Theorem qustgphaus 21926

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 topological 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
qustgp.h	|- H = ( G /.s ( G ~QG Y ) )
qustgphaus.j	|- J = ( TopOpen ` G )
qustgphaus.k	|- K = ( TopOpen ` H )

Assertion

Ref	Expression
qustgphaus	|- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> K e. Haus )

Proof of Theorem qustgphaus

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qustgp.h	. . . . . . . 8 |- H = ( G /.s ( G ~QG Y ) )
2		eqid 2622	. . . . . . . 8 |- ( 0g ` G ) = ( 0g ` G )
3	1, 2	qus0 17652	. . . . . . 7 |- ( Y e. (NrmSGrp ` G ) -> [ ( 0g ` G ) ] ( G ~QG Y ) = ( 0g ` H ) )
4	3	3ad2ant2 1083	. . . . . 6 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> [ ( 0g ` G ) ] ( G ~QG Y ) = ( 0g ` H ) )
5		tgpgrp 21882	. . . . . . . . 9 |- ( G e. TopGrp -> G e. Grp )
6	5	3ad2ant1 1082	. . . . . . . 8 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> G e. Grp )
7		eqid 2622	. . . . . . . . 9 |- ( Base ` G ) = ( Base ` G )
8	7, 2	grpidcl 17450	. . . . . . . 8 |- ( G e. Grp -> ( 0g ` G ) e. ( Base ` G ) )
9	6, 8	syl 17	. . . . . . 7 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> ( 0g ` G ) e. ( Base ` G ) )
10		ovex 6678	. . . . . . . 8 |- ( G ~QG Y ) e. _V
11	10	ecelqsi 7803	. . . . . . 7 |- ( ( 0g ` G ) e. ( Base ` G ) -> [ ( 0g ` G ) ] ( G ~QG Y ) e. ( ( Base ` G ) /. ( G ~QG Y ) ) )
12	9, 11	syl 17	. . . . . 6 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> [ ( 0g ` G ) ] ( G ~QG Y ) e. ( ( Base ` G ) /. ( G ~QG Y ) ) )
13	4, 12	eqeltrrd 2702	. . . . 5 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> ( 0g ` H ) e. ( ( Base ` G ) /. ( G ~QG Y ) ) )
14	13	snssd 4340	. . . 4 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> { ( 0g ` H ) } C_ ( ( Base ` G ) /. ( G ~QG Y ) ) )
15		eqid 2622	. . . . . . 7 |- ( x e. ( Base ` G ) |-> [ x ] ( G ~QG Y ) ) = ( x e. ( Base ` G ) |-> [ x ] ( G ~QG Y ) )
16	15	mptpreima 5628	. . . . . 6 |- ( `' ( x e. ( Base ` G ) |-> [ x ] ( G ~QG Y ) ) " { ( 0g ` H ) } ) = { x e. ( Base ` G ) | [ x ] ( G ~QG Y ) e. { ( 0g ` H ) } }
17		nsgsubg 17626	. . . . . . . . . . 11 |- ( Y e. (NrmSGrp ` G ) -> Y e. (SubGrp ` G ) )
18	17	3ad2ant2 1083	. . . . . . . . . 10 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> Y e. (SubGrp ` G ) )
19		eqid 2622	. . . . . . . . . . 11 |- ( G ~QG Y ) = ( G ~QG Y )
20	7, 19, 2	eqgid 17646	. . . . . . . . . 10 |- ( Y e. (SubGrp ` G ) -> [ ( 0g ` G ) ] ( G ~QG Y ) = Y )
21	18, 20	syl 17	. . . . . . . . 9 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> [ ( 0g ` G ) ] ( G ~QG Y ) = Y )
22	7	subgss 17595	. . . . . . . . . 10 |- ( Y e. (SubGrp ` G ) -> Y C_ ( Base ` G ) )
23	18, 22	syl 17	. . . . . . . . 9 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> Y C_ ( Base ` G ) )
24	21, 23	eqsstrd 3639	. . . . . . . 8 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> [ ( 0g ` G ) ] ( G ~QG Y ) C_ ( Base ` G ) )
25		sseqin2 3817	. . . . . . . 8 |- ( [ ( 0g ` G ) ] ( G ~QG Y ) C_ ( Base ` G ) <-> ( ( Base ` G ) i^i [ ( 0g ` G ) ] ( G ~QG Y ) ) = [ ( 0g ` G ) ] ( G ~QG Y ) )
26	24, 25	sylib 208	. . . . . . 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 ) )
27	7, 19	eqger 17644	. . . . . . . . . . . . 13 |- ( Y e. (SubGrp ` G ) -> ( G ~QG Y ) Er ( Base ` G ) )
28	18, 27	syl 17	. . . . . . . . . . . 12 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> ( G ~QG Y ) Er ( Base ` G ) )
29	28, 9	erth 7791	. . . . . . . . . . 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 ) ) )
30	29	adantr 481	. . . . . . . . . 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 ) ) )
31	4	adantr 481	. . . . . . . . . . 11 |- ( ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) /\ x e. ( Base ` G ) ) -> [ ( 0g ` G ) ] ( G ~QG Y ) = ( 0g ` H ) )
32	31	eqeq1d 2624	. . . . . . . . . 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 ) ) )
33	30, 32	bitrd 268	. . . . . . . . 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 3203	. . . . . . . . . 10 |- x e. _V
35		fvex 6201	. . . . . . . . . 10 |- ( 0g ` G ) e. _V
