rngohom0 - Mathbox for Jeff Madsen

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Theorem rngohom0 38510

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 38448	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3	2	3ad2ant1 1149	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐺 ∈ GrpOp)
4		rnghom0.3	. . . 4 ⊢ 𝐽 = (1^st ‘𝑆)
5	4	rngogrpo 38448	. . 3 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
6	5	3ad2ant2 1150	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐽 ∈ GrpOp)
7	1, 4	rngogrphom 38509	. 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 38427	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) → (𝐹‘𝑍) = 𝑊)
11	3, 6, 7, 10	syl3anc 1396	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑍) = 𝑊)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ‘cfv 6537 (class class class)co 7411 1^st c1st 7983 GrpOpcgr 30781 GIdcgi 30782 GrpOpHom cghomOLD 38421 RingOpscrngo 38432 RingOpsHom crngohom 38498

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5557 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7985 df-2nd 7986 df-map 8825 df-grpo 30785 df-gid 30786 df-ablo 30837 df-ghomOLD 38422 df-rngo 38433 df-rngohom 38501

This theorem is referenced by: keridl 38570

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