MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snclseqg Structured version   Unicode version

Theorem snclseqg 18137
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 5193 . . 3  |-  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" S )  =  ( ( x  e.  X  |->  ( A ( +g  `  G ) x ) ) "
( ( cls `  J
) `  {  .0.  }
) )
3 tgpgrp 18100 . . . . 5  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
43adantr 452 . . . 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 18104 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  X ) )
87adantr 452 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
9 topontop 16983 . . . . . . . 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 14825 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  .0.  e.  X )
134, 12syl 16 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  .0.  e.  X )
1413snssd 3935 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  {  .0.  } 
C_  X )
15 toponuni 16984 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
168, 15syl 16 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  X  =  U. J )
1714, 16sseqtrd 3376 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  {  .0.  } 
C_  U. J )
18 eqid 2435 . . . . . . . 8  |-  U. J  =  U. J
1918clsss3 17115 . . . . . . 7  |-  ( ( J  e.  Top  /\  {  .0.  }  C_  U. J
)  ->  ( ( cls `  J ) `  {  .0.  } )  C_  U. J )
2010, 17, 19syl2anc 643 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  {  .0.  }
)  C_  U. J )
2120, 16sseqtr4d 3377 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  {  .0.  }
)  C_  X )
221, 21syl5eqss 3384 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  S  C_  X )
23 simpr 448 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  A  e.  X )
24 snclseqg.r . . . . 5  |-  .~  =  ( G ~QG  S )
25 eqid 2435 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
266, 24, 25eqglact 14983 . . . 4  |-  ( ( G  e.  Grp  /\  S  C_  X  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" S ) )
274, 22, 23, 26syl3anc 1184 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" S ) )
28 eqid 2435 . . . . 5  |-  ( x  e.  X  |->  ( A ( +g  `  G
) x ) )  =  ( x  e.  X  |->  ( A ( +g  `  G ) x ) )
2928, 6, 25, 5tgplacthmeo 18125 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
x  e.  X  |->  ( A ( +g  `  G
) x ) )  e.  ( J  Homeo  J ) )
3018hmeocls 17792 . . . 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 643 . . 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 2493 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( cls `  J ) `
 ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
) ) )
33 df-ima 4883 . . . . 5  |-  ( ( x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
)  =  ran  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) )  |`  {  .0.  } )
34 resmpt 5183 . . . . . . 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 5089 . . . . 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 2479 . . . 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 5734 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
3911, 38eqeltri 2505 . . . . . . . 8  |-  .0.  e.  _V
40 oveq2 6081 . . . . . . . . 9  |-  ( x  =  .0.  ->  ( A ( +g  `  G
) x )  =  ( A ( +g  `  G )  .0.  )
)
4140eqeq2d 2446 . . . . . . . 8  |-  ( x  =  .0.  ->  (
y  =  ( A ( +g  `  G
) x )  <->  y  =  ( A ( +g  `  G
)  .0.  ) ) )
4239, 41rexsn 3842 . . . . . . 7  |-  ( E. x  e.  {  .0.  } y  =  ( A ( +g  `  G
) x )  <->  y  =  ( A ( +g  `  G
)  .0.  ) )
436, 25, 11grprid 14828 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( A ( +g  `  G )  .0.  )  =  A )
443, 43sylan 458 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ( A ( +g  `  G
)  .0.  )  =  A )
4544eqeq2d 2446 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
y  =  ( A ( +g  `  G
)  .0.  )  <->  y  =  A ) )
4642, 45syl5bb 249 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ( E. x  e.  {  .0.  } y  =  ( A ( +g  `  G
) x )  <->  y  =  A ) )
4746abbidv 2549 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  { y  |  E. x  e. 
{  .0.  } y  =  ( A ( +g  `  G ) x ) }  =  { y  |  y  =  A } )
48 eqid 2435 . . . . . 6  |-  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) )  =  ( x  e.  {  .0.  }  |->  ( A ( +g  `  G ) x ) )
4948rnmpt 5108 . . . . 5  |-  ran  (
x  e.  {  .0.  } 
|->  ( A ( +g  `  G ) x ) )  =  { y  |  E. x  e. 
{  .0.  } y  =  ( A ( +g  `  G ) x ) }
50 df-sn 3812 . . . . 5  |-  { A }  =  { y  |  y  =  A }
5147, 49, 503eqtr4g 2492 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  ran  ( x  e.  {  .0.  } 
|->  ( A ( +g  `  G ) x ) )  =  { A } )
5237, 51eqtrd 2467 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( A ( +g  `  G ) x ) ) " {  .0.  } )  =  { A } )
5352fveq2d 5724 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( cls `  J
) `  ( (
x  e.  X  |->  ( A ( +g  `  G
) x ) )
" {  .0.  }
) )  =  ( ( cls `  J
) `  { A } ) )
5432, 53eqtrd 2467 1  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( cls `  J ) `
 { A }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   _Vcvv 2948    C_ wss 3312   {csn 3806   U.cuni 4007    e. cmpt 4258   ran crn 4871    |` cres 4872   "cima 4873   ` cfv 5446  (class class class)co 6073   [cec 6895   Basecbs 13461   +g cplusg 13521   TopOpenctopn 13641   0gc0g 13715   Grpcgrp 14677   ~QG cqg 14932   Topctop 16950  TopOnctopon 16951   clsccl 17074    Homeo chmeo 17777   TopGrpctgp 18093
This theorem is referenced by:  tgptsmscls  18171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-ec 6899  df-map 7012  df-topgen 13659  df-0g 13719  df-mnd 14682  df-plusf 14683  df-grp 14804  df-minusg 14805  df-eqg 14935  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-cls 17077  df-cn 17283  df-cnp 17284  df-tx 17586  df-hmeo 17779  df-tmd 18094  df-tgp 18095
  Copyright terms: Public domain W3C validator