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Theorem rngogrphom 32940
 Description: A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.)
Hypotheses
Ref Expression
rnggrphom.1 𝐺 = (1st𝑅)
rnggrphom.2 𝐽 = (1st𝑆)
Assertion
Ref Expression
rngogrphom ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))

Proof of Theorem rngogrphom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnggrphom.1 . . 3 𝐺 = (1st𝑅)
2 eqid 2610 . . 3 ran 𝐺 = ran 𝐺
3 rnggrphom.2 . . 3 𝐽 = (1st𝑆)
4 eqid 2610 . . 3 ran 𝐽 = ran 𝐽
51, 2, 3, 4rngohomf 32935 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:ran 𝐺⟶ran 𝐽)
61, 2, 3rngohomadd 32938 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
76eqcomd 2616 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝐹𝑥)𝐽(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
87ralrimivva 2954 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐽(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
91rngogrpo 32879 . . . 4 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
103rngogrpo 32879 . . . 4 (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
112, 4elghomOLD 32856 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐽) ↔ (𝐹:ran 𝐺⟶ran 𝐽 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐽(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
129, 10, 11syl2an 493 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝐺 GrpOpHom 𝐽) ↔ (𝐹:ran 𝐺⟶ran 𝐽 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐽(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
13123adant3 1074 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹 ∈ (𝐺 GrpOpHom 𝐽) ↔ (𝐹:ran 𝐺⟶ran 𝐽 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐽(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
145, 8, 13mpbir2and 959 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  GrpOpcgr 26727   GrpOpHom cghomOLD 32852  RingOpscrngo 32863   RngHom crnghom 32929 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-ablo 26783  df-ghomOLD 32853  df-rngo 32864  df-rngohom 32932 This theorem is referenced by:  rngohom0  32941  rngohomsub  32942  rngokerinj  32944
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