Theorem rnghmghm 20418

Description: A non-unital ring homomorphism is an additive group homomorphism. (Contributed by AV, 23-Feb-2020.)

Assertion

Ref	Expression
rnghmghm	⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))

Proof of Theorem rnghmghm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2739	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2739	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		eqid 2739	. . 3 ⊢ (.r‘𝑆) = (.r‘𝑆)
4	1, 2, 3	isrnghm 20412	. 2 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)))))
5		simprl 776	. 2 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)))) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
6	4, 5	sylbi 218	1 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  ._rcmulr 17212   GrpHom cghm 19178  Rngcrng 20124   RngHom crnghm 20405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8765  df-ghm 19179  df-abl 19749  df-rng 20125  df-rnghm 20407

This theorem is referenced by:  rnghmf  20419  rnghmf1o  20423  rnghmco  20428  zrinitorngc  20614