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Theorem isghmd 14970
Description: Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
Hypotheses
Ref Expression
isghmd.x  |-  X  =  ( Base `  S
)
isghmd.y  |-  Y  =  ( Base `  T
)
isghmd.a  |-  .+  =  ( +g  `  S )
isghmd.b  |-  .+^  =  ( +g  `  T )
isghmd.s  |-  ( ph  ->  S  e.  Grp )
isghmd.t  |-  ( ph  ->  T  e.  Grp )
isghmd.f  |-  ( ph  ->  F : X --> Y )
isghmd.l  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x  .+  y )
)  =  ( ( F `  x ) 
.+^  ( F `  y ) ) )
Assertion
Ref Expression
isghmd  |-  ( ph  ->  F  e.  ( S 
GrpHom  T ) )
Distinct variable groups:    ph, x, y   
x, F, y    x, S, y    x, T, y   
x,  .+ , y    x,  .+^ , y    x, X, y    x, Y, y

Proof of Theorem isghmd
StepHypRef Expression
1 isghmd.s . . 3  |-  ( ph  ->  S  e.  Grp )
2 isghmd.t . . 3  |-  ( ph  ->  T  e.  Grp )
31, 2jca 519 . 2  |-  ( ph  ->  ( S  e.  Grp  /\  T  e.  Grp )
)
4 isghmd.f . . 3  |-  ( ph  ->  F : X --> Y )
5 isghmd.l . . . 4  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x  .+  y )
)  =  ( ( F `  x ) 
.+^  ( F `  y ) ) )
65ralrimivva 2758 . . 3  |-  ( ph  ->  A. x  e.  X  A. y  e.  X  ( F `  ( x 
.+  y ) )  =  ( ( F `
 x )  .+^  ( F `  y ) ) )
74, 6jca 519 . 2  |-  ( ph  ->  ( F : X --> Y  /\  A. x  e.  X  A. y  e.  X  ( F `  ( x  .+  y ) )  =  ( ( F `  x ) 
.+^  ( F `  y ) ) ) )
8 isghmd.x . . 3  |-  X  =  ( Base `  S
)
9 isghmd.y . . 3  |-  Y  =  ( Base `  T
)
10 isghmd.a . . 3  |-  .+  =  ( +g  `  S )
11 isghmd.b . . 3  |-  .+^  =  ( +g  `  T )
128, 9, 10, 11isghm 14961 . 2  |-  ( F  e.  ( S  GrpHom  T )  <->  ( ( S  e.  Grp  /\  T  e.  Grp )  /\  ( F : X --> Y  /\  A. x  e.  X  A. y  e.  X  ( F `  ( x  .+  y ) )  =  ( ( F `  x )  .+^  ( F `
 y ) ) ) ) )
133, 7, 12sylanbrc 646 1  |-  ( ph  ->  F  e.  ( S 
GrpHom  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   -->wf 5409   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   Grpcgrp 14640    GrpHom cghm 14958
This theorem is referenced by:  ghmmhmb  14972  resghm  14977  conjghm  14991  divsghm  14997  galactghm  15061  invoppggim  15111  pj1ghm  15290  frgpup1  15362  mulgghm  15406  invghm  15408  ghmplusg  15416  rnglghm  15666  rngrghm  15667  isrhmd  15785  lmodvsghm  15960  asclghm  16352  cygznlem3  16805  evlslem1  19889  reefgim  20319  qqhghm  24325  pwssplit2  27057  frlmup1  27118  imasgim  27132  psgnghm  27305
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-ghm 14959
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