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

Theorem tgphaus 17815
Description: A topological group is Hausdorff iff the identity subgroup is closed. (Contributed by Mario Carneiro, 18-Sep-2015.)
Hypotheses
Ref Expression
tgphaus.1  |-  .0.  =  ( 0g `  G )
tgphaus.j  |-  J  =  ( TopOpen `  G )
Assertion
Ref Expression
tgphaus  |-  ( G  e.  TopGrp  ->  ( J  e. 
Haus 
<->  {  .0.  }  e.  ( Clsd `  J )
) )

Proof of Theorem tgphaus
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgpgrp 17777 . . . . 5  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
2 eqid 2296 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
3 tgphaus.1 . . . . . 6  |-  .0.  =  ( 0g `  G )
42, 3grpidcl 14526 . . . . 5  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
51, 4syl 15 . . . 4  |-  ( G  e.  TopGrp  ->  .0.  e.  ( Base `  G ) )
6 tgphaus.j . . . . . 6  |-  J  =  ( TopOpen `  G )
76, 2tgptopon 17781 . . . . 5  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  ( Base `  G
) ) )
8 toponuni 16681 . . . . 5  |-  ( J  e.  (TopOn `  ( Base `  G ) )  ->  ( Base `  G
)  =  U. J
)
97, 8syl 15 . . . 4  |-  ( G  e.  TopGrp  ->  ( Base `  G
)  =  U. J
)
105, 9eleqtrd 2372 . . 3  |-  ( G  e.  TopGrp  ->  .0.  e.  U. J
)
11 eqid 2296 . . . . 5  |-  U. J  =  U. J
1211sncld 17115 . . . 4  |-  ( ( J  e.  Haus  /\  .0.  e.  U. J )  ->  {  .0.  }  e.  (
Clsd `  J )
)
1312expcom 424 . . 3  |-  (  .0. 
e.  U. J  ->  ( J  e.  Haus  ->  {  .0.  }  e.  ( Clsd `  J
) ) )
1410, 13syl 15 . 2  |-  ( G  e.  TopGrp  ->  ( J  e. 
Haus  ->  {  .0.  }  e.  ( Clsd `  J
) ) )
15 eqid 2296 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
166, 15tgpsubcn 17789 . . . . 5  |-  ( G  e.  TopGrp  ->  ( -g `  G
)  e.  ( ( J  tX  J )  Cn  J ) )
17 cnclima 17013 . . . . . 6  |-  ( ( ( -g `  G
)  e.  ( ( J  tX  J )  Cn  J )  /\  {  .0.  }  e.  (
Clsd `  J )
)  ->  ( `' ( -g `  G )
" {  .0.  }
)  e.  ( Clsd `  ( J  tX  J
) ) )
1817ex 423 . . . . 5  |-  ( (
-g `  G )  e.  ( ( J  tX  J )  Cn  J
)  ->  ( {  .0.  }  e.  ( Clsd `  J )  ->  ( `' ( -g `  G
) " {  .0.  } )  e.  ( Clsd `  ( J  tX  J
) ) ) )
1916, 18syl 15 . . . 4  |-  ( G  e.  TopGrp  ->  ( {  .0.  }  e.  ( Clsd `  J
)  ->  ( `' ( -g `  G )
" {  .0.  }
)  e.  ( Clsd `  ( J  tX  J
) ) ) )
20 cnvimass 5049 . . . . . . . . 9  |-  ( `' ( -g `  G
) " {  .0.  } )  C_  dom  ( -g `  G )
212, 15grpsubf 14561 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( ( Base `  G
)  X.  ( Base `  G ) ) --> (
Base `  G )
)
221, 21syl 15 . . . . . . . . . 10  |-  ( G  e.  TopGrp  ->  ( -g `  G
) : ( (
Base `  G )  X.  ( Base `  G
) ) --> ( Base `  G ) )
23 fdm 5409 . . . . . . . . . 10  |-  ( (
-g `  G ) : ( ( Base `  G )  X.  ( Base `  G ) ) --> ( Base `  G
)  ->  dom  ( -g `  G )  =  ( ( Base `  G
)  X.  ( Base `  G ) ) )
2422, 23syl 15 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  dom  ( -g `  G )  =  ( ( Base `  G
