rngogrphom - Mathbox for Jeff Madsen

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Theorem rngogrphom 37958

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

**Hypotheses**
Ref	Expression
rnggrphom.1	⊢ 𝐺 = (1^st ‘𝑅)
rnggrphom.2	⊢ 𝐽 = (1^st ‘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 = (1^st ‘𝑅)
2		eqid 2735	. . 3 ⊢ ran 𝐺 = ran 𝐺
3		rnggrphom.2	. . 3 ⊢ 𝐽 = (1^st ‘𝑆)
4		eqid 2735	. . 3 ⊢ ran 𝐽 = ran 𝐽
5	1, 2, 3, 4	rngohomf 37953	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:ran 𝐺⟶ran 𝐽)
6	1, 2, 3	rngohomadd 37956	. . . 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 2741	. . 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 3200	. 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 37897	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
10	3	rngogrpo 37897	. . . 4 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
11	2, 4	elghomOLD 37874	. . . 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 596	. . 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 1131	. 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 713	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1^st c1st 8011 GrpOpcgr 30518 GrpOpHom cghomOLD 37870 RingOpscrngo 37881 RingOpsHom crngohom 37947

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-ablo 30574 df-ghomOLD 37871 df-rngo 37882 df-rngohom 37950

This theorem is referenced by: rngohom0 37959 rngohomsub 37960 rngokerinj 37962

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