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Theorem isghmd 14692
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 518 . 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 2635 . . 3  |-  ( ph  ->  A. x  e.  X  A. y  e.  X  ( F `  ( x 
.+  y ) )  =  ( ( F `
 x )  .+^  ( F `  y ) ) )
74, 6jca 518 . 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 14683 . 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 645 1  |-  ( ph  ->  F  e.  ( S 
GrpHom  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362    GrpHom cghm 14680
This theorem is referenced by:  ghmmhmb  14694  resghm  14699  conjghm  14713  divsghm  14719  galactghm  14783  invoppggim  14833  pj1ghm  15012  frgpup1  15084  mulgghm  15128  invghm  15130  ghmplusg  15138  rnglghm  15388  rngrghm  15389  isrhmd  15507  lmodvsghm  15686  asclghm  16078  cygznlem3  16523  evlslem1  19399  reefgim  19826  pwssplit2  27189  frlmup1  27250  imasgim  27264  psgnghm  27437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-ghm 14681
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