Theorem flimcfil 23112

Description: Every convergent filter in a metric space is a Cauchy filter. (Contributed by Mario Carneiro, 15-Oct-2015.)

Hypothesis

Ref	Expression
lmcau.1	|- J = ( MetOpen ` D )

Assertion

Ref	Expression
flimcfil	|- ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) -> F e. (CauFil ` D ) )

Proof of Theorem flimcfil

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- U. J = U. J
2	1	flimfil 21773	. . . 4 |- ( A e. ( J fLim F ) -> F e. ( Fil ` U. J ) )
3	2	adantl 482	. . 3 |- ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) -> F e. ( Fil ` U. J ) )
4		lmcau.1	. . . . . 6 |- J = ( MetOpen ` D )
5	4	mopnuni 22246	. . . . 5 |- ( D e. ( *Met ` X ) -> X = U. J )
6	5	adantr 481	. . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) -> X = U. J )
7	6	fveq2d 6195	. . 3 |- ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) -> ( Fil ` X ) = ( Fil ` U. J ) )
8	3, 7	eleqtrrd 2704	. 2 |- ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) -> F e. ( Fil ` X ) )
9	1	flimelbas 21772	. . . . . 6 |- ( A e. ( J fLim F ) -> A e. U. J )
10	9	ad2antlr 763	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) /\ x e. RR+ ) -> A e. U. J )
11	5	ad2antrr 762	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) /\ x e. RR+ ) -> X = U. J )
12	10, 11	eleqtrrd 2704	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) /\ x e. RR+ ) -> A e. X )
13		simplr 792	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) /\ x e. RR+ ) -> A e. ( J fLim F ) )
14	4	mopntop 22245	. . . . . . 7 |- ( D e. ( *Met ` X ) -> J e. Top )
15	14	ad2antrr 762	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) /\ x e. RR+ ) -> J e. Top )
16		simpll 790	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) /\ x e. RR+ ) -> D e. ( *Met ` X ) )
17		rpxr 11840	. . . . . . . 8 |- ( x e. RR+ -> x e. RR* )
18	17	adantl 482	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) /\ x e. RR+ ) -> x e. RR* )
19	4	blopn 22305	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ x e. RR* ) -> ( A ( ball ` D ) x ) e. J )
20	16, 12, 18, 19	syl3anc 1326	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) /\ x e. RR+ ) -> ( A ( ball ` D ) x ) e. J )
21		simpr 477	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) /\ x e. RR+ ) -> x e. RR+ )
22		blcntr 22218	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ x e. RR+ ) -> A e. ( A ( ball ` D ) x ) )
23	16, 12, 21, 22	syl3anc 1326	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) /\ x e. RR+ ) -> A e. ( A ( ball ` D ) x ) )
24		opnneip 20923	. . . . . 6 |- ( ( J e. Top /\ ( A ( ball ` D ) x ) e. J /\ A e. ( A ( ball ` D ) x ) ) -> ( A ( ball ` D ) x ) e. ( ( nei ` J ) ` { A } ) )
25	15, 20, 23, 24	syl3anc 1326	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) /\ x e. RR+ ) -> ( A ( ball ` D ) x ) e. ( ( nei ` J ) ` { A } ) )
26		flimnei 21771	. . . . 5 |- ( ( A e. ( J fLim F ) /\ ( A ( ball ` D ) x ) e. ( ( nei ` J ) ` { A } ) ) -> ( A ( ball ` D ) x ) e. F )
27	13, 25, 26	syl2anc 693	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) /\ x e. RR+ ) -> ( A ( ball ` D ) x ) e. F )
28		oveq1 6657	. . . . . 6 |- ( y = A -> ( y ( ball ` D ) x ) = ( A ( ball ` D ) x ) )
29	28	eleq1d 2686	. . . . 5 |- ( y = A -> ( ( y ( ball ` D ) x ) e. F <-> ( A ( ball ` D ) x ) e. F ) )
30	29	rspcev 3309	. . . 4 |- ( ( A e. X /\ ( A ( ball ` D ) x ) e. F ) -> E. y e. X ( y ( ball ` D ) x ) e. F )
31	12, 27, 30	syl2anc 693	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) /\ x e. RR+ ) -> E. y e. X ( y ( ball ` D ) x ) e. F )
32	31	ralrimiva 2966	. 2 |- ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) -> A. x e. RR+ E. y e. X ( y ( ball ` D ) x ) e. F )
33		iscfil3 23071	. . 3 |- ( D e. ( *Met ` X ) -> ( F e. (CauFil ` D ) <-> ( F e. ( Fil ` X ) /\ A. x e. RR+ E. y e. X ( y ( ball ` D ) x ) e. F ) ) )
34	33	adantr 481	. 2 |- ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) -> ( F e. (CauFil ` D ) <-> ( F e. ( Fil ` X ) /\ A. x e. RR+ E. y e. X ( y ( ball ` D ) x ) e. F ) ) )
35	8, 32, 34	mpbir2and 957	1 |- ( ( D e. ( *Met ` X ) /\ A e. ( J fLim F ) ) -> F e. (CauFil ` D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {csn 4177   U.cuni 4436   ` cfv 5888  (class class class)co 6650   RR*cxr 10073   RR+crp 11832   *Metcxmt 19731   ballcbl 19733   MetOpencmopn 19736   Topctop 20698   neicnei 20901   Filcfil 21649   fLim cflim 21738  CauFilccfil 23050

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-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-bl 19741  df-mopn 19742  df-fbas 19743  df-top 20699  df-topon 20716  df-bases 20750  df-nei 20902  df-fil 21650  df-flim 21743  df-cfil 23053

This theorem is referenced by:  cmetss  23113  fmcncfil  29977