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Theorem snclseqg 19685
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 5166 . . 3  |-  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" S )  =  ( ( x  e.  X  |->  ( A ( +g  `  G ) x ) ) "
( ( cls `  J
) `  {  .0.  }
) )
3 tgpgrp 19648 . . . . 5  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
43adantr 465 . . . 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 19652 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  X ) )
87adantr 465 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
9 topontop 18530 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
108, 9syl 16 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  J  e.  Top )
11 snclseqg.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  G )
126, 11grpidcl 15565 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  .0.  e.  X )
134, 12syl 16 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  .0.  e.  X )
1413snssd 4017 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  {  .0.  } 
C_  X )
15 toponuni 18531 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
168, 15syl 16 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  X  =  U. J )
1714, 16sseqtrd 3391 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  {  .0.  } 
C_  U. J )
18 eqid 2442 . . . . . . . 8  |-  U. J  =  U. J
1918clsss3 18662 . . . . . . 7  |-  ( ( J  e.  Top  /\  {  .0.  }  C_  U. J
)  ->  ( ( cls `  J ) `  {  .0.  } )  C_  U. J )
2010, 17, 19syl2anc 661 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  {  .0.  }
)  C_  U. J )
2120, 16sseqtr4d 3392 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  {  .0.  }
)  C_  X )
221, 21syl5eqss 3399 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  S  C_  X )
23 simpr 461 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  A  e.  X )
24 snclseqg.r . . . . 5  |-  .~  =  ( G ~QG  S )
25 eqid 2442 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
266, 24, 25eqglact 15731 . . . 4  |-  ( ( G  e.  Grp  /\  S  C_  X  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" S ) )
274, 22, 23, 26syl3anc 1218 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" S ) )
28 eqid 2442 . . . . 5  |-  ( x  e.  X  |->  ( A ( +g  `  G
) x ) )  =  ( x  e.  X  |->  ( A ( +g  `  G ) x ) )
2928, 6, 25, 5tgplacthmeo 19673 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
x  e.  X  |->  ( A ( +g  `  G
) x ) )  e.  ( J Homeo J ) )
3018hmeocls 19340 . . . 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 661 . . 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 2500 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( cls `  J ) `
 ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
) ) )
33 df-ima 4852 . . . . 5  |-  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
)  =  ran  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) )  |`  {  .0.  } )
34 resmpt 5155 . . . . . . 7  |-  ( {  .0.  }  C_  X  ->  ( ( x  e.  X  |->  ( A ( +g  `  G ) x ) )  |`  {  .0.  } )  =  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) ) )
3514, 34syl 16 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) )  |`  {  .0.  } )  =  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) ) )
3635rneqd 5066 . . . . 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 2486 . . . 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 5700 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
3911, 38eqeltri 2512 . . . . . . . 8  |-  .0.  e.  _V
40 oveq2 6098 . . . . . . . . 9  |-  ( x  =  .0.  ->  ( A ( +g  `  G
) x )  =  ( A ( +g  `  G )  .0.  )
)
4140eqeq2d 2453 . . . . . . . 8  |-  ( x  =  .0.  ->  (
y  =  ( A ( +g  `  G
) x )  <->  y  =  ( A ( +g  `  G
)  .0.  ) ) )
4239, 41rexsn 3915 . . . . . . 7  |-  ( E. x  e.  {  .0.  } y  =  ( A ( +g  `  G
) x )  <->  y  =  ( A ( +g  `  G
)  .0.  ) )
436, 25, 11grprid 15568 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( A ( +g  `  G )  .0.  )  =  A )
443, 43sylan 471 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ( A ( +g  `  G
)  .0.  )  =  A )
4544eqeq2d 2453 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
y  =  ( A ( +g  `  G
)  .0.  )  <->  y  =  A ) )
4642, 45syl5bb 257 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ( E. x  e.  {  .0.  } y  =  ( A ( +g  `  G
) x )  <->  y  =  A ) )
4746abbidv 2556 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  { y  |  E. x  e. 
{  .0.  } y  =  ( A ( +g  `  G ) x ) }  =  { y  |  y  =  A } )
48 eqid 2442 . . . . . 6  |-  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) )  =  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) )
4948rnmpt 5084 . . . . 5  |-  ran  (
x  e.  {  .0.  } 
|->  ( A ( +g  `  G ) x ) )  =  { y  |  E. x  e. 
{  .0.  } y  =  ( A ( +g  `  G ) x ) }
50 df-sn 3877 . . . . 5  |-  { A }  =  { y  |  y  =  A }
5147, 49, 503eqtr4g 2499 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ran  ( x  e.  {  .0.  } 
|->  ( A ( +g  `  G ) x ) )  =  { A } )
5237, 51eqtrd 2474 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) ) " {  .0.  } )  =  { A } )
5352fveq2d 5694 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  ( (
x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
) )  =  ( ( cls `  J
) `  { A } ) )
5432, 53eqtrd 2474 1  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( cls `  J ) `
 { A }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2428   E.wrex 2715   _Vcvv 2971    C_ wss 3327   {csn 3876   U.cuni 4090    e. cmpt 4349   ran crn 4840    |` cres 4841   "cima 4842   ` cfv 5417  (class class class)co 6090   [cec 7098   Basecbs 14173   +g cplusg 14237   TopOpenctopn 14359   0gc0g 14377   Grpcgrp 15409   ~QG cqg 15676   Topctop 18497  TopOnctopon 18498   clsccl 18621   Homeochmeo 19325   TopGrpctgp 19641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-ec 7102  df-map 7215  df-0g 14379  df-topgen 14381  df-mnd 15414  df-plusf 15415  df-grp 15544  df-minusg 15545  df-eqg 15679  df-top 18502  df-bases 18504  df-topon 18505  df-topsp 18506  df-cld 18622  df-cls 18624  df-cn 18830  df-cnp 18831  df-tx 19134  df-hmeo 19327  df-tmd 19642  df-tgp 19643
This theorem is referenced by:  tgptsmscls  19723
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