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Theorem divstgphaus 19692
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 2442 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
31, 2divs0 15738 . . . . . . 7  |-  ( Y  e.  (NrmSGrp `  G
)  ->  [ ( 0g `  G ) ] ( G ~QG  Y )  =  ( 0g `  H ) )
433ad2ant2 1010 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  [ ( 0g
`  G ) ] ( G ~QG  Y )  =  ( 0g `  H ) )
5 tgpgrp 19648 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
653ad2ant1 1009 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  G  e.  Grp )
7 eqid 2442 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
87, 2grpidcl 15565 . . . . . . . 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 6115 . . . . . . . 8  |-  ( G ~QG  Y )  e.  _V
1110ecelqsi 7155 . . . . . . 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 2517 . . . . 5  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( 0g `  H )  e.  ( ( Base `  G
) /. ( G ~QG  Y ) ) )
1413snssd 4017 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  { ( 0g
`  H ) } 
C_  ( ( Base `  G ) /. ( G ~QG  Y ) ) )
15 eqid 2442 . . . . . . 7  |-  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) )  =  ( x  e.  (
Base `  G )  |->  [ x ] ( G ~QG  Y ) )
1615mptpreima 5330 . . . . . 6  |-  ( `' ( x  e.  (
Base `  G )  |->  [ x ] ( G ~QG  Y ) ) " { ( 0g `  H ) } )  =  { x  e.  ( Base `  G
)  |  [ x ] ( G ~QG  Y )  e.  { ( 0g
`  H ) } }
17 nsgsubg 15712 . . . . . . . . . . 11  |-  ( Y  e.  (NrmSGrp `  G
)  ->  Y  e.  (SubGrp `  G ) )
18173ad2ant2 1010 . . . . . . . . . 10  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  Y  e.  (SubGrp `  G ) )
19 eqid 2442 . . . . . . . . . . 11  |-  ( G ~QG  Y )  =  ( G ~QG  Y )
207, 19, 2eqgid 15732 . . . . . . . . . 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 15681 . . . . . . . . . 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 3389 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  [ ( 0g
`  G ) ] ( G ~QG  Y )  C_  ( Base `  G ) )
25 dfss1 3554 . . . . . . . 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 196 . . . . . . 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 15730 . . . . . . . . . . . . 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 7144 . . . . . . . . . . 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 465 . . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G
)  /\  Y  e.  ( Clsd `  J )
)  /\  x  e.  ( Base `  G )
)  ->  [ ( 0g `  G ) ] ( G ~QG  Y )  =  ( 0g `  H ) )
3231eqeq1d 2450 . . . . . . . . . 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 253 . . . . . . . . 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 2974 . . . . . . . . . 10  |-  x  e. 
_V
35 fvex 5700 . . . . . . . . . 10  |-  ( 0g
`  G )  e. 
_V
3634, 35elec 7139 . . . . . . . . 9  |-  ( x  e.  [ ( 0g
`  G ) ] ( G ~QG  Y )  <->  ( 0g `  G ) ( G ~QG  Y ) x )
37 fvex 5700 . . . . . . . . . . 11  |-  ( 0g
`  H )  e. 
_V
3837elsnc2 3907 . . . . . . . . . 10  |-  ( [ x ] ( G ~QG  Y )  e.  { ( 0g `  H ) }  <->  [ x ] ( G ~QG  Y )  =  ( 0g `  H ) )
39 eqcom 2444 . . . . . . . . . 10  |-  ( [ x ] ( G ~QG  Y )  =  ( 0g
`  H )  <->  ( 0g `  H )  =  [
x ] ( G ~QG  Y ) )
4038, 39bitri 249 . . . . . . . . 9  |-  ( [ x ] ( G ~QG  Y )  e.  { ( 0g `  H ) }  <->  ( 0g `  H )  =  [
x ] ( G ~QG  Y ) )
4133, 36, 403bitr4g 288 . . . . . . . 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 3557 . . . . . . 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 2482 . . . . . 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 2486 . . . . 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 990 . . . . 5  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  Y  e.  (
Clsd `  J )
)
4644, 45eqeltrd 2516 . . . 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 19652 . . . . . 6  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  ( Base `  G
) ) )
49483ad2ant1 1009 . . . . 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 2443 . . . . . 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 988 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  G  e.  TopGrp )
5450, 51, 15, 52, 53divslem 14480 . . . . 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 19285 . . . . 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 661 . . . 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 913 . . 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 14479 . . . . 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 19292 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  K  =  ( J qTop  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) ) ) )
6160fveq2d 5694 . . 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 2519 . 2  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  { ( 0g
`  H ) }  e.  ( Clsd `  K
) )
631divstgp 19691 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )
)  ->  H  e.  TopGrp )
64633adant3 1008 . . 3  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  H  e.  TopGrp )
65 eqid 2442 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
6665, 59tgphaus 19686 . . 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 232 1  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  K  e.  Haus )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {crab 2718   _Vcvv 2971    i^i cin 3326    C_ wss 3327   {csn 3876   class class class wbr 4291    e. cmpt 4349   `'ccnv 4838   "cima 4842   -onto->wfo 5415   ` cfv 5417  (class class class)co 6090    Er wer 7097   [cec 7098   /.cqs 7099   Basecbs 14173   TopOpenctopn 14359   0gc0g 14377   qTop cqtop 14440    /.s cqus 14442   Grpcgrp 15409  SubGrpcsubg 15674  NrmSGrpcnsg 15675   ~QG cqg 15676  TopOnctopon 18498   Clsdccld 18619   Hauscha 18911   TopGrpctgp 19641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-ec 7102  df-qs 7106  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-fz 11437  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-sca 14253  df-vsca 14254  df-ip 14255  df-tset 14256  df-ple 14257  df-ds 14259  df-rest 14360  df-topn 14361  df-0g 14379  df-topgen 14381  df-qtop 14444  df-imas 14445  df-divs 14446  df-mnd 15414  df-plusf 15415  df-grp 15544  df-minusg 15545  df-sbg 15546  df-subg 15677  df-nsg 15678  df-eqg 15679  df-oppg 15860  df-top 18502  df-bases 18504  df-topon 18505  df-topsp 18506  df-cld 18622  df-cn 18830  df-cnp 18831  df-t1 18917  df-haus 18918  df-tx 19134  df-hmeo 19327  df-tmd 19642  df-tgp 19643
This theorem is referenced by: (None)
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