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Theorem ghomgrpilem1 23396
Description: Lemma for ghomgrpi 23398. (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 ) ) )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem ghomgrpilem1
StepHypRef Expression
1 fveq2 5485 . . . . . 6  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
21oveq1d 5834 . . . . 5  |-  ( x  =  A  ->  (
( F `  x
) H ( F `
 y ) )  =  ( ( F `
 A ) H ( F `  y
) ) )
3 oveq1 5826 . . . . . 6  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
43fveq2d 5489 . . . . 5  |-  ( x  =  A  ->  ( F `  ( x G y ) )  =  ( F `  ( A G y ) ) )
52, 4eqeq12d 2298 . . . 4  |-  ( x  =  A  ->  (
( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) )  <->  ( ( F `  A ) H ( F `  y ) )  =  ( F `  ( A G y ) ) ) )
65ralbidv 2564 . . 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 21022 . . . . . 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 655 . . . . 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 201 . . . 4  |-  ( F : X --> W  /\  A. x  e.  X  A. y  e.  X  (
( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) ) )
1514simpri 450 . . 3  |-  A. x  e.  X  A. y  e.  X  ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) )
166, 15vtoclri 2859 . 2  |-  ( A  e.  X  ->  A. y  e.  X  ( ( F `  A ) H ( F `  y ) )  =  ( F `  ( A G y ) ) )
17 fveq2 5485 . . . . 5  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
1817oveq2d 5835 . . . 4  |-  ( y  =  B  ->  (
( F `  A
) H ( F `
 y ) )  =  ( ( F `
 A ) H ( F `  B
) ) )
19 oveq2 5827 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
2019fveq2d 5489 . . . 4  |-  ( y  =  B  ->  ( F `  ( A G y ) )  =  ( F `  ( A G B ) ) )
2118, 20eqeq12d 2298 . . 3  |-  ( y  =  B  ->  (
( ( F `  A ) H ( F `  y ) )  =  ( F `
 ( A G y ) )  <->  ( ( F `  A ) H ( F `  B ) )  =  ( F `  ( A G B ) ) ) )
2221rspcv 2881 . 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 457 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 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   A.wral 2544    X. cxp 4686   ran crn 4689    |` cres 4690   -->wf 5217   ` cfv 5221  (class class class)co 5819   GrpOpcgr 20845  GIdcgi 20846   invcgn 20847   GrpOpHom cghom 21016
This theorem is referenced by:  ghomgrpilem2  23397
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-ghom 21017
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