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Theorem rngohom0 37478
Description: A ring homomorphism preserves 0. (Contributed by Jeff Madsen, 2-Jan-2011.)
Hypotheses
Ref Expression
rnghom0.1 𝐺 = (1st𝑅)
rnghom0.2 𝑍 = (GId‘𝐺)
rnghom0.3 𝐽 = (1st𝑆)
rnghom0.4 𝑊 = (GId‘𝐽)
Assertion
Ref Expression
rngohom0 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹𝑍) = 𝑊)

Proof of Theorem rngohom0
StepHypRef Expression
1 rnghom0.1 . . . 4 𝐺 = (1st𝑅)
21rngogrpo 37416 . . 3 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
323ad2ant1 1130 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐺 ∈ GrpOp)
4 rnghom0.3 . . . 4 𝐽 = (1st𝑆)
54rngogrpo 37416 . . 3 (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
653ad2ant2 1131 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐽 ∈ GrpOp)
71, 4rngogrphom 37477 . 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)
108, 9ghomidOLD 37395 . 2 ((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) → (𝐹𝑍) = 𝑊)
113, 6, 7, 10syl3anc 1368 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹𝑍) = 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  cfv 6553  (class class class)co 7426  1st c1st 7997  GrpOpcgr 30319  GIdcgi 30320   GrpOpHom cghomOLD 37389  RingOpscrngo 37400   RingOpsHom crngohom 37466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-map 8853  df-grpo 30323  df-gid 30324  df-ablo 30375  df-ghomOLD 37390  df-rngo 37401  df-rngohom 37469
This theorem is referenced by:  keridl  37538
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