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Theorem ghomgrpilem1 25049
Description: Lemma for ghomgrpi 25051. (Contributed by Paul Chapman, 25-Feb-2008.)
Hypotheses
Ref Expression
ghomgrpilem1.1  |-  G  e. 
GrpOp
ghomgrpilem1.2  |-  H  e. 
GrpOp
ghomgrpilem1.3  |-  F  e.  ( G GrpOpHom  H )
ghomgrpilem1.4  |-  X  =  ran  G
ghomgrpilem1.5  |-  U  =  (GId `  G )
ghomgrpilem1.6  |-  N  =  ( inv `  G
)
ghomgrpilem1.7  |-  W  =  ran  H
ghomgrpilem1.8  |-  T  =  (GId `  H )
ghomgrpilem1.9  |-  M  =  ( inv `  H
)
ghomgrpilem1.10  |-  Z  =  ran  F
ghomgrpilem1.11  |-  S  =  ( H  |`  ( Z  X.  Z ) )
Assertion
Ref Expression
ghomgrpilem1  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( F `  A ) H ( F `  B ) )  =  ( F `
 ( A G B ) ) )

Proof of Theorem ghomgrpilem1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5687 . . . . . 6  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
21oveq1d 6055 . . . . 5  |-  ( x  =  A  ->  (
( F `  x
) H ( F `
 y ) )  =  ( ( F `
 A ) H ( F `  y
) ) )
3 oveq1 6047 . . . . . 6  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
43fveq2d 5691 . . . . 5  |-  ( x  =  A  ->  ( F `  ( x G y ) )  =  ( F `  ( A G y ) ) )
52, 4eqeq12d 2418 . . . 4  |-  ( x  =  A  ->  (
( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) )  <->  ( ( F `  A ) H ( F `  y ) )  =  ( F `  ( A G y ) ) ) )
65ralbidv 2686 . . 3  |-  ( x  =  A  ->  ( A. y  e.  X  ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) )  <->  A. y  e.  X  ( ( F `  A ) H ( F `  y ) )  =  ( F `  ( A G y ) ) ) )
7 ghomgrpilem1.3 . . . . 5  |-  F  e.  ( G GrpOpHom  H )
8 ghomgrpilem1.1 . . . . . 6  |-  G  e. 
GrpOp
9 ghomgrpilem1.2 . . . . . 6  |-  H  e. 
GrpOp
10 ghomgrpilem1.4 . . . . . . 7  |-  X  =  ran  G
11 ghomgrpilem1.7 . . . . . . 7  |-  W  =  ran  H
1210, 11elghom 21904 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : X
--> W  /\  A. x  e.  X  A. y  e.  X  ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) ) ) ) )
138, 9, 12mp2an 654 . . . . 5  |-  ( F  e.  ( G GrpOpHom  H )  <-> 
( F : X --> W  /\  A. x  e.  X  A. y  e.  X  ( ( F `
 x ) H ( F `  y
) )  =  ( F `  ( x G y ) ) ) )
147, 13mpbi 200 . . . 4  |-  ( F : X --> W  /\  A. x  e.  X  A. y  e.  X  (
( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) ) )
1514simpri 449 . . 3  |-  A. x  e.  X  A. y  e.  X  ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) )
166, 15vtoclri 2986 . 2  |-  ( A  e.  X  ->  A. y  e.  X  ( ( F `  A ) H ( F `  y ) )  =  ( F `  ( A G y ) ) )
17 fveq2 5687 . . . . 5  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
1817oveq2d 6056 . . . 4  |-  ( y  =  B  ->  (
( F `  A
) H ( F `
 y ) )  =  ( ( F `
 A ) H ( F `  B
) ) )
19 oveq2 6048 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
2019fveq2d 5691 . . . 4  |-  ( y  =  B  ->  ( F `  ( A G y ) )  =  ( F `  ( A G B ) ) )
2118, 20eqeq12d 2418 . . 3  |-  ( y  =  B  ->  (
( ( F `  A ) H ( F `  y ) )  =  ( F `
 ( A G y ) )  <->  ( ( F `  A ) H ( F `  B ) )  =  ( F `  ( A G B ) ) ) )
2221rspcv 3008 . 2  |-  ( B  e.  X  ->  ( A. y  e.  X  ( ( F `  A ) H ( F `  y ) )  =  ( F `
 ( A G y ) )  -> 
( ( F `  A ) H ( F `  B ) )  =  ( F `
 ( A G B ) ) ) )
2316, 22mpan9 456 1  |-  ( ( 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    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666    X. cxp 4835   ran crn 4838    |` cres 4839   -->wf 5409   ` cfv 5413  (class class class)co 6040   GrpOpcgr 21727  GIdcgi 21728   invcgn 21729   GrpOpHom cghom 21898
This theorem is referenced by:  ghomgrpilem2  25050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-ghom 21899
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