Theorem cmetcusp 23150

Description: The uniform space generated by a complete metric is a complete uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)

Assertion

Ref	Expression
cmetcusp	|- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> (toUnifSp ` (metUnif ` D ) ) e. CUnifSp )

Proof of Theorem cmetcusp

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

Step	Hyp	Ref	Expression
1		cmetmet 23084	. . . . 5 |- ( D e. ( CMet ` X ) -> D e. ( Met ` X ) )
2		metxmet 22139	. . . . 5 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
3		xmetpsmet 22153	. . . . 5 |- ( D e. ( *Met ` X ) -> D e. (PsMet ` X ) )
4	1, 2, 3	3syl 18	. . . 4 |- ( D e. ( CMet ` X ) -> D e. (PsMet ` X ) )
5	4	anim2i 593	. . 3 |- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> ( X =/= (/) /\ D e. (PsMet ` X ) ) )
6		metuust 22365	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (metUnif ` D ) e. (UnifOn ` X ) )
7		eqid 2622	. . . 4 |- (toUnifSp ` (metUnif ` D ) ) = (toUnifSp ` (metUnif ` D ) )
8	7	tususp 22076	. . 3 |- ( (metUnif ` D ) e. (UnifOn ` X ) -> (toUnifSp ` (metUnif ` D ) ) e. UnifSp )
9	5, 6, 8	3syl 18	. 2 |- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> (toUnifSp ` (metUnif ` D ) ) e. UnifSp )
10		simpll 790	. . . . 5 |- ( ( ( ( X =/= (/) /\ D e. ( CMet ` X ) ) /\ c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ) /\ c e. (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) ) -> ( X =/= (/) /\ D e. ( CMet ` X ) ) )
11	10	simprd 479	. . . . . 6 |- ( ( ( ( X =/= (/) /\ D e. ( CMet ` X ) ) /\ c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ) /\ c e. (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) ) -> D e. ( CMet ` X ) )
12	1, 2	syl 17	. . . . . . . 8 |- ( D e. ( CMet ` X ) -> D e. ( *Met ` X ) )
13	12	ad3antlr 767	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. ( CMet ` X ) ) /\ c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ) /\ c e. (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) ) -> D e. ( *Met ` X ) )
14	7	tusbas 22072	. . . . . . . . . . . 12 |- ( (metUnif ` D ) e. (UnifOn ` X ) -> X = ( Base ` (toUnifSp ` (metUnif ` D ) ) ) )
15	14	fveq2d 6195	. . . . . . . . . . 11 |- ( (metUnif ` D ) e. (UnifOn ` X ) -> ( Fil ` X ) = ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) )
16	15	eleq2d 2687	. . . . . . . . . 10 |- ( (metUnif ` D ) e. (UnifOn ` X ) -> ( c e. ( Fil ` X ) <-> c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ) )
17	5, 6, 16	3syl 18	. . . . . . . . 9 |- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> ( c e. ( Fil ` X ) <-> c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ) )
18	17	biimpar 502	. . . . . . . 8 |- ( ( ( X =/= (/) /\ D e. ( CMet ` X ) ) /\ c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ) -> c e. ( Fil ` X ) )
19	18	adantr 481	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. ( CMet ` X ) ) /\ c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ) /\ c e. (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) ) -> c e. ( Fil ` X ) )
20	7	tususs 22074	. . . . . . . . . . . . 13 |- ( (metUnif ` D ) e. (UnifOn ` X ) -> (metUnif ` D ) = (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) )
21	20	fveq2d 6195	. . . . . . . . . . . 12 |- ( (metUnif ` D ) e. (UnifOn ` X ) -> (CauFilu ` (metUnif ` D ) ) = (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) )
22	5, 6, 21	3syl 18	. . . . . . . . . . 11 |- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> (CauFilu ` (metUnif ` D ) ) = (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) )
23	22	eleq2d 2687	. . . . . . . . . 10 |- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> ( c e. (CauFilu ` (metUnif ` D ) ) <-> c e. (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) ) )
24	23	biimpar 502	. . . . . . . . 9 |- ( ( ( X =/= (/) /\ D e. ( CMet ` X ) ) /\ c e. (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) ) -> c e. (CauFilu ` (metUnif ` D ) ) )
25	24	adantlr 751	. . . . . . . 8 |- ( ( ( ( X =/= (/) /\ D e. ( CMet ` X ) ) /\ c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ) /\ c e. (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) ) -> c e. (CauFilu ` (metUnif ` D ) ) )
26		cfilucfil2 22366	. . . . . . . . . 10 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( c e. (CauFilu ` (metUnif ` D ) ) <-> ( c e. ( fBas ` X ) /\ A. x e. RR+ E. y e. c ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) )
27	5, 26	syl 17	. . . . . . . . 9 |- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> ( c e. (CauFilu ` (metUnif ` D ) ) <-> ( c e. ( fBas ` X ) /\ A. x e. RR+ E. y e. c ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) )
28	27	simplbda 654	. . . . . . . 8 |- ( ( ( X =/= (/) /\ D e. ( CMet ` X ) ) /\ c e. (CauFilu ` (metUnif ` D ) ) ) -> A. x e. RR+ E. y e. c ( D " ( y X. y ) ) C_ ( 0 [,) x ) )
29	10, 25, 28	syl2anc 693	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. ( CMet ` X ) ) /\ c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ) /\ c e. (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) ) -> A. x e. RR+ E. y e. c ( D " ( y X. y ) ) C_ ( 0 [,) x ) )
