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Theorem ghomgrplem 28854
Description: Lemma for ghomgrp 28855. (Contributed by Paul Chapman, 25-Feb-2008.)
Hypotheses
Ref Expression
ghomgrplem.1  |-  ( ph  ->  ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
) )
ghomgrplem.2  |-  S  =  { <. <. z ,  z
>. ,  z >. }
ghomgrplem.3  |-  J  =  (  _I  |`  { z } )
Assertion
Ref Expression
ghomgrplem  |-  ( ph  ->  ( H  |`  ( ran  F  X.  ran  F
) )  e.  (
SubGrpOp `  H ) )

Proof of Theorem ghomgrplem
StepHypRef Expression
1 reseq1 5273 . . 3  |-  ( H  =  if ( ph ,  H ,  S )  ->  ( H  |`  ( ran  F  X.  ran  F ) )  =  ( if ( ph ,  H ,  S )  |`  ( ran  F  X.  ran  F ) ) )
2 fveq2 5872 . . 3  |-  ( H  =  if ( ph ,  H ,  S )  ->  ( SubGrpOp `  H
)  =  ( SubGrpOp `  if ( ph ,  H ,  S ) ) )
31, 2eleq12d 2549 . 2  |-  ( H  =  if ( ph ,  H ,  S )  ->  ( ( H  |`  ( ran  F  X.  ran  F ) )  e.  ( SubGrpOp `  H )  <->  ( if ( ph ,  H ,  S )  |`  ( ran  F  X.  ran  F ) )  e.  ( SubGrpOp `  if ( ph ,  H ,  S ) ) ) )
4 rneq 5234 . . . 4  |-  ( F  =  if ( ph ,  F ,  J )  ->  ran  F  =  ran  if ( ph ,  F ,  J )
)
5 xpeq1 5019 . . . . . 6  |-  ( ran 
F  =  ran  if ( ph ,  F ,  J )  ->  ( ran  F  X.  ran  F
)  =  ( ran 
if ( ph ,  F ,  J )  X.  ran  F ) )
6 xpeq2 5020 . . . . . 6  |-  ( ran 
F  =  ran  if ( ph ,  F ,  J )  ->  ( ran  if ( ph ,  F ,  J )  X.  ran  F )  =  ( ran  if (
ph ,  F ,  J )  X.  ran  if ( ph ,  F ,  J ) ) )
75, 6eqtrd 2508 . . . . 5  |-  ( ran 
F  =  ran  if ( ph ,  F ,  J )  ->  ( ran  F  X.  ran  F
)  =  ( ran 
if ( ph ,  F ,  J )  X.  ran  if ( ph ,  F ,  J ) ) )
87reseq2d 5279 . . . 4  |-  ( ran 
F  =  ran  if ( ph ,  F ,  J )  ->  ( if ( ph ,  H ,  S )  |`  ( ran  F  X.  ran  F
) )  =  ( if ( ph ,  H ,  S )  |`  ( ran  if (
ph ,  F ,  J )  X.  ran  if ( ph ,  F ,  J ) ) ) )
94, 8syl 16 . . 3  |-  ( F  =  if ( ph ,  F ,  J )  ->  ( if (
ph ,  H ,  S )  |`  ( ran  F  X.  ran  F
) )  =  ( if ( ph ,  H ,  S )  |`  ( ran  if (
ph ,  F ,  J )  X.  ran  if ( ph ,  F ,  J ) ) ) )
109eleq1d 2536 . 2  |-  ( F  =  if ( ph ,  F ,  J )  ->  ( ( if ( ph ,  H ,  S )  |`  ( ran  F  X.  ran  F
) )  e.  (
SubGrpOp `  if ( ph ,  H ,  S ) )  <->  ( if (
ph ,  H ,  S )  |`  ( ran  if ( ph ,  F ,  J )  X.  ran  if ( ph ,  F ,  J ) ) )  e.  (
SubGrpOp `  if ( ph ,  H ,  S ) ) ) )
11 ghomgrplem.1 . . . . 5  |-  ( ph  ->  ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
) )
1211simp1d 1008 . . . 4  |-  ( ph  ->  G  e.  GrpOp )
13 eleq1 2539 . . . 4  |-  ( G  =  if ( ph ,  G ,  S )  ->  ( G  e. 