)  X.  ( Base `  G ) ) )
2520, 24syl5sseq 3239 . . . . . . . 8  |-  ( G  e.  TopGrp  ->  ( `' (
-g `  G ) " {  .0.  } ) 
C_  ( ( Base `  G )  X.  ( Base `  G ) ) )
26 relxp 4810 . . . . . . . 8  |-  Rel  (
( Base `  G )  X.  ( Base `  G
) )
27 relss 4791 . . . . . . . 8  |-  ( ( `' ( -g `  G
) " {  .0.  } )  C_  ( ( Base `  G )  X.  ( Base `  G
) )  ->  ( Rel  ( ( Base `  G
)  X.  ( Base `  G ) )  ->  Rel  ( `' ( -g `  G ) " {  .0.  } ) ) )
2825, 26, 27ee10 1366 . . . . . . 7  |-  ( G  e.  TopGrp  ->  Rel  ( `' ( -g `  G )
" {  .0.  }
) )
29 dfrel4v 5141 . . . . . . 7  |-  ( Rel  ( `' ( -g `  G ) " {  .0.  } )  <->  ( `' ( -g `  G )
" {  .0.  }
)  =  { <. x ,  y >.  |  x ( `' ( -g `  G ) " {  .0.  } ) y } )
3028, 29sylib 188 . . . . . 6  |-  ( G  e.  TopGrp  ->  ( `' (
-g `  G ) " {  .0.  } )  =  { <. x ,  y >.  |  x ( `' ( -g `  G ) " {  .0.  } ) y } )
31 ffn 5405 . . . . . . . . . . . 12  |-  ( (
-g `  G ) : ( ( Base `  G )  X.  ( Base `  G ) ) --> ( Base `  G
)  ->  ( -g `  G )  Fn  (
( Base `  G )  X.  ( Base `  G
) ) )
3222, 31syl 15 . . . . . . . . . . 11  |-  ( G  e.  TopGrp  ->  ( -g `  G
)  Fn  ( (
Base `  G )  X.  ( Base `  G
) ) )
33 elpreima 5661 . . . . . . . . . . 11  |-  ( (
-g `  G )  Fn  ( ( Base `  G
)  X.  ( Base `  G ) )  -> 
( <. x ,  y
>.  e.  ( `' (
-g `  G ) " {  .0.  } )  <-> 
( <. x ,  y
>.  e.  ( ( Base `  G )  X.  ( Base `  G ) )  /\  ( ( -g `  G ) `  <. x ,  y >. )  e.  {  .0.  } ) ) )
3432, 33syl 15 . . . . . . . . . 10  |-  ( G  e.  TopGrp  ->  ( <. x ,  y >.  e.  ( `' ( -g `  G
) " {  .0.  } )  <->  ( <. x ,  y >.  e.  ( ( Base `  G
)  X.  ( Base `  G ) )  /\  ( ( -g `  G
) `  <. x ,  y >. )  e.  {  .0.  } ) ) )
35 opelxp 4735 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( ( Base `  G
)  X.  ( Base `  G ) )  <->  ( x  e.  ( Base `  G
)  /\  y  e.  ( Base `  G )
) )
3635anbi1i 676 . . . . . . . . . . 11  |-  ( (
<. x ,  y >.  e.  ( ( Base `  G
)  X.  ( Base `  G ) )  /\  ( ( -g `  G
) `  <. x ,  y >. )  e.  {  .0.  } )  <->  ( (
x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  /\  (
( -g `  G ) `
 <. x ,  y
>. )  e.  {  .0.  } ) )
372, 3, 15grpsubeq0 14568 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  Grp  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
( x ( -g `  G ) y )  =  .0.  <->  x  =  y ) )
38373expb 1152 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Grp  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G ) ) )  ->  ( ( x ( -g `  G
) y )  =  .0.  <->  x  =  y
) )
391, 38sylan 457 . . . . . . . . . . . . 13  |-  ( ( G  e.  TopGrp  /\  (
x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) ) )  -> 
( ( x (
-g `  G )
y )  =  .0.  <->  x  =  y ) )
40 df-ov 5877 . . . . . . . . . . . . . . 15  |-  ( x ( -g `  G
) y )  =  ( ( -g `  G
) `  <. x ,  y >. )
4140eleq1i 2359 . . . . . . . . . . . . . 14  |-  ( ( x ( -g `  G
) y )  e. 
