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Theorem ghomgrpilem1 23364
Description: Lemma for ghomgrpi 23366. (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
StepHypRef Expression
1 fveq2 5458 . . . . . 6  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
21oveq1d 5807 . . . . 5  |-  ( x  =  A  ->  (
( F `  x
) H ( F `
 y ) )  =  ( ( F `
 A ) H ( F `  y
) ) )
3 oveq1 5799 . . . . . 6  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
43fveq2d 5462 . . . . 5  |-  ( x  =  A  ->  ( F `  ( x G y ) )  =  ( F `  ( A G y ) ) )
52, 4eqeq12d 2272 . . . 4  |-  ( x  =  A  ->  (
( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) )  <->  ( ( F `  A ) H ( F `  y ) )  =  ( F `  ( A G y ) ) ) )
65ralbidv 2538 . . 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 20990 . . . . . 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 656 . . . . 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 2833 . 2  |-  ( A  e.  X  ->  A. y  e.  X  ( ( F `  A ) H ( F `  y ) )  =  ( F `  ( A G y ) ) )
17 fveq2 5458 . . . . 5  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
1817oveq2d 5808 . . . 4  |-  ( y  =  B  ->  (
( F `  A
) H ( F `
 y ) )  =  ( ( F `
 A ) H ( F `  B
) ) )
19 oveq2 5800 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
2019fveq2d 5462 . . . 4  |-  ( y  =  B  ->  ( F `  ( A G y ) )  =  ( F `  ( A G B ) ) )
2118, 20eqeq12d 2272 . . 3  |-  ( y  =  B  ->  (
( ( F `  A ) H ( F `  y ) )  =  ( F `
 ( A G y ) )  <->  ( ( F `  A ) H ( F `  B ) )  =  ( F `  ( A G B ) ) ) )
2221rcla4v 2855 . 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 1619    e. wcel 1621   A.wral 2518    X. cxp 4659   ran crn 4662    |` cres 4663   -->wf 4669   ` cfv 4673  (class class class)co 5792   GrpOpcgr 20813  GIdcgi 20814   invcgn 20815   GrpOpHom cghom 20984
This theorem is referenced by:  ghomgrpilem2  23365
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-ghom 20985
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