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Theorem distgp 18086
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 444 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  Grp )
2 simpr 448 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  J  =  ~P B )
3 distgp.1 . . . . . 6  |-  B  =  ( Base `  G
)
4 fvex 5705 . . . . . 6  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2478 . . . . 5  |-  B  e. 
_V
6 distopon 17020 . . . . 5  |-  ( B  e.  _V  ->  ~P B  e.  (TopOn `  B
) )
75, 6ax-mp 8 . . . 4  |-  ~P B  e.  (TopOn `  B )
82, 7syl6eqel 2496 . . 3  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  J  e.  (TopOn `  B ) )
9 distgp.2 . . . 4  |-  J  =  ( TopOpen `  G )
103, 9istps 16960 . . 3  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
118, 10sylibr 204 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  TopSp
)
12 eqid 2408 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
133, 12grpsubf 14827 . . . . 5  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( B  X.  B
) --> B )
1413adantr 452 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( -g `  G ) : ( B  X.  B ) --> B )
155, 5xpex 4953 . . . . 5  |-  ( B  X.  B )  e. 
_V
165, 15elmap 7005 . . . 4  |-  ( (
-g `  G )  e.  ( B  ^m  ( B  X.  B ) )  <-> 
( -g `  G ) : ( B  X.  B ) --> B )
1714, 16sylibr 204 . . 3  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( -g `  G )  e.  ( B  ^m  ( B  X.  B ) ) )
182, 2oveq12d 6062 . . . . . 6  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( J  tX  J )  =  ( ~P B  tX  ~P B ) )
19 txdis 17621 . . . . . . 7  |-  ( ( B  e.  _V  /\  B  e.  _V )  ->  ( ~P B  tX  ~P B )  =  ~P ( B  X.  B
) )
205, 5, 19mp2an 654 . . . . . 6  |-  ( ~P B  tX  ~P B
)  =  ~P ( B  X.  B )
2118, 20syl6eq 2456 . . . . 5  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( J  tX  J )  =  ~P ( B  X.  B
) )
2221oveq1d 6059 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ( J  tX  J )  Cn  J )  =  ( ~P ( B  X.  B )  Cn  J
) )
23 cndis 17313 . . . . 5  |-  ( ( ( B  X.  B
)  e.  _V  /\  J  e.  (TopOn `  B
) )  ->  ( ~P ( B  X.  B
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2415, 8, 23sylancr 645 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ~P ( B  X.  B
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2522, 24eqtrd 2440 . . 3  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ( J  tX  J )  Cn  J )  =  ( B  ^m  ( B  X.  B ) ) )
2617, 25eleqtrrd 2485 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( -g `  G )  e.  ( ( J  tX  J
)  Cn  J ) )
279, 12istgp2 18078 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e.  TopSp  /\  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) ) )
281, 11, 26, 27syl3anbrc 1138 1  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  TopGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2920   ~Pcpw 3763    X. cxp 4839   -->wf 5413   ` cfv 5417  (class class class)co 6044    ^m cmap 6981   Basecbs 13428   TopOpenctopn 13608   Grpcgrp 14644   -gcsg 14647  TopOnctopon 16918   TopSpctps 16920    Cn ccn 17246    tX ctx 17549   TopGrpctgp 18058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-map 6983  df-topgen 13626  df-0g 13686  df-mnd 14649  df-plusf 14650  df-grp 14771  df-minusg 14772  df-sbg 14773  df-top 16922  df-bases 16924  df-topon 16925  df-topsp 16926  df-cn 17249  df-cnp 17250  df-tx 17551  df-tmd 18059  df-tgp 18060
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