Users' Mathboxes Mathbox for Paul Chapman < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ghomgrplem Unicode version

Theorem ghomgrplem 24011
Description: Lemma for ghomgrp 24012. (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 4965 . . 3  |-  ( H  =  if ( ph ,  H ,  S )  ->  ( H  |`  ( ran  F  X.  ran  F ) )  =  ( if ( ph ,  H ,  S )  |`  ( ran  F  X.  ran  F ) ) )
2 fveq2 5541 . . 3  |-  ( H  =  if ( ph ,  H ,  S )  ->  ( SubGrpOp `  H
)  =  ( SubGrpOp `  if ( ph ,  H ,  S ) ) )
31, 2eleq12d 2364 . 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 4920 . . . 4  |-  ( F  =  if ( ph ,  F ,  J )  ->  ran  F  =  ran  if ( ph ,  F ,  J )
)
5 xpeq1 4719 . . . . . 6  |-  ( ran 
F  =  ran  if ( ph ,  F ,  J )  ->  ( ran  F  X.  ran  F
)  =  ( ran 
if ( ph ,  F ,  J )  X.  ran  F ) )
6 xpeq2 4720 . . . . . 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 2328 . . . . 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 4971 . . . 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 2362 . 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 2356 . . . 4  |-  ( G  =  if ( ph ,  G ,  S )  ->  ( G  e. 
GrpOp 
<->  if ( ph ,  G ,  S )  e.  GrpOp ) )
14 eleq1 2356 . . . 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 2804 . . . . . 6  |-  z  e. 
_V
1716grposn 20898 . . . . 5  |-  { <. <.
z ,  z >. ,  z >. }  e.  GrpOp
1815, 17eqeltri 2366 . . . 4  |-  S  e. 
GrpOp
1912, 13, 14, 18elimdhyp 3631 . . 3  |-  if (
ph ,  G ,  S )  e.  GrpOp
2011simp2d 968 . . . 4  |-  ( ph  ->  H  e.  GrpOp )
21 eleq1 2356 . . . 4  |-  ( H  =  if ( ph ,  H ,  S )  ->  ( H  e. 
GrpOp 
<->  if ( ph ,  H ,  S )  e.  GrpOp ) )
22 eleq1 2356 . . . 4  |-  ( S  =  if ( ph ,  H ,  S )  ->  ( S  e. 
GrpOp 
<->  if ( ph ,  H ,  S )  e.  GrpOp ) )
2320, 21, 22, 18elimdhyp 3631 . . 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 24010 . . . . 5  |-  (  _I  |`  { z } )  e.  ( S GrpOpHom  S )
2725, 26eqeltri 2366 . . . 4  |-  J  e.  ( S GrpOpHom  S )
2824, 27elimdelov 5943 . . 3  |-  if (
ph ,  F ,  J )  e.  ( if ( ph ,  G ,  S ) GrpOpHom  if ( ph ,  H ,  S ) )
29 eqid 2296 . . 3  |-  ran  if ( ph ,  F ,  J )  =  ran  if ( ph ,  F ,  J )
30 eqid 2296 . . 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 24009 . 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 3623 1  |-  ( ph  ->  ( H  |`  ( ran  F  X.  ran  F
) )  e.  (
SubGrpOp `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   ifcif 3578   {csn 3653   <.cop 3656    _I cid 4320    X. cxp 4703   ran crn 4706    |` cres 4707   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869   SubGrpOpcsubgo 20984   GrpOpHom cghom 21040
This theorem is referenced by:  ghomgrp  24012
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-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876  df-subgo 20985  df-ghom 21041
  Copyright terms: Public domain W3C validator