Theorem lmcau 23111

Description: Every convergent sequence in a metric space is a Cauchy sequence. Theorem 1.4-5 of [Kreyszig] p. 28. (Contributed by NM, 29-Jan-2008.) (Proof shortened by Mario Carneiro, 5-May-2014.)

Hypothesis

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

Assertion

Ref	Expression
lmcau	|- ( D e. ( *Met ` X ) -> dom ( ~~> t ` J ) C_ ( Cau ` D ) )

Proof of Theorem lmcau

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

Step	Hyp	Ref	Expression
1		lmcau.1	. . . . 5 |- J = ( MetOpen ` D )
2	1	methaus 22325	. . . 4 |- ( D e. ( *Met ` X ) -> J e. Haus )
3		lmfun 21185	. . . 4 |- ( J e. Haus -> Fun ( ~~> t ` J ) )
4		funfvbrb 6330	. . . 4 |- ( Fun ( ~~> t ` J ) -> ( f e. dom ( ~~> t ` J ) <-> f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) )
5	2, 3, 4	3syl 18	. . 3 |- ( D e. ( *Met ` X ) -> ( f e. dom ( ~~> t ` J ) <-> f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) )
6		id 22	. . . . . . . 8 |- ( D e. ( *Met ` X ) -> D e. ( *Met ` X ) )
7	1, 6	lmmbr 23056	. . . . . . 7 |- ( D e. ( *Met ` X ) -> ( f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) <-> ( f e. ( X ^pm CC ) /\ ( ( ~~> t ` J ) ` f ) e. X /\ A. y e. RR+ E. u e. ran ZZ>= ( f |` u ) : u --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) y ) ) ) )
8	7	biimpa 501	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) -> ( f e. ( X ^pm CC ) /\ ( ( ~~> t ` J ) ` f ) e. X /\ A. y e. RR+ E. u e. ran ZZ>= ( f |` u ) : u --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) y ) ) )
9	8	simp1d 1073	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) -> f e. ( X ^pm CC ) )
10		simprr 796	. . . . . . . 8 |- ( ( ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) /\ x e. RR+ ) /\ ( j e. ZZ /\ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) ) -> ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) )
11		simplll 798	. . . . . . . . 9 |- ( ( ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) /\ x e. RR+ ) /\ ( j e. ZZ /\ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) ) -> D e. ( *Met ` X ) )
12	8	simp2d 1074	. . . . . . . . . 10 |- ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) -> ( ( ~~> t ` J ) ` f ) e. X )
13	12	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) /\ x e. RR+ ) /\ ( j e. ZZ /\ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) ) -> ( ( ~~> t ` J ) ` f ) e. X )
14		rpre 11839	. . . . . . . . . 10 |- ( x e. RR+ -> x e. RR )
15	14	ad2antlr 763	. . . . . . . . 9 |- ( ( ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) /\ x e. RR+ ) /\ ( j e. ZZ /\ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) ) -> x e. RR )
16		uzid 11702	. . . . . . . . . . . 12 |- ( j e. ZZ -> j e. ( ZZ>= ` j ) )
17	16	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) /\ x e. RR+ ) /\ ( j e. ZZ /\ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) ) -> j e. ( ZZ>= ` j ) )
18		fvres 6207	. . . . . . . . . . 11 |- ( j e. ( ZZ>= ` j ) -> ( ( f |` ( ZZ>= ` j ) ) ` j ) = ( f ` j ) )
19	17, 18	syl 17	. . . . . . . . . 10 |- ( ( ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) /\ x e. RR+ ) /\ ( j e. ZZ /\ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) ) -> ( ( f |` ( ZZ>= ` j ) ) ` j ) = ( f ` j ) )
20	10, 17	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) /\ x e. RR+ ) /\ ( j e. ZZ /\ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) ) -> ( ( f |` ( ZZ>= ` j ) ) ` j ) e. ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) )
21	19, 20	eqeltrrd 2702	. . . . . . . . 9 |- ( ( ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) /\ x e. RR+ ) /\ ( j e. ZZ /\ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) ) -> ( f ` j ) e. ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) )
22		blhalf 22210	. . . . . . . . 9 |- ( ( ( D e. ( *Met ` X ) /\ ( ( ~~> t ` J ) ` f ) e. X ) /\ ( x e. RR /\ ( f ` j ) e. ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) ) -> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) C_ ( ( f ` j ) ( ball ` D ) x ) )
23	11, 13, 15, 21, 22	syl22anc 1327	. . . . . . . 8 |- ( ( ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) /\ x e. RR+ ) /\ ( j e. ZZ /\ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) ) -> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) C_ ( ( f ` j ) ( ball ` D ) x ) )