{  .0.  }  <->  ( ( -g `  G ) `  <. x ,  y >.
)  e.  {  .0.  } )
42 ovex 5899 . . . . . . . . . . . . . . 15  |-  ( x ( -g `  G
) y )  e. 
_V
4342elsnc 3676 . . . . . . . . . . . . . 14  |-  ( ( x ( -g `  G
) y )  e. 
{  .0.  }  <->  ( x
( -g `  G ) y )  =  .0.  )
4441, 43bitr3i 242 . . . . . . . . . . . . 13  |-  ( ( ( -g `  G
) `  <. x ,  y >. )  e.  {  .0.  }  <->  ( x (
-g `  G )
y )  =  .0.  )
45 equcom 1665 . . . . . . . . . . . . 13  |-  ( y  =  x  <->  x  =  y )
4639, 44, 453bitr4g 279 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  (
x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) ) )  -> 
( ( ( -g `  G ) `  <. x ,  y >. )  e.  {  .0.  }  <->  y  =  x ) )
4746pm5.32da 622 . . . . . . . . . . 11  |-  ( G  e.  TopGrp  ->  ( ( ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  /\  (
( -g `  G ) `
 <. x ,  y
>. )  e.  {  .0.  } )  <->  ( ( x  e.  ( Base `  G
)  /\  y  e.  ( Base `  G )
)  /\  y  =  x ) ) )
4836, 47syl5bb 248 . . . . . . . . . 10  |-  ( G  e.  TopGrp  ->  ( ( <.
x ,  y >.  e.  ( ( Base `  G
)  X.  ( Base `  G ) )  /\  ( ( -g `  G
) `  <. x ,  y >. )  e.  {  .0.  } )  <->  ( (
x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  /\  y  =  x ) ) )
4934, 48bitrd 244 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  ( <. x ,  y >.  e.  ( `' ( -g `  G
) " {  .0.  } )  <->  ( ( x  e.  ( Base `  G
)  /\  y  e.  ( Base `  G )
)  /\  y  =  x ) ) )
50 df-br 4040 . . . . . . . . 9  |-  ( x ( `' ( -g `  G ) " {  .0.  } ) y  <->  <. x ,  y >.  e.  ( `' ( -g `  G
) " {  .0.  } ) )
51 eleq1 2356 . . . . . . . . . . . 12  |-  ( y  =  x  ->  (
y  e.  ( Base `  G )  <->  x  e.  ( Base `  G )
) )
5251biimparc 473 . . . . . . . . . . 11  |-  ( ( x  e.  ( Base `  G )  /\  y  =  x )  ->  y  e.  ( Base `  G
) )
5352pm4.71i 613 . . . . . . . . . 10  |-  ( ( x  e.  ( Base `  G )  /\  y  =  x )  <->  ( (
x  e.  ( Base `  G )  /\  y  =  x )  /\  y  e.  ( Base `  G
) ) )
54 an32 773 . . . . . . . . . 10  |-  ( ( ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G ) )  /\  y  =  x )  <->  ( ( x  e.  ( Base `  G
)  /\  y  =  x )  /\  y  e.  ( Base `  G
) ) )
5553, 54bitr4i 243 . . . . . . . . 9  |-  ( ( x  e.  ( Base `  G )  /\  y  =  x )  <->  ( (
x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  /\  y  =  x ) )
5649, 50, 553bitr4g 279 . . . . . . . 8  |-  ( G  e.  TopGrp  ->  ( x ( `' ( -g `  G
) " {  .0.  } ) y  <->  ( x  e.  ( Base `  G
)  /\  y  =  x ) ) )
5756opabbidv 4098 . . . . . . 7  |-  ( G  e.  TopGrp  ->  { <. x ,  y >.  |  x ( `' ( -g `  G ) " {  .0.  } ) y }  =  { <. x ,  y >.  |  ( x  e.  ( Base `  G )  /\  y  =  x ) } )
