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Theorem subgtgp 18096
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 14910 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
32adantl 453 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  Grp )
4 tgptmd 18070 . . 3  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
5 subgsubm 14925 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (SubMnd `  G ) )
61submtmd 18095 . . 3  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e. TopMnd )
74, 5, 6syl2an 464 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  H  e. TopMnd )
81subgbas 14911 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
98adantl 453 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  =  ( Base `  H )
)
109mpteq1d 4258 . . . . 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 2412 . . . . . . . 8  |-  ( inv g `  G )  =  ( inv g `  G )
12 eqid 2412 . . . . . . . 8  |-  ( inv g `  H )  =  ( inv g `  H )
131, 11, 12subginv 14914 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  G )  /\  x  e.  S )  ->  (
( inv g `  G ) `  x
)  =  ( ( inv g `  H
) `  x )
)
1413adantll 695 . . . . . 6  |-  ( ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  S )  ->  ( ( inv g `  G ) `  x
)  =  ( ( inv g `  H
) `  x )
)
1514mpteq2dva 4263 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( x  e.  S  |->  ( ( inv g `  G
) `  x )
)  =  ( x  e.  S  |->  ( ( inv g `  H
) `  x )
) )
16 eqid 2412 . . . . . . . 8  |-  ( Base `  H )  =  (
Base `  H )
1716, 12grpinvf 14812 . . . . . . 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 5746 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  =  ( x  e.  ( Base `  H
)  |->  ( ( inv g `  H ) `
 x ) ) )
2010, 15, 193eqtr4rd 2455 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  =  ( x  e.  S  |->  ( ( inv g `  G ) `
 x ) ) )
21 eqid 2412 . . . . 5  |-  ( (
TopOpen `  G )t  S )  =  ( ( TopOpen `  G )t  S )
22 eqid 2412 . . . . . . 7  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
23 eqid 2412 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
2422, 23tgptopon 18073 . . . . . 6  |-  ( G  e.  TopGrp  ->  ( TopOpen `  G
)  e.  (TopOn `  ( Base `  G )
) )
2524adantr 452 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( TopOpen `  G )  e.  (TopOn `  ( Base `  G
) ) )
2623subgss 14908 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
2726adantl 453 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  ( Base `  G ) )
28 tgpgrp 18069 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
2928adantr 452 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
3023, 11grpinvf 14812 . . . . . . . 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 5746 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  G )  =  ( x  e.  ( Base `  G
)  |->  ( ( inv g `  G ) `
 x ) ) )
3322, 11tgpinv 18076 . . . . . . 7  |-  ( G  e.  TopGrp  ->  ( inv g `  G )  e.  ( ( TopOpen `  G )  Cn  ( TopOpen `  G )
) )
3433adantr 452 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  G )  e.  ( ( TopOpen `  G
)  Cn  ( TopOpen `  G ) ) )
3532, 34eqeltrrd 2487 . . . . 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 17669 . . . 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 2486 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  e.  ( ( ( TopOpen `  G )t  S )  Cn  ( TopOpen
`  G ) ) )
38 frn 5564 . . . . . 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 3353 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ran  ( inv g `  H ) 
C_  S )
41 cnrest2 17312 . . . 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 1184 . . 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 202 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  e.  ( ( ( TopOpen `  G )t  S )  Cn  (
( TopOpen `  G )t  S
) ) )
441, 22resstopn 17212 . . 3  |-  ( (
TopOpen `  G )t  S )  =  ( TopOpen `  H
)
4544, 12istgp 18068 . 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 1138 1  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  TopGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3288    e. cmpt 4234   ran crn 4846   -->wf 5417   ` cfv 5421  (class class class)co 6048   Basecbs 13432   ↾s cress 13433   ↾t crest 13611   TopOpenctopn 13612   Grpcgrp 14648   inv gcminusg 14649  SubMndcsubmnd 14700  SubGrpcsubg 14901  TopOnctopon 16922    Cn ccn 17250  TopMndctmd 18061   TopGrpctgp 18062
This theorem is referenced by:  qqhcn  24336
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-tset 13511  df-rest 13613  df-topn 13614  df-topgen 13630  df-0g 13690  df-mnd 14653  df-plusf 14654  df-submnd 14702  df-grp 14775  df-minusg 14776  df-subg 14904  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cn 17253  df-tx 17555  df-tmd 18063  df-tgp 18064
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