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Theorem ghomgrplem 27472
Description: Lemma for ghomgrp 27473. (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 5215 . . 3  |-  ( H  =  if ( ph ,  H ,  S )  ->  ( H  |`  ( ran  F  X.  ran  F ) )  =  ( if ( ph ,  H ,  S )  |`  ( ran  F  X.  ran  F ) ) )
2 fveq2 5802 . . 3  |-  ( H  =  if ( ph ,  H ,  S )  ->  ( SubGrpOp `  H
)  =  ( SubGrpOp `  if ( ph ,  H ,  S ) ) )
31, 2eleq12d 2536 . 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 5176 . . . 4  |-  ( F  =  if ( ph ,  F ,  J )  ->  ran  F  =  ran  if ( ph ,  F ,  J )
)
5 xpeq1 4965 . . . . . 6  |-  ( ran 
F  =  ran  if ( ph ,  F ,  J )  ->  ( ran  F  X.  ran  F
)  =  ( ran 
if ( ph ,  F ,  J )  X.  ran  F ) )
6 xpeq2 4966 . . . . . 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 2495 . . . . 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 5221 . . . 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 2523 . 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 1000 . . . 4  |-  ( ph  ->  G  e.  GrpOp )
13 eleq1 2526 . . . 4  |-  ( G  =  if ( ph ,  G ,  S )  ->  ( G  e. 
GrpOp 
<->  if ( ph ,  G ,  S )  e.  GrpOp ) )
14 eleq1 2526 . . . 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 3081 . . . . . 6  |-  z  e. 
_V
1716grposn 23874 . . . . 5  |-  { <. <.
z ,  z >. ,  z >. }  e.  GrpOp
1815, 17eqeltri 2538 . . . 4  |-  S  e. 
GrpOp
1912, 13, 14, 18elimdhyp 3964 . . 3  |-  if (
ph ,  G ,  S )  e.  GrpOp
2011simp2d 1001 . . . 4  |-  ( ph  ->  H  e.  GrpOp )
21 eleq1 2526 . . . 4  |-  ( H  =  if ( ph ,  H ,  S )  ->  ( H  e. 
GrpOp 
<->  if ( ph ,  H ,  S )  e.  GrpOp ) )
22 eleq1 2526 . . . 4  |-  ( S  =  if ( ph ,  H ,  S )  ->  ( S  e. 
GrpOp 
<->  if ( ph ,  H ,  S )  e.  GrpOp ) )
2320, 21, 22, 18elimdhyp 3964 . . 3  |-  if (
ph ,  H ,  S )  e.  GrpOp
2411simp3d 1002 . . . 4  |-  ( ph  ->  F  e.  ( G GrpOpHom  H ) )
25 ghomgrplem.3 . . . . 5  |-  J  =  (  _I  |`  { z } )
2616, 15ghomsn 27471 . . . . 5  |-  (  _I  |`  { z } )  e.  ( S GrpOpHom  S )
2725, 26eqeltri 2538 . . . 4  |-  J  e.  ( S GrpOpHom  S )
2824, 27elimdelov 6279 . . 3  |-  if (
ph ,  F ,  J )  e.  ( if ( ph ,  G ,  S ) GrpOpHom  if ( ph ,  H ,  S ) )
29 eqid 2454 . . 3  |-  ran  if ( ph ,  F ,  J )  =  ran  if ( ph ,  F ,  J )
30 eqid 2454 . . 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 27470 . 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 3956 1  |-  ( ph  ->  ( H  |`  ( ran  F  X.  ran  F
) )  e.  (
SubGrpOp `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   ifcif 3902   {csn 3988   <.cop 3994    _I cid 4742    X. cxp 4949   ran crn 4952    |` cres 4953   ` cfv 5529  (class class class)co 6203   GrpOpcgr 23845   SubGrpOpcsubgo 23960   GrpOpHom cghom 24016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-grpo 23850  df-gid 23851  df-ginv 23852  df-subgo 23961  df-ghom 24017
This theorem is referenced by:  ghomgrp  27473
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