Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngohom0 Structured version   Visualization version   GIF version

Theorem rngohom0 37353
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 37291 . . 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 37291 . . 3 (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
653ad2ant2 1131 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐽 ∈ GrpOp)
71, 4rngogrphom 37352 . 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 37270 . 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 6537  (class class class)co 7405  1st c1st 7972  GrpOpcgr 30251  GIdcgi 30252   GrpOpHom cghomOLD 37264  RingOpscrngo 37275   RingOpsHom crngohom 37341
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-map 8824  df-grpo 30255  df-gid 30256  df-ablo 30307  df-ghomOLD 37265  df-rngo 37276  df-rngohom 37344
This theorem is referenced by:  keridl  37413
  Copyright terms: Public domain W3C validator