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Theorem divstgphaus 17805
Description: The quotient of a topological group by a closed normal subgroup is a Hausdorff topological group. In particular, the quotient by the closure of the identity is a Hausdorff group, isomorphic to both the Kolmogorov quotient and the Hausdorff quotient operations on topological spaces (because T0 and Hausdorff coincide for topological groups). (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
divstgp.h  |-  H  =  ( G  /.s  ( G ~QG  Y
) )
divstgphaus.j  |-  J  =  ( TopOpen `  G )
divstgphaus.k  |-  K  =  ( TopOpen `  H )
Assertion
Ref Expression
divstgphaus  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  K  e.  Haus )

Proof of Theorem divstgphaus
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 divstgp.h . . . . . . . 8  |-  H  =  ( G  /.s  ( G ~QG  Y
) )
2 eqid 2283 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
31, 2divs0 14675 . . . . . . 7  |-  ( Y  e.  (NrmSGrp `  G
)  ->  [ ( 0g `  G ) ] ( G ~QG  Y )  =  ( 0g `  H ) )
433ad2ant2 977 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  [ ( 0g
`  G ) ] ( G ~QG  Y )  =  ( 0g `  H ) )
5 tgpgrp 17761 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
653ad2ant1 976 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  G  e.  Grp )
7 eqid 2283 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
87, 2grpidcl 14510 . . . . . . . 8  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  ( Base `  G
) )
96, 8syl 15 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( 0g `  G )  e.  (
Base `  G )
)
10 ovex 5883 . . . . . . . 8  |-  ( G ~QG  Y )  e.  _V
1110ecelqsi 6715 . . . . . . 7  |-  ( ( 0g `  G )  e.  ( Base `  G
)  ->  [ ( 0g `  G ) ] ( G ~QG  Y )  e.  ( ( Base `  G
) /. ( G ~QG  Y ) ) )
129, 11syl 15 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  [ ( 0g
`  G ) ] ( G ~QG  Y )  e.  ( ( Base `  G
) /. ( G ~QG  Y ) ) )
134, 12eqeltrrd 2358 . . . . 5  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( 0g `  H )  e.  ( ( Base `  G
) /. ( G ~QG  Y ) ) )
1413snssd 3760 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  { ( 0g
`  H ) } 
C_  ( ( Base `  G ) /. ( G ~QG  Y ) ) )
15 eqid 2283 . . . . . . 7  |-  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) )  =  ( x  e.  (
Base `  G )  |->  [ x ] ( G ~QG  Y ) )
1615mptpreima 5166 . . . . . 6  |-  ( `' ( x  e.  (
Base `  G )  |->  [ x ] ( G ~QG  Y ) ) " { ( 0g `  H ) } )  =  { x  e.  ( Base `  G
)  |  [ x ] ( G ~QG  Y )  e.  { ( 0g
`  H ) } }
17 nsgsubg 14649 . . . . . . . . . . 11  |-  ( Y  e.  (NrmSGrp `  G
)  ->  Y  e.  (SubGrp `  G ) )
18173ad2ant2 977 . . . . . . . . . 10  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  Y  e.  (SubGrp `  G ) )
19 eqid 2283 . . . . . . . . . . 11  |-  ( G ~QG  Y )  =  ( G ~QG  Y )
207, 19, 2eqgid 14669 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  [ ( 0g `  G ) ] ( G ~QG  Y )  =  Y )
2118, 20syl 15 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  [ ( 0g
`  G ) ] ( G ~QG  Y )  =  Y )
227subgss 14622 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  ( Base `  G ) )
2318, 22syl 15 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  Y  C_  ( Base `  G ) )
2421, 23eqsstrd 3212 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  [ ( 0g
`  G ) ] ( G ~QG  Y )  C_  ( Base `  G ) )
25 dfss1 3373 . . . . . . . 8  |-  ( [ ( 0g `  G
) ] ( G ~QG  Y )  C_  ( Base `  G )  <->  ( ( Base `  G )  i^i 
[ ( 0g `  G ) ] ( G ~QG  Y ) )  =  [ ( 0g `  G ) ] ( G ~QG  Y ) )
2624, 25sylib 188 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( ( Base `  G )  i^i  [
( 0g `  G
