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Theorem divstgphaus 18145
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 2436 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
31, 2divs0 14991 . . . . . . 7  |-  ( Y  e.  (NrmSGrp `  G
)  ->  [ ( 0g `  G ) ] ( G ~QG  Y )  =  ( 0g `  H ) )
433ad2ant2 979 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  [ ( 0g
`  G ) ] ( G ~QG  Y )  =  ( 0g `  H ) )
5 tgpgrp 18101 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
653ad2ant1 978 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  G  e.  Grp )
7 eqid 2436 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
87, 2grpidcl 14826 . . . . . . . 8  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  ( Base `  G
) )
96, 8syl 16 . . . . . . 7  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( 0g `  G )  e.  (
Base `  G )
)
10 ovex 6099 . . . . . . . 8  |-  ( G ~QG  Y )  e.  _V
1110ecelqsi 6953 . . . . . . 7  |-  ( ( 0g `  G )  e.  ( Base `  G
)  ->  [ ( 0g `  G ) ] ( G ~QG  Y )  e.  ( ( Base `  G
) /. ( G ~QG  Y ) ) )
129, 11syl 16 . . . . . 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 2511 . . . . 5  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( 0g `  H )  e.  ( ( Base `  G
) /. ( G ~QG  Y ) ) )
1413snssd 3936 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  { ( 0g
`  H ) } 
C_  ( ( Base `  G ) /. ( G ~QG  Y ) ) )
15 eqid 2436 . . . . . . 7  |-  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) )  =  ( x  e.  (
Base `  G )  |->  [ x ] ( G ~QG  Y ) )
1615mptpreima 5356 . . . . . 6  |-  ( `' ( x  e.  (
Base `  G )  |->  [ x ] ( G ~QG  Y ) ) " { ( 0g `  H ) } )  =  { x  e.  ( Base `  G
)  |  [ x ] ( G ~QG  Y )  e.  { ( 0g
`  H ) } }
17 nsgsubg 14965 . . . . . . . . . . 11  |-  ( Y  e.  (NrmSGrp `  G
)  ->  Y  e.  (SubGrp `  G ) )
18173ad2ant2 979 . . . . . . . . . 10  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  Y  e.  (SubGrp `  G ) )
19 eqid 2436 . . . . . . . . . . 11  |-  ( G ~QG  Y )  =  ( G ~QG  Y )
207, 19, 2eqgid 14985 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  [ ( 0g `  G ) ] ( G ~QG  Y )  =  Y )
2118, 20syl 16 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  [ ( 0g
`  G ) ] ( G ~QG  Y )  =  Y )
227subgss 14938 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  ( Base `  G ) )
2318, 22syl 16 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  Y  C_  ( Base `  G ) )
2421, 23eqsstrd 3375 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  [ ( 0g
`  G ) ] ( G ~QG  Y )  C_  ( Base `  G ) )
25 dfss1 3538 . . . . . . . 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 189 . . . . . . 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 14983 . . . . . . . . . . . . 13  |-  ( Y  e.  (SubGrp `  G
)  ->  ( G ~QG  Y
)  Er  ( Base `  G ) )
2818, 27syl 16 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( G ~QG  Y )  Er  ( Base `  G
) )
2928, 9erth 6942 . . . . . . . . . . 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 452 . . . . . . . . . 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 452 . . . . . . . . . . 11  |-  ( ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G
)  /\  Y  e.  ( Clsd `  J )
)  /\  x  e.  ( Base `  G )
)  ->  [ ( 0g `  G ) ] ( G ~QG  Y )  =  ( 0g `  H ) )
3231eqeq1d 2444 . . . . . . . . . 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 245 . . . . . . . . 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 2952 . . . . . . . . . 10  |-  x  e. 
_V
35 fvex 5735 . . . . . . . . . 10  |-  ( 0g
`  G )  e. 
_V
3634, 35elec 6937 . . . . . . . . 9  |-  ( x  e.  [ ( 0g
`  G ) ] ( G ~QG  Y )  <->  ( 0g `  G ) ( G ~QG  Y ) x )
37 fvex 5735 . . . . . . . . . . 11  |-  ( 0g
`  H )  e. 
