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Theorem snclseqg 17814
Description: The coset of the closure of the identity is the closure of a point. (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
snclseqg.x  |-  X  =  ( Base `  G
)
snclseqg.j  |-  J  =  ( TopOpen `  G )
snclseqg.z  |-  .0.  =  ( 0g `  G )
snclseqg.r  |-  .~  =  ( G ~QG  S )
snclseqg.s  |-  S  =  ( ( cls `  J
) `  {  .0.  }
)
Assertion
Ref Expression
snclseqg  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( cls `  J ) `
 { A }
) )

Proof of Theorem snclseqg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snclseqg.s . . . 4  |-  S  =  ( ( cls `  J
) `  {  .0.  }
)
21imaeq2i 5026 . . 3  |-  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" S )  =  ( ( x  e.  X  |->  ( A ( +g  `  G ) x ) ) "
( ( cls `  J
) `  {  .0.  }
) )
3 tgpgrp 17777 . . . . 5  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
43adantr 451 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  G  e.  Grp )
5 snclseqg.j . . . . . . . . . 10  |-  J  =  ( TopOpen `  G )
6 snclseqg.x . . . . . . . . . 10  |-  X  =  ( Base `  G
)
75, 6tgptopon 17781 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  X ) )
87adantr 451 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
9 topontop 16680 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
108, 9syl 15 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  J  e.  Top )
11 snclseqg.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  G )
126, 11grpidcl 14526 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  .0.  e.  X )
134, 12syl 15 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  .0.  e.  X )
1413snssd 3776 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  {  .0.  } 
C_  X )
15 toponuni 16681 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
168, 15syl 15 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  X  =  U. J )
1714, 16sseqtrd 3227 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  {  .0.  } 
C_  U. J )
18 eqid 2296 . . . . . . . 8  |-  U. J  =  U. J
1918clsss3 16812 . . . . . . 7  |-  ( ( J  e.  Top  /\  {  .0.  }  C_  U. J
)  ->  ( ( cls `  J ) `  {  .0.  } )  C_  U. J )
2010, 17, 19syl2anc 642 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  {  .0.  }
)  C_  U. J )
2120, 16sseqtr4d 3228 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  {  .0.  }
)  C_  X )
221, 21syl5eqss 3235 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  S  C_  X )
23 simpr 447 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  A  e.  X )
24 snclseqg.r . . . . 5  |-  .~  =  ( G ~QG  S )
25 eqid 2296 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
266, 24, 25eqglact 14684 . . . 4  |-  ( ( G  e.  Grp  /\  S  C_  X  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" S ) )
274, 22, 23, 26syl3anc 1182 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" S ) )
28 eqid 2296 . . . . 5  |-  ( x  e.  X  |->  ( A ( +g  `  G
) x ) )  =  ( x  e.  X  |->  ( A ( +g  `  G ) x ) )
2928, 6, 25, 5tgplacthmeo 17802 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
x  e.  X  |->  ( A ( +g  `  G
) x ) )  e.  ( J  Homeo  J ) )
3018hmeocls 17475 . . . 4  |-  ( ( ( x  e.  X  |->  ( A ( +g  `  G ) x ) )  e.  ( J 
Homeo  J )  /\  {  .0.  }  C_  U. J )  ->  ( ( cls `  J ) `  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) ) " {  .0.  } ) )  =  ( ( x  e.  X  |->  ( A ( +g  `  G ) x ) ) " ( ( cls `  J ) `
 {  .0.  }
) ) )
3129, 17, 30syl2anc 642 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  ( (
x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
) )  =  ( ( x  e.  X  |->  ( A ( +g  `  G ) x ) ) " ( ( cls `  J ) `
 {  .0.  }
) ) )
322, 27, 313eqtr4a 2354 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( cls `  J ) `
 ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
) ) )
33 df-ima 4718 . . . . 5  |-  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
)  =  ran  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) )  |`  {  .0.  } )
34 resmpt 5016 . . . . . . 7  |-  ( {  .0.  }  C_  X  ->  ( ( x  e.  X  |->  ( A ( +g  `  G ) x ) )  |`  {  .0.  } )  =  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) ) )
3514, 34syl 15 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) )  |`  {  .0.  } )  =  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) ) )
3635rneqd 4922 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ran  ( ( x  e.  X  |->  ( A ( +g  `  G ) x ) )  |`  {  .0.  } )  =  ran  ( x  e. 
