Theorem snclseqg 21919

Description: The coset of the closure of the identity is the closure of a point. (Contributed by Mario Carneiro, 22-Sep-2015.)

Hypotheses

Ref	Expression
snclseqg.x	|- X = ( Base ` G )
snclseqg.j	|- J = ( TopOpen ` G )
snclseqg.z	|- .0. = ( 0g ` G )
snclseqg.r	|- .~ = ( G ~QG S )
snclseqg.s	|- S = ( ( cls ` J ) ` { .0. } )

Assertion

Ref	Expression
snclseqg	|- ( ( G e. TopGrp /\ A e. X ) -> [ A ] .~ = ( ( cls ` J ) ` { A } ) )

Proof of Theorem snclseqg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snclseqg.s	. . . 4 |- S = ( ( cls ` J ) ` { .0. } )
2	1	imaeq2i 5464	. . 3 |- ( ( x e. X |-> ( A ( +g ` G ) x ) ) " S ) = ( ( x e. X |-> ( A ( +g ` G ) x ) ) " ( ( cls ` J ) ` { .0. } ) )
3		tgpgrp 21882	. . . . 5 |- ( G e. TopGrp -> G e. Grp )
4	3	adantr 481	. . . 4 |- ( ( G e. TopGrp /\ A e. X ) -> G e. Grp )
5		snclseqg.j	. . . . . . . . . 10 |- J = ( TopOpen ` G )
6		snclseqg.x	. . . . . . . . . 10 |- X = ( Base ` G )
7	5, 6	tgptopon 21886	. . . . . . . . 9 |- ( G e. TopGrp -> J e. (TopOn ` X ) )
8	7	adantr 481	. . . . . . . 8 |- ( ( G e. TopGrp /\ A e. X ) -> J e. (TopOn ` X ) )
9		topontop 20718	. . . . . . . 8 |- ( J e. (TopOn ` X ) -> J e. Top )
10	8, 9	syl 17	. . . . . . 7 |- ( ( G e. TopGrp /\ A e. X ) -> J e. Top )
11		snclseqg.z	. . . . . . . . . . 11 |- .0. = ( 0g ` G )
12	6, 11	grpidcl 17450	. . . . . . . . . 10 |- ( G e. Grp -> .0. e. X )
13	4, 12	syl 17	. . . . . . . . 9 |- ( ( G e. TopGrp /\ A e. X ) -> .0. e. X )
14	13	snssd 4340	. . . . . . . 8 |- ( ( G e. TopGrp /\ A e. X ) -> { .0. } C_ X )
15		toponuni 20719	. . . . . . . . 9 |- ( J e. (TopOn ` X ) -> X = U. J )
16	8, 15	syl 17	. . . . . . . 8 |- ( ( G e. TopGrp /\ A e. X ) -> X = U. J )
17	14, 16	sseqtrd 3641	. . . . . . 7 |- ( ( G e. TopGrp /\ A e. X ) -> { .0. } C_ U. J )
18		eqid 2622	. . . . . . . 8 |- U. J = U. J
19	18	clsss3 20863	. . . . . . 7 |- ( ( J e. Top /\ { .0. } C_ U. J ) -> ( ( cls ` J ) ` { .0. } ) C_ U. J )
20	10, 17, 19	syl2anc 693	. . . . . 6 |- ( ( G e. TopGrp /\ A e. X ) -> ( ( cls ` J ) ` { .0. } ) C_ U. J )
21	20, 16	sseqtr4d 3642	. . . . 5 |- ( ( G e. TopGrp /\ A e. X ) -> ( ( cls ` J ) ` { .0. } ) C_ X )
22	1, 21	syl5eqss 3649	. . . 4 |- ( ( G e. TopGrp /\ A e. X ) -> S C_ X )
23		simpr 477	. . . 4 |- ( ( G e. TopGrp /\ A e. X ) -> A e. X )
24		snclseqg.r	. . . . 5 |- .~ = ( G ~QG S )
25		eqid 2622	. . . . 5 |- ( +g ` G ) = ( +g ` G )
26	6, 24, 25	eqglact 17645	. . . 4 |- ( ( G e. Grp /\ S C_ X /\ A e. X ) -> [ A ] .~ = ( ( x e. X |-> ( A ( +g ` G ) x ) ) " S ) )
27	4, 22, 23, 26	syl3anc 1326	. . 3 |- ( ( G e. TopGrp /\ A e. X ) -> [ A ] .~ = ( ( x e. X |-> ( A ( +g ` G ) x ) ) " S ) )
28		eqid 2622	. . . . 5 |- ( x e. X |-> ( A ( +g ` G ) x ) ) = ( x e. X |-> ( A ( +g ` G ) x ) )
29	28, 6, 25, 5	tgplacthmeo 21907	. . . 4 |- ( ( G e. TopGrp /\ A e. X ) -> ( x e. X |-> ( A ( +g ` G ) x ) ) e. ( J Homeo J ) )
30	18	hmeocls 21571	. . . 4 |- ( ( ( x e. X |-> ( A ( +g ` G ) x ) ) e. ( J Homeo J ) /\ { .0. } C_ U. J ) -> ( ( cls ` J ) ` ( ( x e. X |-> ( A ( +g ` G ) x ) ) " { .0. } ) ) = ( ( x e. X |-> ( A ( +g ` G ) x ) ) " ( ( cls ` J ) ` { .0. } ) ) )
31	29, 17, 30	syl2anc 693	. . 3 |- ( ( G e. TopGrp /\ A e. X ) -> ( ( cls ` J ) ` ( ( x e. X |-> ( A ( +g ` G ) x ) ) " { .0. } ) ) = ( ( x e. X |-> ( A ( +g ` G ) x ) ) " ( ( cls ` J ) ` { .0. } ) ) )