) ] ( G ~QG  Y ) )  =  [
( 0g `  G
) ] ( G ~QG  Y ) )
277, 19eqger 14667 . . . . . . . . . . . . 13  |-  ( Y  e.  (SubGrp `  G
)  ->  ( G ~QG  Y
)  Er  ( Base `  G ) )
2818, 27syl 15 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( G ~QG  Y )  Er  ( Base `  G
) )
2928, 9erth 6704 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( ( 0g
`  G ) ( G ~QG  Y ) x  <->  [ ( 0g `  G ) ] ( G ~QG  Y )  =  [
x ] ( G ~QG  Y ) ) )
3029adantr 451 . . . . . . . . . 10  |-  ( ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G
)  /\  Y  e.  ( Clsd `  J )
)  /\  x  e.  ( Base `  G )
)  ->  ( ( 0g `  G ) ( G ~QG  Y ) x  <->  [ ( 0g `  G ) ] ( G ~QG  Y )  =  [
x ] ( G ~QG  Y ) ) )
314adantr 451 . . . . . . . . . . 11  |-  ( ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G
)  /\  Y  e.  ( Clsd `  J )
)  /\  x  e.  ( Base `  G )
)  ->  [ ( 0g `  G ) ] ( G ~QG  Y )  =  ( 0g `  H ) )
3231eqeq1d 2291 . . . . . . . . . 10  |-  ( ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G
)  /\  Y  e.  ( Clsd `  J )
)  /\  x  e.  ( Base `  G )
)  ->  ( [
( 0g `  G
) ] ( G ~QG  Y )  =  [ x ] ( G ~QG  Y )  <-> 
( 0g `  H
)  =  [ x ] ( G ~QG  Y ) ) )
3330, 32bitrd 244 . . . . . . . . 9  |-  ( ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G
)  /\  Y  e.  ( Clsd `  J )
)  /\  x  e.  ( Base `  G )
)  ->  ( ( 0g `  G ) ( G ~QG  Y ) x  <->  ( 0g `  H )  =  [
x ] ( G ~QG  Y ) ) )
34 vex 2791 . . . . . . . . . 10  |-  x  e. 
_V
35 fvex 5539 . . . . . . . . . 10  |-  ( 0g
`  G )  e. 
_V
3634, 35elec 6699 . . . . . . . . 9  |-  ( x  e.  [ ( 0g
`  G ) ] ( G ~QG  Y )  <->  ( 0g `  G ) ( G ~QG  Y ) x )
37 fvex 5539 . . . . . . . . . . 11  |-  ( 0g
`  H )  e. 
_V
3837elsnc2 3669 . . . . . . . . . 10  |-  ( [ x ] ( G ~QG  Y )  e.  { ( 0g `  H ) }  <->  [ x ] ( G ~QG  Y )  =  ( 0g `  H ) )
39 eqcom 2285 . . . . . . . . . 10  |-  ( [ x ] ( G ~QG  Y )  =  ( 0g
`  H )  <->  ( 0g `  H )  =  [
x ] ( G ~QG  Y ) )
4038, 39bitri 240 . . . . . . . . 9  |-  ( [ x ] ( G ~QG  Y )  e.  { ( 0g `  H ) }  <->  ( 0g `  H )  =  [
x ] ( G ~QG  Y ) )
4133, 36, 403bitr4g 279 . . . . . . . 8  |-  ( ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G
)  /\  Y  e.  ( Clsd `  J )
)  /\  x  e.  ( Base `  G )
)  ->  ( x  e.  [ ( 0g `  G ) ] ( G ~QG  Y )  <->  [ x ] ( G ~QG  Y )  e.  { ( 0g
`  H ) } ) )
4241rabbi2dva 3377 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( ( Base `  G )  i^i  [
( 0g `  G
) ] ( G ~QG  Y ) )  =  {
x  e.  ( Base `  G )  |  [
x ] ( G ~QG  Y )  e.  { ( 0g `  H ) } } )
4326, 42, 213eqtr3d 2323 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  { x  e.  ( Base `  G
)  |  [ x ] ( G ~QG  Y )  e.  { ( 0g
`  H ) } }  =  Y )
4416, 43syl5eq 2327 . . . . 5  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( `' ( x  e.  ( Base `  G )  |->  [ x ] ( G ~QG  Y ) ) " { ( 0g `  H ) } )  =  Y )
45 simp3 957 . . . . 5  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  Y  e.  (
Clsd `  J )
)
4644, 45eqeltrd 2357 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( `' ( x  e.  ( Base `  G )  |->  [ x ] ( G ~QG  Y ) ) " { ( 0g `  H ) } )  e.  (
Clsd `  J )
)
47 divstgphaus.j . . . . . . 7  |-  J  =  ( TopOpen `  G )
4847, 7tgptopon 17765 . . . . . 6  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  ( Base `  G
) ) )
49483ad2ant1 976 . . . . 5  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  J  e.  (TopOn `  ( Base `  G
) ) )
501a1i 10 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  H  =  ( G  /.s  ( G ~QG  Y ) ) )
51 eqidd 2284 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( Base `  G
)  =  ( Base `  G ) )
5210a1i 10 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( G ~QG  Y )  e.  _V )