{  .0.  }  |->  ( A ( +g  `  G
) x ) ) )
3733, 36syl5eq 2340 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) ) " {  .0.  } )  =  ran  (
x  e.  {  .0.  } 
|->  ( A ( +g  `  G ) x ) ) )
38 fvex 5555 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
3911, 38eqeltri 2366 . . . . . . . 8  |-  .0.  e.  _V
40 oveq2 5882 . . . . . . . . 9  |-  ( x  =  .0.  ->  ( A ( +g  `  G
) x )  =  ( A ( +g  `  G )  .0.  )
)
4140eqeq2d 2307 . . . . . . . 8  |-  ( x  =  .0.  ->  (
y  =  ( A ( +g  `  G
) x )  <->  y  =  ( A ( +g  `  G
)  .0.  ) ) )
4239, 41rexsn 3688 . . . . . . 7  |-  ( E. x  e.  {  .0.  } y  =  ( A ( +g  `  G
) x )  <->  y  =  ( A ( +g  `  G
)  .0.  ) )
436, 25, 11grprid 14529 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( A ( +g  `  G )  .0.  )  =  A )
443, 43sylan 457 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ( A ( +g  `  G
)  .0.  )  =  A )
4544eqeq2d 2307 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
y  =  ( A ( +g  `  G
)  .0.  )  <->  y  =  A ) )
4642, 45syl5bb 248 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ( E. x  e.  {  .0.  } y  =  ( A ( +g  `  G
) x )  <->  y  =  A ) )
4746abbidv 2410 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  { y  |  E. x  e. 
{  .0.  } y  =  ( A ( +g  `  G ) x ) }  =  { y  |  y  =  A } )
48 eqid 2296 . . . . . 6  |-  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) )  =  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) )
4948rnmpt 4941 . . . . 5  |-  ran  (
x  e.  {  .0.  } 
|->  ( A ( +g  `  G ) x ) )  =  { y  |  E. x  e. 
{  .0.  } y  =  ( A ( +g  `  G ) x ) }
50 df-sn 3659 . . . . 5  |-  { A }  =  { y  |  y  =  A }
5147, 49, 503eqtr4g 2353 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ran  ( x  e.  {  .0.  } 
|->  ( A ( +g  `  G ) x ) )  =  { A } )
5237, 51eqtrd 2328 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) ) " {  .0.  } )  =  { A } )
5352fveq2d 5545 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  ( (
x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
) )  =  ( ( cls `  J
) `  { A } ) )
5432, 53eqtrd 2328 1  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( cls `  J ) `
 { A }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801    C_ wss 3165   {csn 3653   U.cuni 3843    e. cmpt 4093   ran crn 4706    |` cres 4707   "cima 4708   ` cfv 5271  (class class class)co 5874   [cec 6674   Basecbs 13164   +g cplusg 13224   TopOpenctopn 13342   0gc0g 13416   Grpcgrp 14378   ~QG cqg 14633   Topctop 16647  TopOnctopon 16648   clsccl 16771    Homeo chmeo 17460   TopGrpctgp 17770
This theorem is referenced by:  tgptsmscls  17848
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-rmo 2564  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-int 3879  df-iun 3923  df-iin 3924  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-1st 6138  df-2nd 6139  df-riota 6320  df-ec 6678  df-map 6790  df-topgen 13360  df-0g 13420  df-mnd 14383  df-plusf 14384  df-grp 14505  df-minusg 14506  df-eqg 14636  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-cls 16774  df-cn 16973  df-cnp 16974  df-tx 17273  df-hmeo 17462  df-tmd 17771  df-tgp 17772
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