is an additive group homomorphism. (Contributed by AV, 23-Feb-2020.)
Assertion
Ref Expression
rnghmghm (𝐹 ∈ (𝑅 RngHomo 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))

Proof of Theorem rnghmghm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2609 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2609 . . 3 (.r𝑅) = (.r𝑅)
3 eqid 2609 . . 3 (.r𝑆) = (.r𝑆)
41, 2, 3isrnghm 41684 . 2 (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥)(.r𝑆)(𝐹𝑦)))))
5 simprl 789 . 2 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥)(.r𝑆)(𝐹𝑦)))) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
64, 5sylbi 205 1 (𝐹 ∈ (𝑅 RngHomo 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  cfv 5790  (class class class)co 6527  Basecbs 15641  .rcmulr 15715   GrpHom cghm 17426  Rngcrng 41666   RngHomo crngh 41677
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 2032  ax-13 2232  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 41667  df-rnghomo 41679
This theorem is referenced by:  rnghmf  41691  rnghmf1o  41695  rnghmco  41699  zrinitorngc  41794
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