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Theorem subgtgp 17804
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 14640 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
32adantl 452 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  Grp )
4 tgptmd 17778 . . 3  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
5 subgsubm 14655 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (SubMnd `  G ) )
61submtmd 17803 . . 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 14641 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
98adantl 452 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  =  ( Base `  H )
)
10 mpteq1 4116 . . . . . 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 2296 . . . . . . . 8  |-  ( inv g `  G )  =  ( inv g `  G )
13 eqid 2296 . . . . . . . 8  |-  ( inv g `  H )  =  ( inv g `  H )
141, 12, 13subginv 14644 . . . . . . 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 4122 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( x  e.  S  |->  ( ( inv g `  G
) `  x )
)  =  ( x  e.  S  |->  ( ( inv g `  H
) `  x )
) )
17 eqid 2296 . . . . . . . 8  |-  ( Base `  H )  =  (
Base `  H )
1817, 13grpinvf 14542 . . . . . . 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 5591 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  =  ( x  e.  ( Base `  H
)  |->  ( ( inv g `  H ) `
 x ) ) )
2111, 16, 203eqtr4rd 2339 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  =  ( x  e.  S  |->  ( ( inv g `  G ) `
 x ) ) )
22 eqid 2296 . . . . 5  |-  ( (
TopOpen `  G )t  S )  =  ( ( TopOpen `  G )t  S )
23 eqid 2296 . . . . . . 7  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
24 eqid 2296 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
2523, 24tgptopon 17781 . . . . . 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 14638 . . . . . 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 17777 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
3029adantr 451 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
3124, 12grpinvf 14542 . . . . . . . 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 5591 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  G )  =  ( x  e.  ( Base `  G
)  |->  ( ( inv g `  G ) `
 x ) ) )
3423, 12tgpinv 17784 . . . . . . 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 2371 . . . . 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 17386 . . . 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 2370 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( inv g `  H )  e.  ( ( ( TopOpen `  G )t  S )  Cn  ( TopOpen
`  G ) ) )
39 frn 5411 . . . . . 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 3228 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ran  ( inv g `  H ) 
C_  S )
42 cnrest2 17030 . . . 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 16932 . . 3  |-  ( (
TopOpen `  G )t  S )  =  ( TopOpen `  H
)
4645, 13istgp 17776 . 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 1632    e. wcel 1696    C_ wss 3165    e. cmpt 4093   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   ↾t crest 13341   TopOpenctopn 13342   Grpcgrp 14378   inv gcminusg 14379  SubMndcsubmnd 14430  SubGrpcsubg 14631  TopOnctopon 16648    Cn ccn 16970  TopMndctmd 17769   TopGrpctgp 17770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-tset 13243  df-rest 13343  df-topn 13344  df-topgen 13360  df-0g 13420  df-mnd 14383  df-plusf 14384  df-submnd 14432  df-grp 14505  df-minusg 14506  df-subg 14634  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cn 16973  df-tx 17273  df-tmd 17771  df-tgp 17772
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