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Theorem subgtgp 17788
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 14624 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
32adantl 452 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  Grp )
4 tgptmd 17762 . . 3  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
5 subgsubm 14639 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (SubMnd `  G ) )
61submtmd 17787 . . 3  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e. TopMnd )
74, 5, 6syl2an 463 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  H  e. TopMnd )
81subgbas 14625 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
98adantl 452 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  =  ( Base `  H )
)
10 mpteq1 4100 . . . . . 6  |-  ( S  =  ( Base `  H
)  ->  ( x  e.  S  |->  ( ( inv g `  H
) `  x )
)  =  ( x  e.  ( Base `  H
)  |->  ( ( inv g `  H ) `
 x ) ) )
119, 10syl 15 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( x  e.  S  |->  ( ( inv g `  H
) `  x )
)  =  ( x  e.  ( Base `  H
)  |->  ( ( inv g `  H ) `
 x ) ) )
12 eqid 2283 . . . . . . . 8  |-  ( inv g `  G )  =  ( inv g `  G )
13 eqid 2283 . . . . . . . 8  |-  ( inv g `  H )  =  ( inv g `  H )
141, 12, 13subginv 14628 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  G )  /\  x  e.  S )  ->  (
( inv g `  G ) `  x
)  =  ( ( inv g `  H
) `  x )
)
1514adantll 694 . . . . . 6  |-  ( ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  S )  ->  ( ( inv g `  G ) `  x
)  =  ( ( inv g `  H
) `  x )
)
1615mpteq2dva 4106 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( x  e.  S  |->  ( ( inv g `  G
) `  x )
)  =  ( x  e.  S  |->  ( ( inv g `  H
) `  x )
) )
17 eqid 2283 . . . . . . . 8  |-  ( Base `  H )  =  (
Base `  H )
1817, 13grpinvf 14526 . . . . . . 7  |-  ( H  e.  Grp  ->  ( inv g `  H ) : ( Base `  H
) --> ( Base `  H
) )
193, 18syl 15 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H ) : ( Base `  H
) --> ( Base `  H
) )
2019feqmptd 5575 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  =  ( x  e.  ( Base `  H
)  |->  ( ( inv g `  H ) `
 x ) ) )
2111, 16, 203eqtr4rd 2326 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  =  ( x  e.  S  |->  ( ( inv g `  G ) `
 x ) ) )
22 eqid 2283 . . . . 5  |-  ( (
TopOpen `  G )t  S )  =  ( ( TopOpen `  G )t  S )
23 eqid 2283 . . . . . . 7  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
24 eqid 2283 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
2523, 24tgptopon 17765 . . . . . 6  |-  ( G  e.  TopGrp  ->  ( TopOpen `  G
)  e.  (TopOn `  ( Base `  G )
) )
2625adantr 451 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( TopOpen `  G )  e.  (TopOn `  ( Base `  G
) ) )
2724subgss 14622 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
2827adantl 452 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  ( Base `  G ) )
29 tgpgrp 17761 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
3029adantr 451 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
3124, 12grpinvf 14526 . . . . . . . 8  |-  ( G  e.  Grp  ->  ( inv g `  G ) : ( Base `  G
) --> ( Base `  G
) )
3230, 31syl 15 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  G ) : ( Base `  G
) --> ( Base `  G
) )
3332feqmptd 5575 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  G )  =  ( x  e.  ( Base `  G
)  |->  ( ( inv g `  G ) `
 x ) ) )
3423, 12tgpinv 17768 . . . . . . 7  |-  ( G  e.  TopGrp  ->  ( inv g `  G )  e.  ( ( TopOpen `  G )  Cn  ( TopOpen `  G )
) )
3534adantr 451 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  G )  e.  ( ( TopOpen `  G
)  Cn  ( TopOpen `  G ) ) )
3633, 35eqeltrrd 2358 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( x  e.  ( Base `  G
)  |->  ( ( inv g `  G ) `
 x ) )  e.  ( ( TopOpen `  G )  Cn  ( TopOpen
`  G ) ) )
3722, 26, 28, 36cnmpt1res 17370 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( x  e.  S  |->  ( ( inv g `  G
) `  x )
)  e.  ( ( ( TopOpen `  G )t  S
)  Cn  ( TopOpen `  G ) ) )
3821, 37eqeltrd 2357 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  e.  ( ( ( TopOpen `  G )t  S )  Cn  ( TopOpen
`  G ) ) )
39 frn 5395 . . . . . 6  |-  ( ( inv g `  H
) : ( Base `  H ) --> ( Base `  H )  ->  ran  ( inv g `  H
)  C_  ( Base `  H ) )
4019, 39syl 15 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ran  ( inv g `  H ) 
C_  ( Base `  H
) )
4140, 9sseqtr4d 3215 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ran  ( inv g `  H ) 
C_  S )
42 cnrest2 17014 . . . 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
) ) ) )
4326, 41, 28, 42syl3anc 1182 . . 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
) ) ) )
4438, 43mpbid 201 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  e.  ( ( ( TopOpen `  G )t  S )  Cn  (
( TopOpen `  G )t  S
) ) )
451, 23resstopn 16916 . . 3  |-  ( (
TopOpen `  G )t  S )  =  ( TopOpen `  H
)
4645, 13istgp 17760 . 2  |-  ( H  e.  TopGrp 
<->  ( H  e.  Grp  /\  H  e. TopMnd  /\  ( inv g `  H )  e.  ( ( (
TopOpen `  G )t  S )  Cn  ( ( TopOpen `  G )t  S ) ) ) )
473, 7, 44, 46syl3anbrc 1136 1  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  TopGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152    e. cmpt 4077   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   ↾t crest 13325   TopOpenctopn 13326   Grpcgrp 14362   inv gcminusg 14363  SubMndcsubmnd 14414  SubGrpcsubg 14615  TopOnctopon 16632    Cn ccn 16954  TopMndctmd 17753   TopGrpctgp 17754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-tset 13227  df-rest 13327  df-topn 13328  df-topgen 13344  df-0g 13404  df-mnd 14367  df-plusf 14368  df-submnd 14416  df-grp 14489  df-minusg 14490  df-subg 14618  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-tx 17257  df-tmd 17755  df-tgp 17756
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