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Theorem ghomgrpilem1 25098
Description: Lemma for ghomgrpi 25100. (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 5730 . . . . . 6  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
21oveq1d 6098 . . . . 5  |-  ( x  =  A  ->  (
( F `  x
) H ( F `
 y ) )  =  ( ( F `
 A ) H ( F `  y
) ) )
3 oveq1 6090 . . . . . 6  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
43fveq2d 5734 . . . . 5  |-  ( x  =  A  ->  ( F `  ( x G y ) )  =  ( F `  ( A G y ) ) )
52, 4eqeq12d 2452 . . . 4  |-  ( x  =  A  ->  (
( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) )  <->  ( ( F `  A ) H ( F `  y ) )  =  ( F `  ( A G y ) ) ) )
65ralbidv 2727 . . 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 21953 . . . . . 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 3028 . 2  |-  ( A  e.  X  ->  A. y  e.  X  ( ( F `  A ) H ( F `  y ) )  =  ( F `  ( A G y ) ) )
17 fveq2 5730 . . . . 5  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
1817oveq2d 6099 . . . 4  |-  ( y  =  B  ->  (
( F `  A
) H ( F `
 y ) )  =  ( ( F `
 A ) H ( F `  B
) ) )
19 oveq2 6091 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
2019fveq2d 5734 . . . 4  |-  ( y  =  B  ->  ( F `  ( A G y ) )  =  ( F `  ( A G B ) ) )
2118, 20eqeq12d 2452 . . 3  |-  ( y  =  B  ->  (
( ( F `  A ) H ( F `  y ) )  =  ( F `
 ( A G y ) )  <->  ( ( F `  A ) H ( F `  B ) )  =  ( F `  ( A G B ) ) ) )
2221rspcv 3050 . 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 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    X. cxp 4878   ran crn 4881    |` cres 4882   -->wf 5452   ` cfv 5456  (class class class)co 6083   GrpOpcgr 21776  GIdcgi 21777   invcgn 21778   GrpOpHom cghom 21947
This theorem is referenced by:  ghomgrpilem2  25099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-ghom 21948
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