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Theorem ghomlin 21945
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 2436 . . . . 5  |-  ran  H  =  ran  H
31, 2elghom 21944 . . . 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 1283 . . 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 450 . 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 5721 . . . . 5  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
76oveq1d 6089 . . . 4  |-  ( x  =  A  ->  (
( F `  x
) H ( F `
 y ) )  =  ( ( F `
 A ) H ( F `  y
) ) )
8 oveq1 6081 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
98fveq2d 5725 . . . 4  |-  ( x  =  A  ->  ( F `  ( x G y ) )  =  ( F `  ( A G y ) ) )
107, 9eqeq12d 2450 . . 3  |-  ( x  =  A  ->  (
( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) )  <->  ( ( F `  A ) H ( F `  y ) )  =  ( F `  ( A G y ) ) ) )
11 fveq2 5721 . . . . 5  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
1211oveq2d 6090 . . . 4  |-  ( y  =  B  ->  (
( F `  A
) H ( F `
 y ) )  =  ( ( F `
 A ) H ( F `  B
) ) )
13 oveq2 6082 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1413fveq2d 5725 . . . 4  |-  ( y  =  B  ->  ( F `  ( A G y ) )  =  ( F `  ( A G B ) ) )
1512, 14eqeq12d 2450 . . 3  |-  ( y  =  B  ->  (
( ( F `  A ) H ( F `  y ) )  =  ( F `
 ( A G y ) )  <->  ( ( F `  A ) H ( F `  B ) )  =  ( F `  ( A G B ) ) ) )
1610, 15rspc2v 3051 . 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 456 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2698   ran crn 4872   -->wf 5443   ` cfv 5447  (class class class)co 6074   GrpOpcgr 21767   GrpOpHom cghom 21938
This theorem is referenced by:  ghomid  21946  ghomf1olem  25098  ghomdiv  26551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-ghom 21939
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