Theorem rngogrphom 38475

Description: A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.)

Hypotheses

Ref	Expression
rnggrphom.1	⊢ 𝐺 = (1st ‘𝑅)
rnggrphom.2	⊢ 𝐽 = (1st ‘𝑆)

Assertion

Ref	Expression
rngogrphom	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))

Proof of Theorem rngogrphom

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

Step	Hyp	Ref	Expression
1		rnggrphom.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2		eqid 2764	. . 3 ⊢ ran 𝐺 = ran 𝐺
3		rnggrphom.2	. . 3 ⊢ 𝐽 = (1st ‘𝑆)
4		eqid 2764	. . 3 ⊢ ran 𝐽 = ran 𝐽
5	1, 2, 3, 4	rngohomf 38470	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:ran 𝐺⟶ran 𝐽)
6	1, 2, 3	rngohomadd 38473	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → (𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
7	6	eqcomd 2770	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
8	7	ralrimivva 3207	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
9	1	rngogrpo 38414	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
10	3	rngogrpo 38414	. . . 4 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
11	2, 4	elghomOLD 38391	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐽) ↔ (𝐹:ran 𝐺⟶ran 𝐽 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
12	9, 10, 11	syl2an 605	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝐺 GrpOpHom 𝐽) ↔ (𝐹:ran 𝐺⟶ran 𝐽 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
13	12	3adant3 1146	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹 ∈ (𝐺 GrpOpHom 𝐽) ↔ (𝐹:ran 𝐺⟶ran 𝐽 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
14	5, 8, 13	mpbir2and 723	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ran crn 5650  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398  1^st c1st 7970  GrpOpcgr 30694   GrpOpHom cghomOLD 38387  RingOpscrngo 38398   RingOpsHom crngohom 38464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-map 8812  df-ablo 30750  df-ghomOLD 38388  df-rngo 38399  df-rngohom 38467

This theorem is referenced by:  rngohom0  38476  rngohomsub  38477  rngokerinj  38479