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Theorem rngohom0 26280
Description: A ring homomorphism preserves  0. (Contributed by Jeff Madsen, 2-Jan-2011.)
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  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  Z )  =  W )

Proof of Theorem rngohom0
StepHypRef Expression
1 rnghom0.1 . . . 4  |-  G  =  ( 1st `  R
)
21rngogrpo 21827 . . 3  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
323ad2ant1 978 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  G  e.  GrpOp
)
4 rnghom0.3 . . . 4  |-  J  =  ( 1st `  S
)
54rngogrpo 21827 . . 3  |-  ( S  e.  RingOps  ->  J  e.  GrpOp )
653ad2ant2 979 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  J  e.  GrpOp
)
71, 4rngogrphom 26279 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F  e.  ( G GrpOpHom  J ) )
8 rnghom0.2 . . 3  |-  Z  =  (GId `  G )
9 rnghom0.4 . . 3  |-  W  =  (GId `  J )
108, 9ghomid 21802 . 2  |-  ( ( G  e.  GrpOp  /\  J  e.  GrpOp  /\  F  e.  ( G GrpOpHom  J ) )  ->  ( F `  Z )  =  W )
113, 6, 7, 10syl3anc 1184 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  Z )  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021   1stc1st 6287   GrpOpcgr 21623  GIdcgi 21624   GrpOpHom cghom 21794   RingOpscrngo 21812    RngHom crnghom 26268
This theorem is referenced by:  keridl  26334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-map 6957  df-grpo 21628  df-gid 21629  df-ablo 21719  df-ghom 21795  df-rngo 21813  df-rngohom 26271
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