Theorem cmetcusp1 23149

Description: If the uniform set of a complete metric space is the uniform structure generated by its metric, then it is a complete uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)

Hypotheses

Ref	Expression
cmetcusp1.x	|- X = ( Base ` F )
cmetcusp1.d	|- D = ( ( dist ` F ) |` ( X X. X ) )
cmetcusp1.u	|- U = (UnifSt ` F )

Assertion

Ref	Expression
cmetcusp1	|- ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) -> F e. CUnifSp )

Proof of Theorem cmetcusp1

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmsms 23145	. . . 4 |- ( F e. CMetSp -> F e. MetSp )
2		msxms 22259	. . . 4 |- ( F e. MetSp -> F e. *MetSp )
3	1, 2	syl 17	. . 3 |- ( F e. CMetSp -> F e. *MetSp )
4		cmetcusp1.x	. . . 4 |- X = ( Base ` F )
5		cmetcusp1.d	. . . 4 |- D = ( ( dist ` F ) |` ( X X. X ) )
6		cmetcusp1.u	. . . 4 |- U = (UnifSt ` F )
7	4, 5, 6	xmsusp 22374	. . 3 |- ( ( X =/= (/) /\ F e. *MetSp /\ U = (metUnif ` D ) ) -> F e. UnifSp )
8	3, 7	syl3an2 1360	. 2 |- ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) -> F e. UnifSp )
9		simpl3 1066	. . . . . . 7 |- ( ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) /\ c e. ( Fil ` X ) ) -> U = (metUnif ` D ) )
10	9	fveq2d 6195	. . . . . 6 |- ( ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) /\ c e. ( Fil ` X ) ) -> (CauFilu ` U ) = (CauFilu ` (metUnif ` D ) ) )
11	10	eleq2d 2687	. . . . 5 |- ( ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) /\ c e. ( Fil ` X ) ) -> ( c e. (CauFilu ` U ) <-> c e. (CauFilu ` (metUnif ` D ) ) ) )
12		simpl1 1064	. . . . . 6 |- ( ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) /\ c e. ( Fil ` X ) ) -> X =/= (/) )
13	4, 5	cmscmet 23143	. . . . . . . . 9 |- ( F e. CMetSp -> D e. ( CMet ` X ) )
14		cmetmet 23084	. . . . . . . . 9 |- ( D e. ( CMet ` X ) -> D e. ( Met ` X ) )
15		metxmet 22139	. . . . . . . . 9 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
16	13, 14, 15	3syl 18	. . . . . . . 8 |- ( F e. CMetSp -> D e. ( *Met ` X ) )
17	16	3ad2ant2 1083	. . . . . . 7 |- ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) -> D e. ( *Met ` X ) )
18	17	adantr 481	. . . . . 6 |- ( ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) /\ c e. ( Fil ` X ) ) -> D e. ( *Met ` X ) )
19		simpr 477	. . . . . 6 |- ( ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) /\ c e. ( Fil ` X ) ) -> c e. ( Fil ` X ) )
20		cfilucfil4 23118	. . . . . 6 |- ( ( X =/= (/) /\ D e. ( *Met ` X ) /\ c e. ( Fil ` X ) ) -> ( c e. (CauFilu ` (metUnif ` D ) ) <-> c e. (CauFil ` D ) ) )
21	12, 18, 19, 20	syl3anc 1326	. . . . 5 |- ( ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) /\ c e. ( Fil ` X ) ) -> ( c e. (CauFilu ` (metUnif ` D ) ) <-> c e. (CauFil ` D ) ) )
22	11, 21	bitrd 268	. . . 4 |- ( ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) /\ c e. ( Fil ` X ) ) -> ( c e. (CauFilu ` U ) <-> c e. (CauFil ` D ) ) )
23		eqid 2622	. . . . . . . . . . . 12 |- ( MetOpen ` D ) = ( MetOpen ` D )
24	23	iscmet 23082	. . . . . . . . . . 11 |- ( D e. ( CMet ` X ) <-> ( D e. ( Met ` X ) /\ A. c e. (CauFil ` D ) ( ( MetOpen ` D ) fLim c ) =/= (/) ) )
