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Theorem subgtgp 18166
Description: A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
Hypothesis
Ref Expression
subgtgp.h  |-  H  =  ( Gs  S )
Assertion
Ref Expression
subgtgp  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  TopGrp )

Proof of Theorem subgtgp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 subgtgp.h . . . 4  |-  H  =  ( Gs  S )
21subggrp 14978 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
32adantl 454 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  Grp )
4 tgptmd 18140 . . 3  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
5 subgsubm 14993 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (SubMnd `  G ) )
61submtmd 18165 . . 3  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e. TopMnd )
74, 5, 6syl2an 465 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  H  e. TopMnd )
81subgbas 14979 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
98adantl 454 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  =  ( Base `  H )
)
109mpteq1d 4315 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( x  e.  S  |->  ( ( inv g `  H
) `  x )
)  =  ( x  e.  ( Base `  H
)  |->  ( ( inv g `  H ) `
 x ) ) )
11 eqid 2442 . . . . . . . 8  |-  ( inv g `  G )  =  ( inv g `  G )
12 eqid 2442 . . . . . . . 8  |-  ( inv g `  H )  =  ( inv g `  H )
131, 11, 12subginv 14982 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  G )  /\  x  e.  S )  ->  (
( inv g `  G ) `  x
)  =  ( ( inv g `  H
) `  x )
)
1413adantll 696 . . . . . 6  |-  ( ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  S )  ->  ( ( inv g `  G ) `  x
)  =  ( ( inv g `  H
) `  x )
)
1514mpteq2dva 4320 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( x  e.  S  |->  ( ( inv g `  G
) `  x )
)  =  ( x  e.  S  |->  ( ( inv g `  H
) `  x )
) )
16 eqid 2442 . . . . . . . 8  |-  ( Base `  H )  =  (
Base `  H )
1716, 12grpinvf 14880 . . . . . . 7  |-  ( H  e.  Grp  ->  ( inv g `  H ) : ( Base `  H
) --> ( Base `  H
) )
183, 17syl 16 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H ) : ( Base `  H
) --> ( Base `  H
) )
1918feqmptd 5808 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  =  ( x  e.  ( Base `  H
)  |->  ( ( inv g `  H ) `
 x ) ) )
2010, 15, 193eqtr4rd 2485 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  =  ( x  e.  S  |->  ( ( inv g `  G ) `
 x ) ) )
21 eqid 2442 . . . . 5  |-  ( (
TopOpen `  G )t  S )  =  ( ( TopOpen `  G )t  S )
22 eqid 2442 . . . . . . 7  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
23 eqid 2442 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
2422, 23tgptopon 18143 . . . . . 6  |-  ( G  e.  TopGrp  ->  ( TopOpen `  G
)  e.  (TopOn `  ( Base `  G )
) )
2524adantr 453 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( TopOpen `  G )  e.  (TopOn `  ( Base `  G
) ) )
2623subgss 14976 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
2726adantl 454 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  ( Base `  G ) )
28 tgpgrp 18139 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
2928adantr 453 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
3023, 11grpinvf 14880 . . . . . . . 8  |-  ( G  e.  Grp  ->  ( inv g `  G ) : ( Base `  G
) --> ( Base `  G
) )
3129, 30syl 16 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  G ) : ( Base `  G
) --> ( Base `  G
) )
3231feqmptd 5808 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  G )  =  ( x  e.  ( Base `  G
)  |->  ( ( inv g `  G ) `
 x ) ) )
3322, 11tgpinv 18146 . . . . . . 7  |-  ( G  e.  TopGrp  ->  ( inv g `  G )  e.  ( ( TopOpen `  G )  Cn  ( TopOpen `  G )
) )
3433adantr 453 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  G )  e.  ( ( TopOpen `  G
)  Cn  ( TopOpen `  G ) ) )
3532, 34eqeltrrd 2517 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( x  e.  ( Base `  G
)  |->  ( ( inv g `  G ) `
 x ) )  e.  ( ( TopOpen `  G )  Cn  ( TopOpen
`  G ) ) )
3621, 25, 27, 35cnmpt1res 17739 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( x  e.  S  |->  ( ( inv g `  G
) `  x )
)  e.  ( ( ( TopOpen `  G )t  S
)  Cn  ( TopOpen `  G ) ) )
3720, 36eqeltrd 2516 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  e.  ( ( ( TopOpen `  G )t  S )  Cn  ( TopOpen
`  G ) ) )
38 frn 5626 . . . . . 6  |-  ( ( inv g `  H
) : ( Base `  H ) --> ( Base `  H )  ->  ran  ( inv g `  H
)  C_  ( Base `  H ) )
3918, 38syl 16 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ran  ( inv g `  H ) 
C_  ( Base `  H
) )
4039, 9sseqtr4d 3371 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ran  ( inv g `  H ) 
C_  S )
41 cnrest2 17381 . . . 4  |-  ( ( ( TopOpen `  G )  e.  (TopOn `  ( Base `  G ) )  /\  ran  ( inv g `  H )  C_  S  /\  S  C_  ( Base `  G ) )  -> 
( ( inv g `  H )  e.  ( ( ( TopOpen `  G
)t 
S )  Cn  ( TopOpen
`  G ) )  <-> 
( inv g `  H )  e.  ( ( ( TopOpen `  G
)t 
S )  Cn  (
( TopOpen `  G )t  S
) ) ) )
4225, 40, 27, 41syl3anc 1185 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( inv g `  H )  e.  ( ( (
TopOpen `  G )t  S )  Cn  ( TopOpen `  G
) )  <->  ( inv g `  H )  e.  ( ( ( TopOpen `  G )t  S )  Cn  (
( TopOpen `  G )t  S
) ) ) )
4337, 42mpbid 203 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  e.  ( ( ( TopOpen `  G )t  S )  Cn  (
( TopOpen `  G )t  S
) ) )
441, 22resstopn 17281 . . 3  |-  ( (
TopOpen `  G )t  S )  =  ( TopOpen `  H
)
4544, 12istgp 18138 . 2  |-  ( H  e.  TopGrp 
<->  ( H  e.  Grp  /\  H  e. TopMnd  /\  ( inv g `  H )  e.  ( ( (
TopOpen `  G )t  S )  Cn  ( ( TopOpen `  G )t  S ) ) ) )
463, 7, 43, 45syl3anbrc 1139 1  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  TopGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727    C_ wss 3306    e. cmpt 4291   ran crn 4908   -->wf 5479   ` cfv 5483  (class class class)co 6110   Basecbs 13500   ↾s cress 13501   ↾t crest 13679   TopOpenctopn 13680   Grpcgrp 14716   inv gcminusg 14717  SubMndcsubmnd 14768  SubGrpcsubg 14969  TopOnctopon 16990    Cn ccn 17319  TopMndctmd 18131   TopGrpctgp 18132
This theorem is referenced by:  qqhcn  24406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-oadd 6757  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-fi 7445  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-tset 13579  df-rest 13681  df-topn 13682  df-topgen 13698  df-0g 13758  df-mnd 14721  df-plusf 14722  df-submnd 14770  df-grp 14843  df-minusg 14844  df-subg 14972  df-top 16994  df-bases 16996  df-topon 16997  df-topsp 16998  df-cn 17322  df-tx 17625  df-tmd 18133  df-tgp 18134
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