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Theorem rngohom0 36434
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 36372 . . 3 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
323ad2ant1 1134 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐺 ∈ GrpOp)
4 rnghom0.3 . . . 4 𝐽 = (1st𝑆)
54rngogrpo 36372 . . 3 (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
653ad2ant2 1135 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐽 ∈ GrpOp)
71, 4rngogrphom 36433 . 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 36351 . 2 ((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) → (𝐹𝑍) = 𝑊)
113, 6, 7, 10syl3anc 1372 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹𝑍) = 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  cfv 6497  (class class class)co 7358  1st c1st 7920  GrpOpcgr 29434  GIdcgi 29435   GrpOpHom cghomOLD 36345  RingOpscrngo 36356   RngHom crnghom 36422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8768  df-grpo 29438  df-gid 29439  df-ablo 29490  df-ghomOLD 36346  df-rngo 36357  df-rngohom 36425
This theorem is referenced by:  keridl  36494
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