30		iscfil 23063	. . . . . . . 8 |- ( D e. ( *Met ` X ) -> ( c e. (CauFil ` D ) <-> ( c e. ( Fil ` X ) /\ A. x e. RR+ E. y e. c ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) )
31	30	biimpar 502	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ ( c e. ( Fil ` X ) /\ A. x e. RR+ E. y e. c ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) -> c e. (CauFil ` D ) )
32	13, 19, 29, 31	syl12anc 1324	. . . . . 6 |- ( ( ( ( X =/= (/) /\ D e. ( CMet ` X ) ) /\ c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ) /\ c e. (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) ) -> c e. (CauFil ` D ) )
33		eqid 2622	. . . . . . 7 |- ( MetOpen ` D ) = ( MetOpen ` D )
34	33	cmetcvg 23083	. . . . . 6 |- ( ( D e. ( CMet ` X ) /\ c e. (CauFil ` D ) ) -> ( ( MetOpen ` D ) fLim c ) =/= (/) )
35	11, 32, 34	syl2anc 693	. . . . 5 |- ( ( ( ( X =/= (/) /\ D e. ( CMet ` X ) ) /\ c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ) /\ c e. (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) ) -> ( ( MetOpen ` D ) fLim c ) =/= (/) )
36		eqid 2622	. . . . . . . . . . 11 |- (unifTop ` (metUnif ` D ) ) = (unifTop ` (metUnif ` D ) )
37	7, 36	tustopn 22075	. . . . . . . . . 10 |- ( (metUnif ` D ) e. (UnifOn ` X ) -> (unifTop ` (metUnif ` D ) ) = ( TopOpen ` (toUnifSp ` (metUnif ` D ) ) ) )
38	5, 6, 37	3syl 18	. . . . . . . . 9 |- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> (unifTop ` (metUnif ` D ) ) = ( TopOpen ` (toUnifSp ` (metUnif ` D ) ) ) )
39	12	anim2i 593	. . . . . . . . . 10 |- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> ( X =/= (/) /\ D e. ( *Met ` X ) ) )
40		xmetutop 22373	. . . . . . . . . 10 |- ( ( X =/= (/) /\ D e. ( *Met ` X ) ) -> (unifTop ` (metUnif ` D ) ) = ( MetOpen ` D ) )
41	39, 40	syl 17	. . . . . . . . 9 |- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> (unifTop ` (metUnif ` D ) ) = ( MetOpen ` D ) )
42	38, 41	eqtr3d 2658	. . . . . . . 8 |- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> ( TopOpen ` (toUnifSp ` (metUnif ` D ) ) ) = ( MetOpen ` D ) )
43	42	oveq1d 6665	. . . . . . 7 |- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> ( ( TopOpen ` (toUnifSp ` (metUnif ` D ) ) ) fLim c ) = ( ( MetOpen ` D ) fLim c ) )
44	43	neeq1d 2853	. . . . . 6 |- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> ( ( ( TopOpen ` (toUnifSp ` (metUnif ` D ) ) ) fLim c ) =/= (/) <-> ( ( MetOpen ` D ) fLim c ) =/= (/) ) )
45	44	biimpar 502	. . . . 5 |- ( ( ( X =/= (/) /\ D e. ( CMet ` X ) ) /\ ( ( MetOpen ` D ) fLim c ) =/= (/) ) -> ( ( TopOpen ` (toUnifSp ` (metUnif ` D ) ) ) fLim c ) =/= (/) )
46	10, 35, 45	syl2anc 693	. . . 4 |- ( ( ( ( X =/= (/) /\ D e. ( CMet ` X ) ) /\ c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ) /\ c e. (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) ) -> ( ( TopOpen ` (toUnifSp ` (metUnif ` D ) ) ) fLim c ) =/= (/) )
47	46	ex 450	. . 3 |- ( ( ( X =/= (/) /\ D e. ( CMet ` X ) ) /\ c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ) -> ( c e. (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) -> ( ( TopOpen ` (toUnifSp ` (metUnif ` D ) ) ) fLim c ) =/= (/) ) )
48	47	ralrimiva 2966	. 2 |- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> A. c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ( c e. (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) -> ( ( TopOpen ` (toUnifSp ` (metUnif ` D ) ) ) fLim c ) =/= (/) ) )
49		iscusp 22103	. 2 |- ( (toUnifSp ` (metUnif ` D ) ) e. CUnifSp <-> ( (toUnifSp ` (metUnif ` D ) ) e. UnifSp /\ A. c e. ( Fil ` ( Base ` (toUnifSp ` (metUnif ` D ) ) ) ) ( c e. (CauFilu ` (UnifSt ` (toUnifSp ` (metUnif ` D ) ) ) ) -> ( ( TopOpen ` (toUnifSp ` (metUnif ` D ) ) ) fLim c ) =/= (/) ) ) )
50	9, 48, 49	sylanbrc 698	1 |- ( ( X =/= (/) /\ D e. ( CMet ` X ) ) -> (toUnifSp ` (metUnif ` D ) ) e. CUnifSp )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   X. cxp 5112   "cima 5117   ` cfv 5888  (class class class)co 6650   0cc0 9936   RR+crp 11832   [,)cico 12177   Basecbs 15857   TopOpenctopn 16082  PsMetcpsmet 19730   *Metcxmt 19731   Metcme 19732   fBascfbas 19734   MetOpencmopn 19736  metUnifcmetu 19737   Filcfil 21649   fLim cflim 21738  UnifOncust 22003  unifTopcutop 22034  UnifStcuss 22057  UnifSpcusp 22058  toUnifSpctus 22059  CauFil_uccfilu 22090  CUnifSpccusp 22101  CauFilccfil 23050   CMetcms 23052

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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  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-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-ico 12181  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-tset 15960  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-metu 19745  df-fil 21650  df-ust 22004  df-utop 22035  df-uss 22060  df-usp 22061  df-tus 22062  df-cfilu 22091  df-cusp 22102  df-cfil 23053  df-cmet 23055

This theorem is referenced by: (None)