rngogrphom - Mathbox for Jeff Madsen

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Theorem rngogrphom 38312

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 38307	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:ran 𝐺⟶ran 𝐽)
6	1, 2, 3	rngohomadd 38310	. . . 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 3181	. 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 38251	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
10	3	rngogrpo 38251	. . . 4 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
11	2, 4	elghomOLD 38228	. . . 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 5627 ⟶wf 6490 ‘cfv 6494 (class class class)co 7362 1^st c1st 7935 GrpOpcgr 30579 GrpOpHom cghomOLD 38224 RingOpscrngo 38235 RingOpsHom crngohom 38301

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684

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 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5521 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7937 df-2nd 7938 df-map 8770 df-ablo 30635 df-ghomOLD 38225 df-rngo 38236 df-rngohom 38304

This theorem is referenced by: rngohom0 38313 rngohomsub 38314 rngokerinj 38316

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