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Theorem isghmd 15876
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 532 . 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 2914 . . 3  |-  ( ph  ->  A. x  e.  X  A. y  e.  X  ( F `  ( x 
.+  y ) )  =  ( ( F `
 x )  .+^  ( F `  y ) ) )
74, 6jca 532 . 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 15867 . 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 664 1  |-  ( ph  ->  F  e.  ( S 
GrpHom  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   -->wf 5523   ` cfv 5527  (class class class)co 6201   Basecbs 14293   +g cplusg 14358   Grpcgrp 15530    GrpHom cghm 15864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-ghm 15865
This theorem is referenced by:  ghmmhmb  15878  resghm  15883  conjghm  15897  divsghm  15903  invoppggim  15995  galactghm  16028  pj1ghm  16322  frgpup1  16394  mulgghm  16438  invghm  16440  ghmplusg  16450  rnglghm  16817  rngrghm  16818  isrhmd  16943  lmodvsghm  17130  pwssplit2  17265  asclghm  17533  evlslem1  17726  cygznlem3  18128  psgnghm  18136  frlmup1  18352  reefgim  22049  qqhghm  26563  imasgim  29604  mat2pmatghm  31220  pm2mpghm  31304
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