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Theorem rngohom0 35893
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 RngHom 𝑆)) → (𝐹𝑍) = 𝑊)

Proof of Theorem rngohom0
StepHypRef Expression
1 rnghom0.1 . . . 4 𝐺 = (1st𝑅)
21rngogrpo 35831 . . 3 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
323ad2ant1 1135 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐺 ∈ GrpOp)
4 rnghom0.3 . . . 4 𝐽 = (1st𝑆)
54rngogrpo 35831 . . 3 (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
653ad2ant2 1136 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐽 ∈ GrpOp)
71, 4rngogrphom 35892 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))
8 rnghom0.2 . . 3 𝑍 = (GId‘𝐺)
9 rnghom0.4 . . 3 𝑊 = (GId‘𝐽)
108, 9ghomidOLD 35810 . 2 ((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) → (𝐹𝑍) = 𝑊)
113, 6, 7, 10syl3anc 1373 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹𝑍) = 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089   = wceq 1543  wcel 2111  cfv 6397  (class class class)co 7231  1st c1st 7777  GrpOpcgr 28594  GIdcgi 28595   GrpOpHom cghomOLD 35804  RingOpscrngo 35815   RngHom crnghom 35881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5272  ax-pr 5336  ax-un 7541
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3422  df-sbc 3709  df-csb 3826  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-pw 4529  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-iun 4920  df-br 5068  df-opab 5130  df-mpt 5150  df-id 5469  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-res 5577  df-ima 5578  df-iota 6355  df-fun 6399  df-fn 6400  df-f 6401  df-f1 6402  df-fo 6403  df-f1o 6404  df-fv 6405  df-riota 7188  df-ov 7234  df-oprab 7235  df-mpo 7236  df-1st 7779  df-2nd 7780  df-map 8530  df-grpo 28598  df-gid 28599  df-ablo 28650  df-ghomOLD 35805  df-rngo 35816  df-rngohom 35884
This theorem is referenced by:  keridl  35953
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