rngohom0 - Mathbox for Jeff Madsen

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Theorem rngohom0 37966

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 37904	. . 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 37904	. . 3 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
6	5	3ad2ant2 1134	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐽 ∈ GrpOp)
7	1, 4	rngogrphom 37965	. 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 37883	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) → (𝐹‘𝑍) = 𝑊)
11	3, 6, 7, 10	syl3anc 1373	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 1540 ∈ wcel 2109 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 GrpOpcgr 30418 GIdcgi 30419 GrpOpHom cghomOLD 37877 RingOpscrngo 37888 RingOpsHom crngohom 37954

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-1st 7968 df-2nd 7969 df-map 8801 df-grpo 30422 df-gid 30423 df-ablo 30474 df-ghomOLD 37878 df-rngo 37889 df-rngohom 37957

This theorem is referenced by: keridl 38026

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