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Theorem isghmd 15736
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 529 . 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 2798 . . 3  |-  ( ph  ->  A. x  e.  X  A. y  e.  X  ( F `  ( x 
.+  y ) )  =  ( ( F `
 x )  .+^  ( F `  y ) ) )
74, 6jca 529 . 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 15727 . 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 657 1  |-  ( ph  ->  F  e.  ( S 
GrpHom  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   A.wral 2705   -->wf 5402   ` cfv 5406  (class class class)co 6080   Basecbs 14157   +g cplusg 14221   Grpcgrp 15393    GrpHom cghm 15724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-ghm 15725
This theorem is referenced by:  ghmmhmb  15738  resghm  15743  conjghm  15757  divsghm  15763  invoppggim  15855  galactghm  15888  pj1ghm  16180  frgpup1  16252  mulgghm  16296  invghm  16298  ghmplusg  16308  rnglghm  16628  rngrghm  16629  isrhmd  16751  lmodvsghm  16930  pwssplit2  17063  asclghm  17331  cygznlem3  17844  psgnghm  17852  frlmup1  18068  evlslem1  21367  reefgim  21800  qqhghm  26271  imasgim  29300
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