_V
3837elsnc2 3836 . . . . . . . . . 10  |-  ( [ x ] ( G ~QG  Y )  e.  { ( 0g `  H ) }  <->  [ x ] ( G ~QG  Y )  =  ( 0g `  H ) )
39 eqcom 2438 . . . . . . . . . 10  |-  ( [ x ] ( G ~QG  Y )  =  ( 0g
`  H )  <->  ( 0g `  H )  =  [
x ] ( G ~QG  Y ) )
4038, 39bitri 241 . . . . . . . . 9  |-  ( [ x ] ( G ~QG  Y )  e.  { ( 0g `  H ) }  <->  ( 0g `  H )  =  [
x ] ( G ~QG  Y ) )
4133, 36, 403bitr4g 280 . . . . . . . 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 3542 . . . . . . 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 2476 . . . . . 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 2480 . . . . 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 959 . . . . 5  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  Y  e.  (
Clsd `  J )
)
4644, 45eqeltrd 2510 . . . 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 18105 . . . . . 6  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  ( Base `  G
) ) )
49483ad2ant1 978 . . . . 5  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  J  e.  (TopOn `  ( Base `  G
) ) )
501a1i 11 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  H  =  ( G  /.s  ( G ~QG  Y ) ) )
51 eqidd 2437 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( Base `  G
)  =  ( Base `  G ) )
5210a1i 11 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( G ~QG  Y )  e.  _V )
53 simp1 957 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  G  e.  TopGrp )
5450, 51, 15, 52, 53divslem 13761 . . . . 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 17738 . . . . 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 643 . . . 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 889 . . 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 13760 . . . . 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 17745 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  K  =  ( J qTop  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) ) ) )
6160fveq2d 5725 . . 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 2513 . 2  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  { ( 0g
`  H ) }  e.  ( Clsd `  K
) )
631divstgp 18144 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )
)  ->  H  e.  TopGrp )
64633adant3 977 . . 3  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  H  e.  TopGrp )
65 eqid 2436 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
6665, 59tgphaus 18139 . . 3  |-  ( H  e.  TopGrp  ->  ( K  e. 
Haus 
<->  { ( 0g `  H ) }  e.  ( Clsd `  K )
) )
6764, 66syl 16 . 2  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( K  e. 
Haus 
<->  { ( 0g `  H ) }  e.  ( Clsd `  K )
) )
6862, 67mpbird 224 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2702   _Vcvv 2949    i^i cin 3312    C_ wss 3313   {csn 3807   class class class wbr 4205    e. cmpt 4259   `'ccnv 4870   "cima 4874   -onto->wfo 5445   ` cfv 5447  (class class class)co 6074    Er wer 6895   [cec 6896   /.cqs 6897   Basecbs 13462   TopOpenctopn 13642   0gc0g 13716   qTop cqtop 13722    /.s cqus 13724   Grpcgrp 14678  SubGrpcsubg 14931  NrmSGrpcnsg 14932   ~QG cqg 14933  TopOnctopon 16952   Clsdccld 17073   Hauscha 17365   TopGrpctgp 18094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-tpos 6472  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-oadd 6721  df-er 6898  df-ec 6900  df-qs 6904  df-map 7013  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-sup 7439  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-2 10051  df-3 10052  df-4 10053  df-5 10054  df-6 10055  df-7 10056  df-8 10057  df-9 10058  df-10 10059  df-n0 10215  df-z 10276  df-dec 10376  df-uz 10482  df-fz 11037  df-struct 13464  df-ndx 13465  df-slot 13466  df-base 13467  df-sets 13468  df-ress 13469  df-plusg 13535  df-mulr 13536  df-sca 13538  df-vsca 13539  df-tset 13541  df-ple 13542  df-ds 13544  df-rest 13643  df-topn 13644  df-topgen 13660  df-0g 13720  df-qtop 13726  df-imas 13727  df-divs 13728  df-mnd 14683  df-plusf 14684  df-grp 14805  df-minusg 14806  df-sbg 14807  df-subg 14934  df-nsg 14935  df-eqg 14936  df-oppg 15135  df-top 16956  df-bases 16958  df-topon 16959  df-topsp 16960  df-cld 17076  df-cn 17284  df-cnp 17285  df-t1 17371  df-haus 17372  df-tx 17587  df-hmeo 17780  df-tmd 18095  df-tgp 18096
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