GrpOp 
<->  if ( ph ,  G ,  S )  e.  GrpOp ) )
14 eleq1 2539 . . . 4  |-  ( S  =  if ( ph ,  G ,  S )  ->  ( S  e. 
GrpOp 
<->  if ( ph ,  G ,  S )  e.  GrpOp ) )
15 ghomgrplem.2 . . . . 5  |-  S  =  { <. <. z ,  z
>. ,  z >. }
16 vex 3121 . . . . . 6  |-  z  e. 
_V
1716grposn 25040 . . . . 5  |-  { <. <.
z ,  z >. ,  z >. }  e.  GrpOp
1815, 17eqeltri 2551 . . . 4  |-  S  e. 
GrpOp
1912, 13, 14, 18elimdhyp 4009 . . 3  |-  if (
ph ,  G ,  S )  e.  GrpOp
2011simp2d 1009 . . . 4  |-  ( ph  ->  H  e.  GrpOp )
21 eleq1 2539 . . . 4  |-  ( H  =  if ( ph ,  H ,  S )  ->  ( H  e. 
GrpOp 
<->  if ( ph ,  H ,  S )  e.  GrpOp ) )
22 eleq1 2539 . . . 4  |-  ( S  =  if ( ph ,  H ,  S )  ->  ( S  e. 
GrpOp 
<->  if ( ph ,  H ,  S )  e.  GrpOp ) )
2320, 21, 22, 18elimdhyp 4009 . . 3  |-  if (
ph ,  H ,  S )  e.  GrpOp
2411simp3d 1010 . . . 4  |-  ( ph  ->  F  e.  ( G GrpOpHom  H ) )
25 ghomgrplem.3 . . . . 5  |-  J  =  (  _I  |`  { z } )
2616, 15ghomsn 28853 . . . . 5  |-  (  _I  |`  { z } )  e.  ( S GrpOpHom  S )
2725, 26eqeltri 2551 . . . 4  |-  J  e.  ( S GrpOpHom  S )
2824, 27elimdelov 6373 . . 3  |-  if (
ph ,  F ,  J )  e.  ( if ( ph ,  G ,  S ) GrpOpHom  if ( ph ,  H ,  S ) )
29 eqid 2467 . . 3  |-  ran  if ( ph ,  F ,  J )  =  ran  if ( ph ,  F ,  J )
30 eqid 2467 . . 3  |-  ( if ( ph ,  H ,  S )  |`  ( ran  if ( ph ,  F ,  J )  X.  ran  if ( ph ,  F ,  J ) ) )  =  ( if ( ph ,  H ,  S )  |`  ( ran  if (
ph ,  F ,  J )  X.  ran  if ( ph ,  F ,  J ) ) )
3119, 23, 28, 29, 30ghomgrpi 28852 . 2  |-  ( if ( ph ,  H ,  S )  |`  ( ran  if ( ph ,  F ,  J )  X.  ran  if ( ph ,  F ,  J ) ) )  e.  (
SubGrpOp `  if ( ph ,  H ,  S ) )
323, 10, 31dedth2v 4001 1  |-  ( ph  ->  ( H  |`  ( ran  F  X.  ran  F
) )  e.  (
SubGrpOp `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   ifcif 3945   {csn 4033   <.cop 4039    _I cid 4796    X. cxp 5003   ran crn 5006    |` cres 5007   ` cfv 5594  (class class class)co 6295   GrpOpcgr 25011   SubGrpOpcsubgo 25126   GrpOpHom cghom 25182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-grpo 25016  df-gid 25017  df-ginv 25018  df-subgo 25127  df-ghom 25183
This theorem is referenced by:  ghomgrp  28855
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