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Theorem divstgphaus 18157
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 2438 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
31, 2divs0 15003 . . . . . . 7  |-  ( Y  e.  (NrmSGrp `  G
)  ->  [ ( 0g `  G ) ] ( G ~QG  Y )  =  ( 0g `  H ) )
433ad2ant2 980 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  [ ( 0g
`  G ) ] ( G ~QG  Y )  =  ( 0g `  H ) )
5 tgpgrp 18113 . . . . . . . . 9  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
653ad2ant1 979 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  G  e.  Grp )
7 eqid 2438 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
87, 2grpidcl 14838 . . . . . . . 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 6109 . . . . . . . 8  |-  ( G ~QG  Y )  e.  _V
1110ecelqsi 6963 . . . . . . 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 2513 . . . . 5  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( 0g `  H )  e.  ( ( Base `  G
) /. ( G ~QG  Y ) ) )
1413snssd 3945 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  { ( 0g
`  H ) } 
C_  ( ( Base `  G ) /. ( G ~QG  Y ) ) )
15 eqid 2438 . . . . . . 7  |-  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) )  =  ( x  e.  (
Base `  G )  |->  [ x ] ( G ~QG  Y ) )
1615mptpreima 5366 . . . . . 6  |-  ( `' ( x  e.  (
Base `  G )  |->  [ x ] ( G ~QG  Y ) ) " { ( 0g `  H ) } )  =  { x  e.  ( Base `  G
)  |  [ x ] ( G ~QG  Y )  e.  { ( 0g
`  H ) } }
17 nsgsubg 14977 . . . . . . . . . . 11  |-  ( Y  e.  (NrmSGrp `  G
)  ->  Y  e.  (SubGrp `  G ) )
18173ad2ant2 980 . . . . . . . . . 10  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  Y  e.  (SubGrp `  G ) )
19 eqid 2438 . . . . . . . . . . 11  |-  ( G ~QG  Y )  =  ( G ~QG  Y )
207, 19, 2eqgid 14997 . . . . . . . . . 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 14950 . . . . . . . . . 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 3384 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  [ ( 0g
`  G ) ] ( G ~QG  Y )  C_  ( Base `  G ) )
25 dfss1 3547 . . . . . . . 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 190 . . . . . . 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 14995 . . . . . . . . . . . . 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 6952 . . . . . . . . . . 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 453 . . . . . . . . . 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 453 . . . . . . . . . . 11  |-  ( ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G
)  /\  Y  e.  ( Clsd `  J )
)  /\  x  e.  ( Base `  G )
)  ->  [ ( 0g `  G ) ] ( G ~QG  Y )  =  ( 0g `  H ) )
3231eqeq1d 2446 . . . . . . . . . 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 246 . . . . . . . . 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 2961 . . . . . . . . . 10  |-  x  e. 
_V
35 fvex 5745 . . . . . . . . . 10  |-  ( 0g
`  G )  e. 
_V
3634, 35elec 6947 . . . . . . . . 9  |-  ( x  e.  [ ( 0g
`  G ) ] ( G ~QG  Y )  <->  ( 0g `  G ) ( G ~QG  Y ) x )
37 fvex 5745 . . . . . . . . . . 11  |-  ( 0g
`  H )  e. 
_V
3837elsnc2 3845 . . . . . . . . . 10  |-  ( [ x ] ( G ~QG  Y )  e.  { ( 0g `  H ) }  <->  [ x ] ( G ~QG  Y )  =  ( 0g `  H ) )
39 eqcom 2440 . . . . . . . . . 10  |-  ( [ x ] ( G ~QG  Y )  =  ( 0g
`  H )  <->  ( 0g `  H )  =  [
x ] ( G ~QG  Y ) )
4038, 39bitri 242 . . . . . . . . 9  |-  ( [ x ] ( G ~QG  Y )  e.  { ( 0g `  H ) }  <->  ( 0g `  H )  =  [
x ] ( G ~QG  Y ) )
4133, 36, 403bitr4g 281 . . . . . . . 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 3551 . . . . . . 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 2478 . . . . . 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 2482 . . . . 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 960 . . . . 5  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  Y  e.  (
Clsd `  J )
)
4644, 45eqeltrd 2512 . . . 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 18117 . . . . . 6  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  ( Base `  G
) ) )
49483ad2ant1 979 . . . . 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 2439 . . . . . 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 958 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  G  e.  TopGrp )
5450, 51, 15, 52, 53divslem 13773 . . . . 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 17750 . . . . 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 644 . . . 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 890 . . 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 13772 . . . . 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 17757 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  K  =  ( J qTop  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) ) ) )
6160fveq2d 5735 . . 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 2515 . 2  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  { ( 0g
`  H ) }  e.  ( Clsd `  K
) )
631divstgp 18156 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )
)  ->  H  e.  TopGrp )
64633adant3 978 . . 3  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  H  e.  TopGrp )
65 eqid 2438 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
6665, 59tgphaus 18151 . . 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 225 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958    i^i cin 3321    C_ wss 3322   {csn 3816   class class class wbr 4215    e. cmpt 4269   `'ccnv 4880   "cima 4884   -onto->wfo 5455   ` cfv 5457  (class class class)co 6084    Er wer 6905   [cec 6906   /.cqs 6907   Basecbs 13474   TopOpenctopn 13654   0gc0g 13728   qTop cqtop 13734    /.s cqus 13736   Grpcgrp 14690  SubGrpcsubg 14943  NrmSGrpcnsg 14944   ~QG cqg 14945  TopOnctopon 16964   Clsdccld 17085   Hauscha 17377   TopGrpctgp 18106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-ec 6910  df-qs 6914  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-rest 13655  df-topn 13656  df-topgen 13672  df-0g 13732  df-qtop 13738  df-imas 13739  df-divs 13740  df-mnd 14695  df-plusf 14696  df-grp 14817  df-minusg 14818  df-sbg 14819  df-subg 14946  df-nsg 14947  df-eqg 14948  df-oppg 15147  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-cn 17296  df-cnp 17297  df-t1 17383  df-haus 17384  df-tx 17599  df-hmeo 17792  df-tmd 18107  df-tgp 18108
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