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Theorem ghomgrplem 23996
Description: Lemma for ghomgrp 23997. (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 4949 . . 3  |-  ( H  =  if ( ph ,  H ,  S )  ->  ( H  |`  ( ran  F  X.  ran  F ) )  =  ( if ( ph ,  H ,  S )  |`  ( ran  F  X.  ran  F ) ) )
2 fveq2 5525 . . 3  |-  ( H  =  if ( ph ,  H ,  S )  ->  ( SubGrpOp `  H
)  =  ( SubGrpOp `  if ( ph ,  H ,  S ) ) )
31, 2eleq12d 2351 . 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 4904 . . . 4  |-  ( F  =  if ( ph ,  F ,  J )  ->  ran  F  =  ran  if ( ph ,  F ,  J )
)
5 xpeq1 4703 . . . . . 6  |-  ( ran 
F  =  ran  if ( ph ,  F ,  J )  ->  ( ran  F  X.  ran  F
)  =  ( ran 
if ( ph ,  F ,  J )  X.  ran  F ) )
6 xpeq2 4704 . . . . . 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 2315 . . . . 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 4955 . . . 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 15 . . 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 2349 . 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 967 . . . 4  |-  ( ph  ->  G  e.  GrpOp )
13 eleq1 2343 . . . 4  |-  ( G  =  if ( ph ,  G ,  S )  ->  ( G  e. 
GrpOp 
<->  if ( ph ,  G ,  S )  e.  GrpOp ) )
14 eleq1 2343 . . . 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 2791 . . . . . 6  |-  z  e. 
_V
1716grposn 20882 . . . . 5  |-  { <. <.
z ,  z >. ,  z >. }  e.  GrpOp
1815, 17eqeltri 2353 . . . 4  |-  S  e. 
GrpOp
1912, 13, 14, 18elimdhyp 3618 . . 3  |-  if (
ph ,  G ,  S )  e.  GrpOp
2011simp2d 968 . . . 4  |-  ( ph  ->  H  e.  GrpOp )
21 eleq1 2343 . . . 4  |-  ( H  =  if ( ph ,  H ,  S )  ->  ( H  e. 
GrpOp 
<->  if ( ph ,  H ,  S )  e.  GrpOp ) )
22 eleq1 2343 . . . 4  |-  ( S  =  if ( ph ,  H ,  S )  ->  ( S  e. 
GrpOp 
<->  if ( ph ,  H ,  S )  e.  GrpOp ) )
2320, 21, 22, 18elimdhyp 3618 . . 3  |-  if (
ph ,  H ,  S )  e.  GrpOp
2411simp3d 969 . . . 4  |-  ( ph  ->  F  e.  ( G GrpOpHom  H ) )
25 ghomgrplem.3 . . . . 5  |-  J  =  (  _I  |`  { z } )
2616, 15ghomsn 23995 . . . . 5  |-  (  _I  |`  { z } )  e.  ( S GrpOpHom  S )
2725, 26eqeltri 2353 . . . 4  |-  J  e.  ( S GrpOpHom  S )
2824, 27elimdelov 5927 . . 3  |-  if (
ph ,  F ,  J )  e.  ( if ( ph ,  G ,  S ) GrpOpHom  if ( ph ,  H ,  S ) )
29 eqid 2283 . . 3  |-  ran  if ( ph ,  F ,  J )  =  ran  if ( ph ,  F ,  J )
30 eqid 2283 . . 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 23994 . 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 3610 1  |-  ( ph  ->  ( H  |`  ( ran  F  X.  ran  F
) )  e.  (
SubGrpOp `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   ifcif 3565   {csn 3640   <.cop 3643    _I cid 4304    X. cxp 4687   ran crn 4690    |` cres 4691   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853   SubGrpOpcsubgo 20968   GrpOpHom cghom 21024
This theorem is referenced by:  ghomgrp  23997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-subgo 20969  df-ghom 21025
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