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Theorem rngohomsub 26280
Description: Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)
Hypotheses
Ref Expression
rnghomsub.1  |-  G  =  ( 1st `  R
)
rnghomsub.2  |-  X  =  ran  G
rnghomsub.3  |-  H  =  (  /g  `  G
)
rnghomsub.4  |-  J  =  ( 1st `  S
)
rnghomsub.5  |-  K  =  (  /g  `  J
)
Assertion
Ref Expression
rngohomsub  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  ( A H B ) )  =  ( ( F `
 A ) K ( F `  B
) ) )

Proof of Theorem rngohomsub
StepHypRef Expression
1 rnghomsub.1 . . . . 5  |-  G  =  ( 1st `  R
)
21rngogrpo 21826 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
323ad2ant1 978 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  G  e.  GrpOp
)
4 rnghomsub.4 . . . . 5  |-  J  =  ( 1st `  S
)
54rngogrpo 21826 . . . 4  |-  ( S  e.  RingOps  ->  J  e.  GrpOp )
653ad2ant2 979 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  J  e.  GrpOp
)
71, 4rngogrphom 26278 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F  e.  ( G GrpOpHom  J ) )
83, 6, 73jca 1134 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( G  e.  GrpOp  /\  J  e.  GrpOp  /\  F  e.  ( G GrpOpHom  J ) ) )
9 rnghomsub.2 . . 3  |-  X  =  ran  G
10 rnghomsub.3 . . 3  |-  H  =  (  /g  `  G
)
11 rnghomsub.5 . . 3  |-  K  =  (  /g  `  J
)
129, 10, 11ghomdiv 26250 . 2  |-  ( ( ( G  e.  GrpOp  /\  J  e.  GrpOp  /\  F  e.  ( G GrpOpHom  J )
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  ( A H B ) )  =  ( ( F `
 A ) K ( F `  B
) ) )
138, 12sylan 458 1  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  ( A H B ) )  =  ( ( F `
 A ) K ( F `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ran crn 4819   ` cfv 5394  (class class class)co 6020   1stc1st 6286   GrpOpcgr 21622    /g cgs 21625   GrpOpHom cghom 21793   RingOpscrngo 21811    RngHom crnghom 26267
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-map 6956  df-grpo 21627  df-gid 21628  df-ginv 21629  df-gdiv 21630  df-ablo 21718  df-ghom 21794  df-rngo 21812  df-rngohom 26270
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