25	24	simprbi 480	. . . . . . . . . 10 |- ( D e. ( CMet ` X ) -> A. c e. (CauFil ` D ) ( ( MetOpen ` D ) fLim c ) =/= (/) )
26	13, 25	syl 17	. . . . . . . . 9 |- ( F e. CMetSp -> A. c e. (CauFil ` D ) ( ( MetOpen ` D ) fLim c ) =/= (/) )
27		eqid 2622	. . . . . . . . . . . . . 14 |- ( TopOpen ` F ) = ( TopOpen ` F )
28	27, 4, 5	xmstopn 22256	. . . . . . . . . . . . 13 |- ( F e. *MetSp -> ( TopOpen ` F ) = ( MetOpen ` D ) )
29	3, 28	syl 17	. . . . . . . . . . . 12 |- ( F e. CMetSp -> ( TopOpen ` F ) = ( MetOpen ` D ) )
30	29	oveq1d 6665	. . . . . . . . . . 11 |- ( F e. CMetSp -> ( ( TopOpen ` F ) fLim c ) = ( ( MetOpen ` D ) fLim c ) )
31	30	neeq1d 2853	. . . . . . . . . 10 |- ( F e. CMetSp -> ( ( ( TopOpen ` F ) fLim c ) =/= (/) <-> ( ( MetOpen ` D ) fLim c ) =/= (/) ) )
32	31	ralbidv 2986	. . . . . . . . 9 |- ( F e. CMetSp -> ( A. c e. (CauFil ` D ) ( ( TopOpen ` F ) fLim c ) =/= (/) <-> A. c e. (CauFil ` D ) ( ( MetOpen ` D ) fLim c ) =/= (/) ) )
33	26, 32	mpbird 247	. . . . . . . 8 |- ( F e. CMetSp -> A. c e. (CauFil ` D ) ( ( TopOpen ` F ) fLim c ) =/= (/) )
34	33	r19.21bi 2932	. . . . . . 7 |- ( ( F e. CMetSp /\ c e. (CauFil ` D ) ) -> ( ( TopOpen ` F ) fLim c ) =/= (/) )
35	34	ex 450	. . . . . 6 |- ( F e. CMetSp -> ( c e. (CauFil ` D ) -> ( ( TopOpen ` F ) fLim c ) =/= (/) ) )
36	35	3ad2ant2 1083	. . . . 5 |- ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) -> ( c e. (CauFil ` D ) -> ( ( TopOpen ` F ) fLim c ) =/= (/) ) )
37	36	adantr 481	. . . 4 |- ( ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) /\ c e. ( Fil ` X ) ) -> ( c e. (CauFil ` D ) -> ( ( TopOpen ` F ) fLim c ) =/= (/) ) )
38	22, 37	sylbid 230	. . 3 |- ( ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) /\ c e. ( Fil ` X ) ) -> ( c e. (CauFilu ` U ) -> ( ( TopOpen ` F ) fLim c ) =/= (/) ) )
39	38	ralrimiva 2966	. 2 |- ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) -> A. c e. ( Fil ` X ) ( c e. (CauFilu ` U ) -> ( ( TopOpen ` F ) fLim c ) =/= (/) ) )
40	4, 6, 27	iscusp2 22106	. 2 |- ( F e. CUnifSp <-> ( F e. UnifSp /\ A. c e. ( Fil ` X ) ( c e. (CauFilu ` U ) -> ( ( TopOpen ` F ) fLim c ) =/= (/) ) ) )
41	8, 39, 40	sylanbrc 698	1 |- ( ( X =/= (/) /\ F e. CMetSp /\ U = (metUnif ` D ) ) -> F e. CUnifSp )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   (/)c0 3915   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950   TopOpenctopn 16082   *Metcxmt 19731   Metcme 19732   MetOpencmopn 19736  metUnifcmetu 19737   Filcfil 21649   fLim cflim 21738  UnifStcuss 22057  UnifSpcusp 22058  CauFil_uccfilu 22090  CUnifSpccusp 22101   *MetSpcxme 22122   MetSpcmt 22123  CauFilccfil 23050   CMetcms 23052  CMetSpccms 23129

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-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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  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-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-fil 21650  df-ust 22004  df-utop 22035  df-usp 22061  df-cfilu 22091  df-cusp 22102  df-xms 22125  df-ms 22126  df-cfil 23053  df-cmet 23055  df-cms 23132

This theorem is referenced by:  cnfldcusp  23153  recusp  23170