Theorem tngtopn 22454

Description: The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

Ref	Expression
tngbas.t	|- T = ( G toNrmGrp N )
tngtset.2	|- D = ( dist ` T )
tngtset.3	|- J = ( MetOpen ` D )

Assertion

Ref	Expression
tngtopn	|- ( ( G e. V /\ N e. W ) -> J = ( TopOpen ` T ) )

Proof of Theorem tngtopn

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

Step	Hyp	Ref	Expression
1		tngbas.t	. . 3 |- T = ( G toNrmGrp N )
2		tngtset.2	. . 3 |- D = ( dist ` T )
3		tngtset.3	. . 3 |- J = ( MetOpen ` D )
4	1, 2, 3	tngtset 22453	. 2 |- ( ( G e. V /\ N e. W ) -> J = (TopSet ` T ) )
5		df-mopn 19742	. . . . . . . . 9 |- MetOpen = ( x e. U. ran *Met |-> ( topGen ` ran ( ball ` x ) ) )
6	5	dmmptss 5631	. . . . . . . 8 |- dom MetOpen C_ U. ran *Met
7	6	sseli 3599	. . . . . . 7 |- ( D e. dom MetOpen -> D e. U. ran *Met )
8		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( -g ` G ) = ( -g ` G )
9	1, 8	tngds 22452	. . . . . . . . . . . . . . . . 17 |- ( N e. W -> ( N o. ( -g ` G ) ) = ( dist ` T ) )
10	9, 2	syl6eqr 2674	. . . . . . . . . . . . . . . 16 |- ( N e. W -> ( N o. ( -g ` G ) ) = D )
11	10	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( G e. V /\ N e. W ) -> ( N o. ( -g ` G ) ) = D )
12	11	dmeqd 5326	. . . . . . . . . . . . . 14 |- ( ( G e. V /\ N e. W ) -> dom ( N o. ( -g ` G ) ) = dom D )
13		dmcoss 5385	. . . . . . . . . . . . . . 15 |- dom ( N o. ( -g ` G ) ) C_ dom ( -g ` G )
14		eqid 2622	. . . . . . . . . . . . . . . . 17 |- ( Base ` G ) = ( Base ` G )
15		eqid 2622	. . . . . . . . . . . . . . . . 17 |- ( +g ` G ) = ( +g ` G )
16		eqid 2622	. . . . . . . . . . . . . . . . 17 |- ( invg ` G ) = ( invg ` G )
17	14, 15, 16, 8	grpsubfval 17464	. . . . . . . . . . . . . . . 16 |- ( -g ` G ) = ( x e. ( Base ` G ) , y e. ( Base ` G ) |-> ( x ( +g ` G ) ( ( invg ` G ) ` y ) ) )
18		ovex 6678	. . . . . . . . . . . . . . . 16 |- ( x ( +g ` G ) ( ( invg ` G ) ` y ) ) e. _V
19	17, 18	dmmpt2 7240	. . . . . . . . . . . . . . 15 |- dom ( -g ` G ) = ( ( Base ` G ) X. ( Base ` G ) )
20	13, 19	sseqtri 3637	. . . . . . . . . . . . . 14 |- dom ( N o. ( -g ` G ) ) C_ ( ( Base ` G ) X. ( Base ` G ) )
21	12, 20	syl6eqssr 3656	. . . . . . . . . . . . 13 |- ( ( G e. V /\ N e. W ) -> dom D C_ ( ( Base ` G ) X. ( Base ` G ) ) )
22	21	adantr 481	. . . . . . . . . . . 12 |- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> dom D C_ ( ( Base ` G ) X. ( Base ` G ) ) )
23		dmss 5323	. . . . . . . . . . . 12 |- ( dom D C_ ( ( Base ` G ) X. ( Base ` G ) ) -> dom dom D C_ dom ( ( Base ` G ) X. ( Base ` G ) ) )
24	22, 23	syl 17	. . . . . . . . . . 11 |- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> dom dom D C_ dom ( ( Base ` G ) X. ( Base ` G ) ) )
25		dmxpid 5345	. . . . . . . . . . 11 |- dom ( ( Base ` G ) X. ( Base ` G ) ) = ( Base ` G )
26	24, 25	syl6sseq 3651	. . . . . . . . . 10 |- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> dom dom D C_ ( Base ` G ) )
27		simpr 477	. . . . . . . . . . . 12 |- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> D e. U. ran *Met )
