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Theorem indistgp 20467
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 457 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  Grp )
2 simpr 461 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  J  =  { (/) ,  B }
)
3 distgp.1 . . . . . 6  |-  B  =  ( Base `  G
)
4 fvex 5882 . . . . . 6  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2551 . . . . 5  |-  B  e. 
_V
6 indistopon 19370 . . . . 5  |-  ( B  e.  _V  ->  { (/) ,  B }  e.  (TopOn `  B ) )
75, 6ax-mp 5 . . . 4  |-  { (/) ,  B }  e.  (TopOn `  B )
82, 7syl6eqel 2563 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  J  e.  (TopOn `  B )
)
9 distgp.2 . . . 4  |-  J  =  ( TopOpen `  G )
103, 9istps 19306 . . 3  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
118, 10sylibr 212 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  TopSp )
12 eqid 2467 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
133, 12grpsubf 15989 . . . . 5  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( B  X.  B
) --> B )
1413adantr 465 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G ) : ( B  X.  B
) --> B )
155, 5xpex 6599 . . . . 5  |-  ( B  X.  B )  e. 
_V
165, 15elmap 7459 . . . 4  |-  ( (
-g `  G )  e.  ( B  ^m  ( B  X.  B ) )  <-> 
( -g `  G ) : ( B  X.  B ) --> B )
1714, 16sylibr 212 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G )  e.  ( B  ^m  ( B  X.  B ) ) )
182oveq2d 6311 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  J )  =  ( ( J 
tX  J )  Cn 
{ (/) ,  B }
) )
19 txtopon 19960 . . . . . 6  |-  ( ( J  e.  (TopOn `  B )  /\  J  e.  (TopOn `  B )
)  ->  ( J  tX  J )  e.  (TopOn `  ( B  X.  B
) ) )
208, 8, 19syl2anc 661 . . . . 5  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( J  tX  J )  e.  (TopOn `  ( B  X.  B ) ) )
21 cnindis 19661 . . . . 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 662 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  { (/) ,  B } )  =  ( B  ^m  ( B  X.  B ) ) )
2318, 22eqtrd 2508 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2417, 23eleqtrrd 2558 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) )
259, 12istgp2 20458 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e.  TopSp  /\  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) ) )
261, 11, 24, 25syl3anbrc 1180 1  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  TopGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790   {cpr 4035    X. cxp 5003   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   Basecbs 14507   TopOpenctopn 14694   Grpcgrp 15925   -gcsg 15927  TopOnctopon 19264   TopSpctps 19266    Cn ccn 19593    tX ctx 19929   TopGrpctgp 20438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-map 7434  df-0g 14714  df-topgen 14716  df-plusf 15745  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cn 19596  df-cnp 19597  df-tx 19931  df-tmd 20439  df-tgp 20440
This theorem is referenced by: (None)
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