Theorem rngohom0 38623

Description: Obsolete theorem, use rhm0 20571 instead. A ring homomorphism preserves 0. (Contributed by Jeff Madsen, 2-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
rnghom0.1	⊢ 𝐺 = (1st ‘𝑅)
rnghom0.2	⊢ 𝑍 = (GId‘𝐺)
rnghom0.3	⊢ 𝐽 = (1st ‘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 = (1st ‘𝑅)
2	1	rngogrpo 38561	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3	2	3ad2ant1 1151	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐺 ∈ GrpOp)
4		rnghom0.3	. . . 4 ⊢ 𝐽 = (1st ‘𝑆)
5	4	rngogrpo 38561	. . 3 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
6	5	3ad2ant2 1152	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐽 ∈ GrpOp)
7	1, 4	rngogrphom 38622	. 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 38540	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) → (𝐹‘𝑍) = 𝑊)
11	3, 6, 7, 10	syl3anc 1398	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑍) = 𝑊)

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  GrpOpcgr 30841  GIdcgi 30842   GrpOpHom cghomOLD 38534  RingOpscrngo 38545   RingOpsHom crngohom 38611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-grpo 30845  df-gid 30846  df-ablo 30897  df-ghomOLD 38535  df-rngo 38546  df-rngohom 38614

This theorem is referenced by:  keridl  38683