32	2, 27, 31	3eqtr4a 2682	. 2 |- ( ( G e. TopGrp /\ A e. X ) -> [ A ] .~ = ( ( cls ` J ) ` ( ( x e. X |-> ( A ( +g ` G ) x ) ) " { .0. } ) ) )
33		df-ima 5127	. . . . 5 |- ( ( x e. X |-> ( A ( +g ` G ) x ) ) " { .0. } ) = ran ( ( x e. X |-> ( A ( +g ` G ) x ) ) |` { .0. } )
34	14	resmptd 5452	. . . . . 6 |- ( ( G e. TopGrp /\ A e. X ) -> ( ( x e. X |-> ( A ( +g ` G ) x ) ) |` { .0. } ) = ( x e. { .0. } |-> ( A ( +g ` G ) x ) ) )
35	34	rneqd 5353	. . . . 5 |- ( ( G e. TopGrp /\ A e. X ) -> ran ( ( x e. X |-> ( A ( +g ` G ) x ) ) |` { .0. } ) = ran ( x e. { .0. } |-> ( A ( +g ` G ) x ) ) )
36	33, 35	syl5eq 2668	. . . 4 |- ( ( G e. TopGrp /\ A e. X ) -> ( ( x e. X |-> ( A ( +g ` G ) x ) ) " { .0. } ) = ran ( x e. { .0. } |-> ( A ( +g ` G ) x ) ) )
37		fvex 6201	. . . . . . . . 9 |- ( 0g ` G ) e. _V
38	11, 37	eqeltri 2697	. . . . . . . 8 |- .0. e. _V
39		oveq2 6658	. . . . . . . . 9 |- ( x = .0. -> ( A ( +g ` G ) x ) = ( A ( +g ` G ) .0. ) )
40	39	eqeq2d 2632	. . . . . . . 8 |- ( x = .0. -> ( y = ( A ( +g ` G ) x ) <-> y = ( A ( +g ` G ) .0. ) ) )
41	38, 40	rexsn 4223	. . . . . . 7 |- ( E. x e. { .0. } y = ( A ( +g ` G ) x ) <-> y = ( A ( +g ` G ) .0. ) )
42	6, 25, 11	grprid 17453	. . . . . . . . 9 |- ( ( G e. Grp /\ A e. X ) -> ( A ( +g ` G ) .0. ) = A )
43	3, 42	sylan 488	. . . . . . . 8 |- ( ( G e. TopGrp /\ A e. X ) -> ( A ( +g ` G ) .0. ) = A )
44	43	eqeq2d 2632	. . . . . . 7 |- ( ( G e. TopGrp /\ A e. X ) -> ( y = ( A ( +g ` G ) .0. ) <-> y = A ) )
45	41, 44	syl5bb 272	. . . . . 6 |- ( ( G e. TopGrp /\ A e. X ) -> ( E. x e. { .0. } y = ( A ( +g ` G ) x ) <-> y = A ) )
46	45	abbidv 2741	. . . . 5 |- ( ( G e. TopGrp /\ A e. X ) -> { y | E. x e. { .0. } y = ( A ( +g ` G ) x ) } = { y | y = A } )
47		eqid 2622	. . . . . 6 |- ( x e. { .0. } |-> ( A ( +g ` G ) x ) ) = ( x e. { .0. } |-> ( A ( +g ` G ) x ) )
48	47	rnmpt 5371	. . . . 5 |- ran ( x e. { .0. } |-> ( A ( +g ` G ) x ) ) = { y | E. x e. { .0. } y = ( A ( +g ` G ) x ) }
49		df-sn 4178	. . . . 5 |- { A } = { y | y = A }
50	46, 48, 49	3eqtr4g 2681	. . . 4 |- ( ( G e. TopGrp /\ A e. X ) -> ran ( x e. { .0. } |-> ( A ( +g ` G ) x ) ) = { A } )
51	36, 50	eqtrd 2656	. . 3 |- ( ( G e. TopGrp /\ A e. X ) -> ( ( x e. X |-> ( A ( +g ` G ) x ) ) " { .0. } ) = { A } )
52	51	fveq2d 6195	. 2 |- ( ( G e. TopGrp /\ A e. X ) -> ( ( cls ` J ) ` ( ( x e. X |-> ( A ( +g ` G ) x ) ) " { .0. } ) ) = ( ( cls ` J ) ` { A } ) )
53	32, 52	eqtrd 2656	1 |- ( ( G e. TopGrp /\ A e. X ) -> [ A ] .~ = ( ( cls ` J ) ` { A } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   C_ wss 3574   {csn 4177   U.cuni 4436   |-> cmpt 4729   ran crn 5115   |` cres 5116   "cima 5117   ` cfv 5888  (class class class)co 6650   [cec 7740   Basecbs 15857   +g cplusg 15941   TopOpenctopn 16082   0gc0g 16100   Grpcgrp 17422   ~_QG cqg 17590   Topctop 20698  TopOnctopon 20715   clsccl 20822   Homeochmeo 21556   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-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-1st 7168  df-2nd 7169  df-ec 7744  df-map 7859  df-0g 16102  df-topgen 16104  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-eqg 17593  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-cls 20825  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-tmd 21876  df-tgp 21877

This theorem is referenced by:  tgptsmscls  21953