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

Theorem indistgp 17783
Description: Any group equipped with the indiscrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
distgp.1  |-  B  =  ( Base `  G
)
distgp.2  |-  J  =  ( TopOpen `  G )
Assertion
Ref Expression
indistgp  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  TopGrp )

Proof of Theorem indistgp
StepHypRef Expression
1 simpl 443 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  Grp )
2 simpr 447 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  J  =  { (/) ,  B }
)
3 distgp.1 . . . . . 6  |-  B  =  ( Base `  G
)
4 fvex 5539 . . . . . 6  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2353 . . . . 5  |-  B  e. 
_V
6 indistopon 16738 . . . . 5  |-  ( B  e.  _V  ->  { (/) ,  B }  e.  (TopOn `  B ) )
75, 6ax-mp 8 . . . 4  |-  { (/) ,  B }  e.  (TopOn `  B )
82, 7syl6eqel 2371 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  J  e.  (TopOn `  B )
)
9 distgp.2 . . . 4  |-  J  =  ( TopOpen `  G )
103, 9istps 16674 . . 3  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
118, 10sylibr 203 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  TopSp )
12 eqid 2283 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
133, 12grpsubf 14545 . . . . 5  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( B  X.  B
) --> B )
1413adantr 451 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G ) : ( B  X.  B
) --> B )
155, 5xpex 4801 . . . . 5  |-  ( B  X.  B )  e. 
_V
165, 15elmap 6796 . . . 4  |-  ( (
-g `  G )  e.  ( B  ^m  ( B  X.  B ) )  <-> 
( -g `  G ) : ( B  X.  B ) --> B )
1714, 16sylibr 203 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G )  e.  ( B  ^m  ( B  X.  B ) ) )
182oveq2d 5874 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  J )  =  ( ( J 
tX  J )  Cn 
{ (/) ,  B }
) )
19 txtopon 17286 . . . . . 6  |-  ( ( J  e.  (TopOn `  B )  /\  J  e.  (TopOn `  B )
)  ->  ( J  tX  J )  e.  (TopOn `  ( B  X.  B
) ) )
208, 8, 19syl2anc 642 . . . . 5  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( J  tX  J )  e.  (TopOn `  ( B  X.  B ) ) )
21 cnindis 17020 . . . . 5  |-  ( ( ( J  tX  J
)  e.  (TopOn `  ( B  X.  B
) )  /\  B  e.  _V )  ->  (
( J  tX  J
)  Cn  { (/) ,  B } )  =  ( B  ^m  ( B  X.  B ) ) )
2220, 5, 21sylancl 643 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  { (/) ,  B } )  =  ( B  ^m  ( B  X.  B ) ) )
2318, 22eqtrd 2315 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2417, 23eleqtrrd 2360 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) )
259, 12istgp2 17774 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e.  TopSp  /\  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) ) )
261, 11, 24, 25syl3anbrc 1136 1  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  TopGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   {cpr 3641    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Basecbs 13148   TopOpenctopn 13326   Grpcgrp 14362   -gcsg 14365  TopOnctopon 16632   TopSpctps 16634    Cn ccn 16954    tX ctx 17255   TopGrpctgp 17754
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-iun 3907  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-map 6774  df-topgen 13344  df-0g 13404  df-mnd 14367  df-plusf 14368  df-grp 14489  df-minusg 14490  df-sbg 14491  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-cnp 16958  df-tx 17257  df-tmd 17755  df-tgp 17756
  Copyright terms: Public domain W3C validator