36	34, 35	elec 7786	. . . . . . . . 9 |- ( x e. [ ( 0g ` G ) ] ( G ~QG Y ) <-> ( 0g ` G ) ( G ~QG Y ) x )
37		fvex 6201	. . . . . . . . . . 11 |- ( 0g ` H ) e. _V
38	37	elsn2 4211	. . . . . . . . . 10 |- ( [ x ] ( G ~QG Y ) e. { ( 0g ` H ) } <-> [ x ] ( G ~QG Y ) = ( 0g ` H ) )
39		eqcom 2629	. . . . . . . . . 10 |- ( [ x ] ( G ~QG Y ) = ( 0g ` H ) <-> ( 0g ` H ) = [ x ] ( G ~QG Y ) )
40	38, 39	bitri 264	. . . . . . . . 9 |- ( [ x ] ( G ~QG Y ) e. { ( 0g ` H ) } <-> ( 0g ` H ) = [ x ] ( G ~QG Y ) )
41	33, 36, 40	3bitr4g 303	. . . . . . . 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 ) } ) )
42	41	rabbi2dva 3821	. . . . . . 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 ) } } )
43	26, 42, 21	3eqtr3d 2664	. . . . . 6 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> { x e. ( Base ` G ) | [ x ] ( G ~QG Y ) e. { ( 0g ` H ) } } = Y )
44	16, 43	syl5eq 2668	. . . . 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 1063	. . . . 5 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> Y e. ( Clsd ` J ) )
46	44, 45	eqeltrd 2701	. . . 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		qustgphaus.j	. . . . . . 7 |- J = ( TopOpen ` G )
48	47, 7	tgptopon 21886	. . . . . 6 |- ( G e. TopGrp -> J e. (TopOn ` ( Base ` G ) ) )
49	48	3ad2ant1 1082	. . . . 5 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> J e. (TopOn ` ( Base ` G ) ) )
50	1	a1i 11	. . . . . 6 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> H = ( G /.s ( G ~QG Y ) ) )
51		eqidd 2623	. . . . . 6 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> ( Base ` G ) = ( Base ` G ) )
52	10	a1i 11	. . . . . 6 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> ( G ~QG Y ) e. _V )
53		simp1 1061	. . . . . 6 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> G e. TopGrp )
54	50, 51, 15, 52, 53	quslem 16203	. . . . 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 21516	. . . . 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 ) ) ) )
56	49, 54, 55	syl2anc 693	. . . 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 ) ) ) )
57	14, 46, 56	mpbir2and 957	. . 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 ) ) ) ) )
58	50, 51, 15, 52, 53	qusval 16202	. . . . 5 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> H = ( ( x e. ( Base ` G ) |-> [ x ] ( G ~QG Y ) ) "s G ) )
59		qustgphaus.k	. . . . 5 |- K = ( TopOpen ` H )
60	58, 51, 54, 53, 47, 59	imastopn 21523	. . . 4 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> K = ( J qTop ( x e. ( Base ` G ) |-> [ x ] ( G ~QG Y ) ) ) )
61	60	fveq2d 6195	. . 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 ) ) ) ) )
62	57, 61	eleqtrrd 2704	. 2 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> { ( 0g ` H ) } e. ( Clsd ` K ) )
63	1	qustgp 21925	. . . 4 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) ) -> H e. TopGrp )
64	63	3adant3 1081	. . 3 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> H e. TopGrp )
65		eqid 2622	. . . 4 |- ( 0g ` H ) = ( 0g ` H )
66	65, 59	tgphaus 21920	. . 3 |- ( H e. TopGrp -> ( K e. Haus <-> { ( 0g ` H ) } e. ( Clsd ` K ) ) )
67	64, 66	syl 17	. 2 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> ( K e. Haus <-> { ( 0g ` H ) } e. ( Clsd ` K ) ) )
68	62, 67	mpbird 247	1 |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> K e. Haus )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   "cima 5117   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   Er wer 7739   [cec 7740   /.cqs 7741   Basecbs 15857   TopOpenctopn 16082   0gc0g 16100   qTop cqtop 16163   /._s cqus 16165   Grpcgrp 17422  SubGrpcsubg 17588  NrmSGrpcnsg 17589   ~_QG cqg 17590  TopOnctopon 20715   Clsdccld 20820   Hauscha 21112   TopGrpctgp 21875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-rest 16083  df-topn 16084  df-0g 16102  df-topgen 16104  df-qtop 16167  df-imas 16168  df-qus 16169  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-nsg 17592  df-eqg 17593  df-oppg 17776  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-cn 21031  df-cnp 21032  df-t1 21118  df-haus 21119  df-tx 21365  df-hmeo 21558  df-tmd 21876  df-tgp 21877

This theorem is referenced by: (None)