MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  divstgphaus Unicode version

Theorem divstgphaus 17821
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 2296 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
31, 2divs0 14691 . . . . . . 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 17777 . . . . . . . . 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 2296 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
87, 2grpidcl 14526 . . . . . . . 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 5899 . . . . . . . 8  |-  ( G ~QG  Y )  e.  _V
1110ecelqsi 6731 . . . . . . 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 2371 . . . . 5  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  ( 0g `  H )  e.  ( ( Base `  G
) /. ( G ~QG  Y ) ) )
1413snssd 3776 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  { ( 0g
`  H ) } 
C_  ( ( Base `  G ) /. ( G ~QG  Y ) ) )
15 eqid 2296 . . . . . . 7  |-  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) )  =  ( x  e.  (
Base `  G )  |->  [ x ] ( G ~QG  Y ) )
1615mptpreima 5182 . . . . . 6  |-  ( `' ( x  e.  (
Base `  G )  |->  [ x ] ( G ~QG  Y ) ) " { ( 0g `  H ) } )  =  { x  e.  ( Base `  G
)  |  [ x ] ( G ~QG  Y )  e.  { ( 0g
`  H ) } }
17 nsgsubg 14665 . . . . . . . . . . 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 2296 . . . . . . . . . . 11  |-  ( G ~QG  Y )  =  ( G ~QG  Y )
207, 19, 2eqgid 14685 . . . . . . . . . 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 14638 . . . . . . . . . 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 3225 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  [ ( 0g
`  G ) ] ( G ~QG  Y )  C_  ( Base `  G ) )
25 dfss1 3386 . . . . . . . 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 14683 . . . . . . . . . . . . 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 6720 . . . . . . . . . . 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 2304 . . . . . . . . . 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 2804 . . . . . . . . . 10  |-  x  e. 
_V
35 fvex 5555 . . . . . . . . . 10  |-  ( 0g
`  G )  e. 
_V
3634, 35elec 6715 . . . . . . . . 9  |-  ( x  e.  [ ( 0g
`  G ) ] ( G ~QG  Y )  <->  ( 0g `  G ) ( G ~QG  Y ) x )
37 fvex 5555 . . . . . . . . . . 11  |-  ( 0g
`  H )  e. 
_V
3837elsnc2 3682 . . . . . . . . . 10  |-  ( [ x ] ( G ~QG  Y )  e.  { ( 0g `  H ) }  <->  [ x ] ( G ~QG  Y )  =  ( 0g `  H ) )
39 eqcom 2298 . . . . . . . . . 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 3390 . . . . . . 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 2336 . . . . . 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 2340 . . . . 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 2370 . . . 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 17781 . . . . . 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 2297 . . . . . 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 13461 . . . . 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 17420 . . . . 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 13460 . . . . 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 17427 . . . 4  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  K  =  ( J qTop  ( x  e.  ( Base `  G
)  |->  [ x ]
( G ~QG  Y ) ) ) )
6160fveq2d 5545 . . 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 2373 . 2  |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) )  ->  { ( 0g
`  H ) }  e.  ( Clsd `  K
) )
631divstgp 17820 . . . 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 2296 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
6665, 59tgphaus 17815 . . 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 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    i^i cin 3164    C_ wss 3165   {csn 3653   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   "cima 4708   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874    Er wer 6673   [cec 6674   /.cqs 6675   Basecbs 13164   TopOpenctopn 13342   0gc0g 13416   qTop cqtop 13422    /.s cqus 13424   Grpcgrp 14378  SubGrpcsubg 14631  NrmSGrpcnsg 14632   ~QG cqg 14633  TopOnctopon 16648   Clsdccld 16769   Hauscha 17052   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-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  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-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-0g 13420  df-qtop 13426  df-imas 13427  df-divs 13428  df-mnd 14383  df-plusf 14384  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-nsg 14635  df-eqg 14636  df-oppg 14835  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-cn 16973  df-cnp 16974  df-t1 17058  df-haus 17059  df-tx 17273  df-hmeo 17462  df-tmd 17771  df-tgp 17772
  Copyright terms: Public domain W3C validator