53 simp1 955 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  G  e.  TopGrp )
5450, 51, 15, 52, 53divslem 13445 . . . . 5  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) ) : ( Base `  G
) -onto-> ( ( Base `  G ) /. ( G ~QG  Y ) ) )
55 qtopcld 17404 . . . . 5  |-  ( ( J  e.  (TopOn `  ( Base `  G )
)  /\  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) ) : ( Base `  G
) -onto-> ( ( Base `  G ) /. ( G ~QG  Y ) ) )  ->  ( { ( 0g `  H ) }  e.  ( Clsd `  ( J qTop  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) ) ) )  <->  ( { ( 0g `  H ) }  C_  ( ( Base `  G ) /. ( G ~QG  Y ) )  /\  ( `' ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) ) " { ( 0g `  H ) } )  e.  ( Clsd `  J
) ) ) )
5649, 54, 55syl2anc 642 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( { ( 0g `  H ) }  e.  ( Clsd `  ( J qTop  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) ) ) )  <->  ( { ( 0g `  H ) }  C_  ( ( Base `  G ) /. ( G ~QG  Y ) )  /\  ( `' ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) ) " { ( 0g `  H ) } )  e.  ( Clsd `  J
) ) ) )
5714, 46, 56mpbir2and 888 . . 3  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  { ( 0g
`  H ) }  e.  ( Clsd `  ( J qTop  ( x  e.  (
Base `  G )  |->  [ x ] ( G ~QG  Y ) ) ) ) )
5850, 51, 15, 52, 53divsval 13444 . . . . 5  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  H  =  ( ( x  e.  (
Base `  G )  |->  [ x ] ( G ~QG  Y ) )  "s  G
) )
59 divstgphaus.k . . . . 5  |-  K  =  ( TopOpen `  H )
6058, 51, 54, 53, 47, 59imastopn 17411 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  K  =  ( J qTop  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) ) ) )
6160fveq2d 5529 . . 3  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( Clsd `  K
)  =  ( Clsd `  ( J qTop  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) ) ) ) )
6257, 61eleqtrrd 2360 . 2  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  { ( 0g
`  H ) }  e.  ( Clsd `  K
) )
631divstgp 17804 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )
)  ->  H  e.  TopGrp )
64633adant3 975 . . 3  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  H  e.  TopGrp )
65 eqid 2283 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
6665, 59tgphaus 17799 . . 3  |-  ( H  e.  TopGrp  ->  ( K  e. 
Haus 
<->  { ( 0g `  H ) }  e.  ( Clsd `  K )
) )
6764, 66syl 15 . 2  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( K  e. 
Haus 
<->  { ( 0g `  H ) }  e.  ( Clsd `  K )
) )
6862, 67mpbird 223 1  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  K  e.  Haus )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152   {csn 3640   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   "cima 4692   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858    Er wer 6657   [cec 6658   /.cqs 6659   Basecbs 13148   TopOpenctopn 13326   0gc0g 13400   qTop cqtop 13406    /.s cqus 13408   Grpcgrp 14362  SubGrpcsubg 14615  NrmSGrpcnsg 14616   ~QG cqg 14617  TopOnctopon 16632   Clsdccld 16753   Hauscha 17036   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-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  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-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-0g 13404  df-qtop 13410  df-imas 13411  df-divs 13412  df-mnd 14367  df-plusf 14368  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-nsg 14619  df-eqg 14620  df-oppg 14819  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-cn 16957  df-cnp 16958  df-t1 17042  df-haus 17043  df-tx 17257  df-hmeo 17446  df-tmd 17755  df-tgp 17756
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