Theorem rnghmghm 20529

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 2769	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2769	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		eqid 2769	. . 3 ⊢ (.r‘𝑆) = (.r‘𝑆)
4	1, 2, 3	isrnghm 20523	. 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 782	. 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 220	1 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  ._rcmulr 17311   GrpHom cghm 19283  Rngcrng 20230   RngHom crnghm 20516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-map 8826  df-ghm 19284  df-abl 19853  df-rng 20231  df-rnghm 20518

This theorem is referenced by:  rnghmf  20530  rnghmf1o  20534  rnghmco  20539  zrinitorngc  20727