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Theorem distgp 20464
Description: Any group equipped with the discrete 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
distgp  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  TopGrp )

Proof of Theorem distgp
StepHypRef Expression
1 simpl 457 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  Grp )
2 simpr 461 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  J  =  ~P B )
3 distgp.1 . . . . . 6  |-  B  =  ( Base `  G
)
4 fvex 5882 . . . . . 6  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2551 . . . . 5  |-  B  e. 
_V
6 distopon 19364 . . . . 5  |-  ( B  e.  _V  ->  ~P B  e.  (TopOn `  B
) )
75, 6ax-mp 5 . . . 4  |-  ~P B  e.  (TopOn `  B )
82, 7syl6eqel 2563 . . 3  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  J  e.  (TopOn `  B ) )
9 distgp.2 . . . 4  |-  J  =  ( TopOpen `  G )
103, 9istps 19304 . . 3  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
118, 10sylibr 212 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  TopSp
)
12 eqid 2467 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
133, 12grpsubf 15988 . . . . 5  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( B  X.  B
) --> B )
1413adantr 465 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P 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  =  ~P B
)  ->  ( -g `  G )  e.  ( B  ^m  ( B  X.  B ) ) )
182, 2oveq12d 6313 . . . . . 6  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( J  tX  J )  =  ( ~P B  tX  ~P B ) )
19 txdis 19999 . . . . . . 7  |-  ( ( B  e.  _V  /\  B  e.  _V )  ->  ( ~P B  tX  ~P B )  =  ~P ( B  X.  B
) )
205, 5, 19mp2an 672 . . . . . 6  |-  ( ~P B  tX  ~P B
)  =  ~P ( B  X.  B )
2118, 20syl6eq 2524 . . . . 5  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( J  tX  J )  =  ~P ( B  X.  B
) )
2221oveq1d 6310 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ( J  tX  J )  Cn  J )  =  ( ~P ( B  X.  B )  Cn  J
) )
23 cndis 19658 . . . . 5  |-  ( ( ( B  X.  B
)  e.  _V  /\  J  e.  (TopOn `  B
) )  ->  ( ~P ( B  X.  B
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2415, 8, 23sylancr 663 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ~P ( B  X.  B
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2522, 24eqtrd 2508 . . 3  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ( J  tX  J )  Cn  J )  =  ( B  ^m  ( B  X.  B ) ) )
2617, 25eleqtrrd 2558 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( -g `  G )  e.  ( ( J  tX  J
)  Cn  J ) )
279, 12istgp2 20456 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e.  TopSp  /\  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) ) )
281, 11, 26, 27syl3anbrc 1180 1  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  TopGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   ~Pcpw 4016    X. cxp 5003   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   Basecbs 14506   TopOpenctopn 14693   Grpcgrp 15924   -gcsg 15926  TopOnctopon 19262   TopSpctps 19264    Cn ccn 19591    tX ctx 19927   TopGrpctgp 20436
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 14713  df-topgen 14715  df-plusf 15744  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-grp 15928  df-minusg 15929  df-sbg 15930  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cn 19594  df-cnp 19595  df-tx 19929  df-tmd 20437  df-tgp 20438
This theorem is referenced by: (None)
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