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Theorem snclseqg 21130
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 21093 . . . . 5  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
43adantr 467 . . . 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 21097 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  X ) )
87adantr 467 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
9 topontop 19941 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
108, 9syl 17 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  J  e.  Top )
11 snclseqg.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  G )
126, 11grpidcl 16694 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  .0.  e.  X )
134, 12syl 17 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  .0.  e.  X )
1413snssd 4117 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  {  .0.  } 
C_  X )
15 toponuni 19942 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
168, 15syl 17 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  X  =  U. J )
1714, 16sseqtrd 3468 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  {  .0.  } 
C_  U. J )
18 eqid 2451 . . . . . . . 8  |-  U. J  =  U. J
1918clsss3 20074 . . . . . . 7  |-  ( ( J  e.  Top  /\  {  .0.  }  C_  U. J
)  ->  ( ( cls `  J ) `  {  .0.  } )  C_  U. J )
2010, 17, 19syl2anc 667 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  {  .0.  }
)  C_  U. J )
2120, 16sseqtr4d 3469 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  {  .0.  }
)  C_  X )
221, 21syl5eqss 3476 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  S  C_  X )
23 simpr 463 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  A  e.  X )
24 snclseqg.r . . . . 5  |-  .~  =  ( G ~QG  S )
25 eqid 2451 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
266, 24, 25eqglact 16868 . . . 4  |-  ( ( G  e.  Grp  /\  S  C_  X  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" S ) )
274, 22, 23, 26syl3anc 1268 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" S ) )
28 eqid 2451 . . . . 5  |-  ( x  e.  X  |->  ( A ( +g  `  G
) x ) )  =  ( x  e.  X  |->  ( A ( +g  `  G ) x ) )
2928, 6, 25, 5tgplacthmeo 21118 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
x  e.  X  |->  ( A ( +g  `  G
) x ) )  e.  ( J Homeo J ) )
3018hmeocls 20783 . . . 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 667 . . 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 2511 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( cls `  J ) `
 ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
) ) )
33 df-ima 4847 . . . . 5  |-  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
)  =  ran  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) )  |`  {  .0.  } )
3414resmptd 5156 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) )  |`  {  .0.  } )  =  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) ) )
3534rneqd 5062 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ran  ( ( x  e.  X  |->  ( A ( +g  `  G ) x ) )  |`  {  .0.  } )  =  ran  ( x  e. 
{  .0.  }  |->  ( A ( +g  `  G
) x ) ) )
3633, 35syl5eq 2497 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) ) " {  .0.  } )  =  ran  (
x  e.  {  .0.  } 
|->  ( A ( +g  `  G ) x ) ) )
37 fvex 5875 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
3811, 37eqeltri 2525 . . . . . . . 8  |-  .0.  e.  _V
39 oveq2 6298 . . . . . . . . 9  |-  ( x  =  .0.  ->  ( A ( +g  `  G
) x )  =  ( A ( +g  `  G )  .0.  )
)
4039eqeq2d 2461 . . . . . . . 8  |-  ( x  =  .0.  ->  (
y  =  ( A ( +g  `  G
) x )  <->  y  =  ( A ( +g  `  G
)  .0.  ) ) )
4138, 40rexsn 4011 . . . . . . 7  |-  ( E. x  e.  {  .0.  } y  =  ( A ( +g  `  G
) x )  <->  y  =  ( A ( +g  `  G
)  .0.  ) )
426, 25, 11grprid 16697 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( A ( +g  `  G )  .0.  )  =  A )
433, 42sylan 474 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ( A ( +g  `  G
)  .0.  )  =  A )
4443eqeq2d 2461 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
y  =  ( A ( +g  `  G
)  .0.  )  <->  y  =  A ) )
4541, 44syl5bb 261 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ( E. x  e.  {  .0.  } y  =  ( A ( +g  `  G
) x )  <->  y  =  A ) )
4645abbidv 2569 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  { y  |  E. x  e. 
{  .0.  } y  =  ( A ( +g  `  G ) x ) }  =  { y  |  y  =  A } )
47 eqid 2451 . . . . . 6  |-  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) )  =  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) )
4847rnmpt 5080 . . . . 5  |-  ran  (
x  e.  {  .0.  } 
|->  ( A ( +g  `  G ) x ) )  =  { y  |  E. x  e. 
{  .0.  } y  =  ( A ( +g  `  G ) x ) }
49 df-sn 3969 . . . . 5  |-  { A }  =  { y  |  y  =  A }
5046, 48, 493eqtr4g 2510 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ran  ( x  e.  {  .0.  } 
|->  ( A ( +g  `  G ) x ) )  =  { A } )
5136, 50eqtrd 2485 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) ) " {  .0.  } )  =  { A } )
5251fveq2d 5869 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  ( (
x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
) )  =  ( ( cls `  J
) `  { A } ) )
5332, 52eqtrd 2485 1  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( cls `  J ) `
 { A }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   {cab 2437   E.wrex 2738   _Vcvv 3045    C_ wss 3404   {csn 3968   U.cuni 4198    |-> cmpt 4461   ran crn 4835    |` cres 4836   "cima 4837   ` cfv 5582  (class class class)co 6290   [cec 7361   Basecbs 15121   +g cplusg 15190   TopOpenctopn 15320   0gc0g 15338   Grpcgrp 16669   ~QG cqg 16813   Topctop 19917  TopOnctopon 19918   clsccl 20033   Homeochmeo 20768   TopGrpctgp 21086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-ec 7365  df-map 7474  df-0g 15340  df-topgen 15342  df-plusf 16487  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-minusg 16674  df-eqg 16816  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-cls 20036  df-cn 20243  df-cnp 20244  df-tx 20577  df-hmeo 20770  df-tmd 21087  df-tgp 21088
This theorem is referenced by:  tgptsmscls  21164
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