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Theorem ghomgrpilem1 24007
Description: Lemma for ghomgrpi 24009. (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 5541 . . . . . 6  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
21oveq1d 5889 . . . . 5  |-  ( x  =  A  ->  (
( F `  x
) H ( F `
 y ) )  =  ( ( F `
 A ) H ( F `  y
) ) )
3 oveq1 5881 . . . . . 6  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
43fveq2d 5545 . . . . 5  |-  ( x  =  A  ->  ( F `  ( x G y ) )  =  ( F `  ( A G y ) ) )
52, 4eqeq12d 2310 . . . 4  |-  ( x  =  A  ->  (
( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) )  <->  ( ( F `  A ) H ( F `  y ) )  =  ( F `  ( A G y ) ) ) )
65ralbidv 2576 . . 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 21046 . . . . . 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 653 . . . . 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 199 . . . 4  |-  ( F : X --> W  /\  A. x  e.  X  A. y  e.  X  (
( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) ) )
1514simpri 448 . . 3  |-  A. x  e.  X  A. y  e.  X  ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) )
166, 15vtoclri 2871 . 2  |-  ( A  e.  X  ->  A. y  e.  X  ( ( F `  A ) H ( F `  y ) )  =  ( F `  ( A G y ) ) )
17 fveq2 5541 . . . . 5  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
1817oveq2d 5890 . . . 4  |-  ( y  =  B  ->  (
( F `  A
) H ( F `
 y ) )  =  ( ( F `
 A ) H ( F `  B
) ) )
19 oveq2 5882 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
2019fveq2d 5545 . . . 4  |-  ( y  =  B  ->  ( F `  ( A G y ) )  =  ( F `  ( A G B ) ) )
2118, 20eqeq12d 2310 . . 3  |-  ( y  =  B  ->  (
( ( F `  A ) H ( F `  y ) )  =  ( F `
 ( A G y ) )  <->  ( ( F `  A ) H ( F `  B ) )  =  ( F `  ( A G B ) ) ) )
2221rspcv 2893 . 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 455 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    X. cxp 4703   ran crn 4706    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871   GrpOpHom cghom 21040
This theorem is referenced by:  ghomgrpilem2  24008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-ghom 21041
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