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Theorem snclseqg 17798
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 5010 . . 3  |-  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" S )  =  ( ( x  e.  X  |->  ( A ( +g  `  G ) x ) ) "
( ( cls `  J
) `  {  .0.  }
) )
3 tgpgrp 17761 . . . . 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 17765 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  X ) )
87adantr 451 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
9 topontop 16664 . . . . . . . 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 14510 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  .0.  e.  X )
134, 12syl 15 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  .0.  e.  X )
1413snssd 3760 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  {  .0.  } 
C_  X )
15 toponuni 16665 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
168, 15syl 15 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  X  =  U. J )
1714, 16sseqtrd 3214 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  {  .0.  } 
C_  U. J )
18 eqid 2283 . . . . . . . 8  |-  U. J  =  U. J
1918clsss3 16796 . . . . . . 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 3215 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  {  .0.  }
)  C_  X )
221, 21syl5eqss 3222 . . . 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 2283 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
266, 24, 25eqglact 14668 . . . 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 2283 . . . . 5  |-  ( x  e.  X  |->  ( A ( +g  `  G
) x ) )  =  ( x  e.  X  |->  ( A ( +g  `  G ) x ) )
2928, 6, 25, 5tgplacthmeo 17786 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
x  e.  X  |->  ( A ( +g  `  G
) x ) )  e.  ( J  Homeo  J ) )
3018hmeocls 17459 . . . 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 2341 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( cls `  J ) `
 ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
) ) )
33 df-ima 4702 . . . . 5  |-  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
)  =  ran  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) )  |`  {  .0.  } )
34 resmpt 5000 . . . . . . 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 4906 . . . . 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 2327 . . . 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 5539 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
3911, 38eqeltri 2353 . . . . . . . 8  |-  .0.  e.  _V
40 oveq2 5866 . . . . . . . . 9  |-  ( x  =  .0.  ->  ( A ( +g  `  G
) x )  =  ( A ( +g  `  G )  .0.  )
)
4140eqeq2d 2294 . . . . . . . 8  |-  ( x  =  .0.  ->  (
y  =  ( A ( +g  `  G
) x )  <->  y  =  ( A ( +g  `  G
)  .0.  ) ) )
4239, 41rexsn 3675 . . . . . . 7  |-  ( E. x  e.  {  .0.  } y  =  ( A ( +g  `  G
) x )  <->  y  =  ( A ( +g  `  G
)  .0.  ) )
436, 25, 11grprid 14513 . . . . . . . . 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 2294 . . . . . . 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 2397 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  { y  |  E. x  e. 
{  .0.  } y  =  ( A ( +g  `  G ) x ) }  =  { y  |  y  =  A } )
48 eqid 2283 . . . . . 6  |-  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) )  =  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) )
4948rnmpt 4925 . . . . 5  |-  ran  (
x  e.  {  .0.  } 
|->  ( A ( +g  `  G ) x ) )  =  { y  |  E. x  e. 
{  .0.  } y  =  ( A ( +g  `  G ) x ) }
50 df-sn 3646 . . . . 5  |-  { A }  =  { y  |  y  =  A }
5147, 49, 503eqtr4g 2340 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ran  ( x  e.  {  .0.  } 
|->  ( A ( +g  `  G ) x ) )  =  { A } )
5237, 51eqtrd 2315 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) ) " {  .0.  } )  =  { A } )
5352fveq2d 5529 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  ( (
x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
) )  =  ( ( cls `  J
) `  { A } ) )
5432, 53eqtrd 2315 1  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( cls `  J ) `
 { A }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788    C_ wss 3152   {csn 3640   U.cuni 3827    e. cmpt 4077   ran crn 4690    |` cres 4691   "cima 4692   ` cfv 5255  (class class class)co 5858   [cec 6658   Basecbs 13148   +g cplusg 13208   TopOpenctopn 13326   0gc0g 13400   Grpcgrp 14362   ~QG cqg 14617   Topctop 16631  TopOnctopon 16632   clsccl 16755    Homeo chmeo 17444   TopGrpctgp 17754
This theorem is referenced by:  tgptsmscls  17832
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-rmo 2551  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-int 3863  df-iun 3907  df-iin 3908  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-1st 6122  df-2nd 6123  df-riota 6304  df-ec 6662  df-map 6774  df-topgen 13344  df-0g 13404  df-mnd 14367  df-plusf 14368  df-grp 14489  df-minusg 14490  df-eqg 14620  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-cls 16758  df-cn 16957  df-cnp 16958  df-tx 17257  df-hmeo 17446  df-tmd 17755  df-tgp 17756
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