rngogrphom - Mathbox for Jeff Madsen

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Theorem rngogrphom 36056

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 RngHom 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 2738	. . 3 ⊢ ran 𝐺 = ran 𝐺
3		rnggrphom.2	. . 3 ⊢ 𝐽 = (1^st ‘𝑆)
4		eqid 2738	. . 3 ⊢ ran 𝐽 = ran 𝐽
5	1, 2, 3, 4	rngohomf 36051	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:ran 𝐺⟶ran 𝐽)
6	1, 2, 3	rngohomadd 36054	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → (𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
7	6	eqcomd 2744	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
8	7	ralrimivva 3114	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
9	1	rngogrpo 35995	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
10	3	rngogrpo 35995	. . . 4 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
11	2, 4	elghomOLD 35972	. . . 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 595	. . 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 1130	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 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 709	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ran crn 5581 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 GrpOpcgr 28752 GrpOpHom cghomOLD 35968 RingOpscrngo 35979 RngHom crnghom 36045

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-map 8575 df-ablo 28808 df-ghomOLD 35969 df-rngo 35980 df-rngohom 36048

This theorem is referenced by: rngohom0 36057 rngohomsub 36058 rngokerinj 36060

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