rngogrphom - Mathbox for Jeff Madsen

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Theorem rngogrphom 38174

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 2737	. . 3 ⊢ ran 𝐺 = ran 𝐺
3		rnggrphom.2	. . 3 ⊢ 𝐽 = (1^st ‘𝑆)
4		eqid 2737	. . 3 ⊢ ran 𝐽 = ran 𝐽
5	1, 2, 3, 4	rngohomf 38169	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:ran 𝐺⟶ran 𝐽)
6	1, 2, 3	rngohomadd 38172	. . . 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 2743	. . 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 3180	. 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 38113	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
10	3	rngogrpo 38113	. . . 4 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
11	2, 4	elghomOLD 38090	. . . 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 597	. . 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 1133	. 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 714	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 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ran crn 5626 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 1^st c1st 7933 GrpOpcgr 30568 GrpOpHom cghomOLD 38086 RingOpscrngo 38097 RingOpsHom crngohom 38163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-map 8769 df-ablo 30624 df-ghomOLD 38087 df-rngo 38098 df-rngohom 38166

This theorem is referenced by: rngohom0 38175 rngohomsub 38176 rngokerinj 38178

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