28		xmetunirn 22142	. . . . . . . . . . . 12 |- ( D e. U. ran *Met <-> D e. ( *Met ` dom dom D ) )
29	27, 28	sylib 208	. . . . . . . . . . 11 |- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> D e. ( *Met ` dom dom D ) )
30		eqid 2622	. . . . . . . . . . . 12 |- ( MetOpen ` D ) = ( MetOpen ` D )
31	30	mopnuni 22246	. . . . . . . . . . 11 |- ( D e. ( *Met ` dom dom D ) -> dom dom D = U. ( MetOpen ` D ) )
32	29, 31	syl 17	. . . . . . . . . 10 |- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> dom dom D = U. ( MetOpen ` D ) )
33	1, 14	tngbas 22445	. . . . . . . . . . 11 |- ( N e. W -> ( Base ` G ) = ( Base ` T ) )
34	33	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> ( Base ` G ) = ( Base ` T ) )
35	26, 32, 34	3sstr3d 3647	. . . . . . . . 9 |- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> U. ( MetOpen ` D ) C_ ( Base ` T ) )
36		sspwuni 4611	. . . . . . . . 9 |- ( ( MetOpen ` D ) C_ ~P ( Base ` T ) <-> U. ( MetOpen ` D ) C_ ( Base ` T ) )
37	35, 36	sylibr 224	. . . . . . . 8 |- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> ( MetOpen ` D ) C_ ~P ( Base ` T ) )
38	37	ex 450	. . . . . . 7 |- ( ( G e. V /\ N e. W ) -> ( D e. U. ran *Met -> ( MetOpen ` D ) C_ ~P ( Base ` T ) ) )
39	7, 38	syl5 34	. . . . . 6 |- ( ( G e. V /\ N e. W ) -> ( D e. dom MetOpen -> ( MetOpen ` D ) C_ ~P ( Base ` T ) ) )
40		ndmfv 6218	. . . . . . 7 |- ( -. D e. dom MetOpen -> ( MetOpen ` D ) = (/) )
41		0ss 3972	. . . . . . 7 |- (/) C_ ~P ( Base ` T )
42	40, 41	syl6eqss 3655	. . . . . 6 |- ( -. D e. dom MetOpen -> ( MetOpen ` D ) C_ ~P ( Base ` T ) )
43	39, 42	pm2.61d1 171	. . . . 5 |- ( ( G e. V /\ N e. W ) -> ( MetOpen ` D ) C_ ~P ( Base ` T ) )
44	3, 43	syl5eqss 3649	. . . 4 |- ( ( G e. V /\ N e. W ) -> J C_ ~P ( Base ` T ) )
45	4, 44	eqsstr3d 3640	. . 3 |- ( ( G e. V /\ N e. W ) -> (TopSet ` T ) C_ ~P ( Base ` T ) )
46		eqid 2622	. . . 4 |- ( Base ` T ) = ( Base ` T )
47		eqid 2622	. . . 4 |- (TopSet ` T ) = (TopSet ` T )
48	46, 47	topnid 16096	. . 3 |- ( (TopSet ` T ) C_ ~P ( Base ` T ) -> (TopSet ` T ) = ( TopOpen ` T ) )
49	45, 48	syl 17	. 2 |- ( ( G e. V /\ N e. W ) -> (TopSet ` T ) = ( TopOpen ` T ) )
50	4, 49	eqtrd 2656	1 |- ( ( G e. V /\ N e. W ) -> J = ( TopOpen ` T ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   X. cxp 5112   dom cdm 5114   ran crn 5115   o. ccom 5118   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  TopSetcts 15947   distcds 15950   TopOpenctopn 16082   topGenctg 16098   invgcminusg 17423   -gcsg 17424   *Metcxmt 19731   ballcbl 19733   MetOpencmopn 19736   toNrmGrp ctng 22383

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  ax-pre-sup 10014

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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-div 10685  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-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-tset 15960  df-ds 15964  df-rest 16083  df-topn 16084  df-topgen 16104  df-sbg 17427  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-tng 22389

This theorem is referenced by:  tngngp2  22456  tchtopn  23025