rngohom0 - Mathbox for Jeff Madsen

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Theorem rngohom0 37938

Description: A ring homomorphism preserves 0. (Contributed by Jeff Madsen, 2-Jan-2011.)

**Hypotheses**
Ref	Expression
rnghom0.1	⊢ 𝐺 = (1^st ‘𝑅)
rnghom0.2	⊢ 𝑍 = (GId‘𝐺)
rnghom0.3	⊢ 𝐽 = (1^st ‘𝑆)
rnghom0.4	⊢ 𝑊 = (GId‘𝐽)

**Assertion**
Ref	Expression
rngohom0	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑍) = 𝑊)

**Proof of Theorem rngohom0**
Step	Hyp	Ref	Expression
1		rnghom0.1	. . . 4 ⊢ 𝐺 = (1^st ‘𝑅)
2	1	rngogrpo 37876	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3	2	3ad2ant1 1133	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐺 ∈ GrpOp)
4		rnghom0.3	. . . 4 ⊢ 𝐽 = (1^st ‘𝑆)
5	4	rngogrpo 37876	. . 3 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
6	5	3ad2ant2 1134	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐽 ∈ GrpOp)
7	1, 4	rngogrphom 37937	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))
8		rnghom0.2	. . 3 ⊢ 𝑍 = (GId‘𝐺)
9		rnghom0.4	. . 3 ⊢ 𝑊 = (GId‘𝐽)
10	8, 9	ghomidOLD 37855	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) → (𝐹‘𝑍) = 𝑊)
11	3, 6, 7, 10	syl3anc 1372	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑍) = 𝑊)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ‘cfv 6541 (class class class)co 7413 1^st c1st 7994 GrpOpcgr 30436 GIdcgi 30437 GrpOpHom cghomOLD 37849 RingOpscrngo 37860 RingOpsHom crngohom 37926

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7996 df-2nd 7997 df-map 8850 df-grpo 30440 df-gid 30441 df-ablo 30492 df-ghomOLD 37850 df-rngo 37861 df-rngohom 37929

This theorem is referenced by: keridl 37998

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