58 opabresid 5019 . . . . . . 7  |-  { <. x ,  y >.  |  ( x  e.  ( Base `  G )  /\  y  =  x ) }  =  (  _I  |`  ( Base `  G ) )
5957, 58syl6eq 2344 . . . . . 6  |-  ( G  e.  TopGrp  ->  { <. x ,  y >.  |  x ( `' ( -g `  G ) " {  .0.  } ) y }  =  (  _I  |`  ( Base `  G ) ) )
609reseq2d 4971 . . . . . 6  |-  ( G  e.  TopGrp  ->  (  _I  |`  ( Base `  G ) )  =  (  _I  |`  U. J
) )
6130, 59, 603eqtrd 2332 . . . . 5  |-  ( G  e.  TopGrp  ->  ( `' (
-g `  G ) " {  .0.  } )  =  (  _I  |`  U. J
) )
6261eleq1d 2362 . . . 4  |-  ( G  e.  TopGrp  ->  ( ( `' ( -g `  G
) " {  .0.  } )  e.  ( Clsd `  ( J  tX  J
) )  <->  (  _I  |` 
U. J )  e.  ( Clsd `  ( J  tX  J ) ) ) )
6319, 62sylibd 205 . . 3  |-  ( G  e.  TopGrp  ->  ( {  .0.  }  e.  ( Clsd `  J
)  ->  (  _I  |` 
U. J )  e.  ( Clsd `  ( J  tX  J ) ) ) )
64 topontop 16680 . . . . 5  |-  ( J  e.  (TopOn `  ( Base `  G ) )  ->  J  e.  Top )
657, 64syl 15 . . . 4  |-  ( G  e.  TopGrp  ->  J  e.  Top )
6611hausdiag 17355 . . . . 5  |-  ( J  e.  Haus  <->  ( J  e. 
Top  /\  (  _I  |` 
U. J )  e.  ( Clsd `  ( J  tX  J ) ) ) )
6766baib 871 . . . 4  |-  ( J  e.  Top  ->  ( J  e.  Haus  <->  (  _I  |` 
U. J )  e.  ( Clsd `  ( J  tX  J ) ) ) )
6865, 67syl 15 . . 3  |-  ( G  e.  TopGrp  ->  ( J  e. 
Haus 
<->  (  _I  |`  U. J
)  e.  ( Clsd `  ( J  tX  J
) ) ) )
6963, 68sylibrd 225 . 2  |-  ( G  e.  TopGrp  ->  ( {  .0.  }  e.  ( Clsd `  J
)  ->  J  e.  Haus ) )
7014, 69impbid 183 1  |-  ( G  e.  TopGrp  ->  ( J  e. 
Haus 
<->  {  .0.  }  e.  ( Clsd `  J )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   {csn 3653   <.cop 3656   U.cuni 3843   class class class wbr 4039   {copab 4092    _I cid 4320    X. cxp 4703   `'ccnv 4704   dom cdm 4705    |` cres 4707   "cima 4708   Rel wrel 4710    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   TopOpenctopn 13342   0gc0g 13416   Grpcgrp 14378   -gcsg 14381   Topctop 16647  TopOnctopon 16648   Clsdccld 16769    Cn ccn 16970   Hauscha 17052    tX ctx 17271   TopGrpctgp 17770
This theorem is referenced by:  tgpt1  17816  divstgphaus  17821
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-iun 3923  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-map 6790  df-topgen 13360  df-0g 13420  df-mnd 14383  df-plusf 14384  df-grp 14505  df-minusg 14506  df-sbg 14507  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-cn 16973  df-t1 17058  df-haus 17059  df-tx 17273  df-tmd 17771  df-tgp 17772
  Copyright terms: Public domain W3C validator