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Theorem ghomlin 25139
Description: Linearity of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
ghomlin.1  |-  X  =  ran  G
Assertion
Ref Expression
ghomlin  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( ( F `  A ) H ( F `  B ) )  =  ( F `
 ( A G B ) ) )

Proof of Theorem ghomlin
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghomlin.1 . . . . 5  |-  X  =  ran  G
2 eqid 2467 . . . . 5  |-  ran  H  =  ran  H
31, 2elghom 25138 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : X
--> ran  H  /\  A. x  e.  X  A. y  e.  X  (
( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) ) ) ) )
43biimp3a 1328 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F : X
--> ran  H  /\  A. x  e.  X  A. y  e.  X  (
( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) ) ) )
54simprd 463 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  A. x  e.  X  A. y  e.  X  ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) )
6 fveq2 5866 . . . . 5  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
76oveq1d 6300 . . . 4  |-  ( x  =  A  ->  (
( F `  x
) H ( F `
 y ) )  =  ( ( F `
 A ) H ( F `  y
) ) )
8 oveq1 6292 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
98fveq2d 5870 . . . 4  |-  ( x  =  A  ->  ( F `  ( x G y ) )  =  ( F `  ( A G y ) ) )
107, 9eqeq12d 2489 . . 3  |-  ( x  =  A  ->  (
( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) )  <->  ( ( F `  A ) H ( F `  y ) )  =  ( F `  ( A G y ) ) ) )
11 fveq2 5866 . . . . 5  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
1211oveq2d 6301 . . . 4  |-  ( y  =  B  ->  (
( F `  A
) H ( F `
 y ) )  =  ( ( F `
 A ) H ( F `  B
) ) )
13 oveq2 6293 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1413fveq2d 5870 . . . 4  |-  ( y  =  B  ->  ( F `  ( A G y ) )  =  ( F `  ( A G B ) ) )
1512, 14eqeq12d 2489 . . 3  |-  ( y  =  B  ->  (
( ( F `  A ) H ( F `  y ) )  =  ( F `
 ( A G y ) )  <->  ( ( F `  A ) H ( F `  B ) )  =  ( F `  ( A G B ) ) ) )
1610, 15rspc2v 3223 . 2  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( ( F `
 x ) H ( F `  y
) )  =  ( F `  ( x G y ) )  ->  ( ( F `
 A ) H ( F `  B
) )  =  ( F `  ( A G B ) ) ) )
175, 16mpan9 469 1  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( ( F `  A ) H ( F `  B ) )  =  ( F `
 ( A G B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   ran crn 5000   -->wf 5584   ` cfv 5588  (class class class)co 6285   GrpOpcgr 24961   GrpOpHom cghom 25132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-ghom 25133
This theorem is referenced by:  ghomid  25140  ghomf1olem  28785  ghomdiv  30176
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