24	10, 23	fssd 6057	. . . . . . 7 |- ( ( ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) /\ x e. RR+ ) /\ ( j e. ZZ /\ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) ) -> ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( f ` j ) ( ball ` D ) x ) )
25		rphalfcl 11858	. . . . . . . . . 10 |- ( x e. RR+ -> ( x / 2 ) e. RR+ )
26	8	simp3d 1075	. . . . . . . . . 10 |- ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) -> A. y e. RR+ E. u e. ran ZZ>= ( f |` u ) : u --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) y ) )
27		oveq2 6658	. . . . . . . . . . . . 13 |- ( y = ( x / 2 ) -> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) y ) = ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) )
28	27	feq3d 6032	. . . . . . . . . . . 12 |- ( y = ( x / 2 ) -> ( ( f |` u ) : u --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) y ) <-> ( f |` u ) : u --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) )
29	28	rexbidv 3052	. . . . . . . . . . 11 |- ( y = ( x / 2 ) -> ( E. u e. ran ZZ>= ( f |` u ) : u --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) y ) <-> E. u e. ran ZZ>= ( f |` u ) : u --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) )
30	29	rspcv 3305	. . . . . . . . . 10 |- ( ( x / 2 ) e. RR+ -> ( A. y e. RR+ E. u e. ran ZZ>= ( f |` u ) : u --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) y ) -> E. u e. ran ZZ>= ( f |` u ) : u --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) )
31	25, 26, 30	syl2im 40	. . . . . . . . 9 |- ( x e. RR+ -> ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) -> E. u e. ran ZZ>= ( f |` u ) : u --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) )
32	31	impcom 446	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) /\ x e. RR+ ) -> E. u e. ran ZZ>= ( f |` u ) : u --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) )
33		uzf 11690	. . . . . . . . 9 |- ZZ>= : ZZ --> ~P ZZ
34		ffn 6045	. . . . . . . . 9 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
35		reseq2 5391	. . . . . . . . . . 11 |- ( u = ( ZZ>= ` j ) -> ( f |` u ) = ( f |` ( ZZ>= ` j ) ) )
36		id 22	. . . . . . . . . . 11 |- ( u = ( ZZ>= ` j ) -> u = ( ZZ>= ` j ) )
37	35, 36	feq12d 6033	. . . . . . . . . 10 |- ( u = ( ZZ>= ` j ) -> ( ( f |` u ) : u --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) <-> ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) )
38	37	rexrn 6361	. . . . . . . . 9 |- ( ZZ>= Fn ZZ -> ( E. u e. ran ZZ>= ( f |` u ) : u --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) <-> E. j e. ZZ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) ) )
39	33, 34, 38	mp2b 10	. . . . . . . 8 |- ( E. u e. ran ZZ>= ( f |` u ) : u --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) <-> E. j e. ZZ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) )
40	32, 39	sylib 208	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) /\ x e. RR+ ) -> E. j e. ZZ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( ( ~~> t ` J ) ` f ) ( ball ` D ) ( x / 2 ) ) )
41	24, 40	reximddv 3018	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) /\ x e. RR+ ) -> E. j e. ZZ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( f ` j ) ( ball ` D ) x ) )
42	41	ralrimiva 2966	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) -> A. x e. RR+ E. j e. ZZ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( f ` j ) ( ball ` D ) x ) )
43		iscau 23074	. . . . . 6 |- ( D e. ( *Met ` X ) -> ( f e. ( Cau ` D ) <-> ( f e. ( X ^pm CC ) /\ A. x e. RR+ E. j e. ZZ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( f ` j ) ( ball ` D ) x ) ) ) )
44	43	adantr 481	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) -> ( f e. ( Cau ` D ) <-> ( f e. ( X ^pm CC ) /\ A. x e. RR+ E. j e. ZZ ( f |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> ( ( f ` j ) ( ball ` D ) x ) ) ) )
45	9, 42, 44	mpbir2and 957	. . . 4 |- ( ( D e. ( *Met ` X ) /\ f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) ) -> f e. ( Cau ` D ) )
46	45	ex 450	. . 3 |- ( D e. ( *Met ` X ) -> ( f ( ~~> t ` J ) ( ( ~~> t ` J ) ` f ) -> f e. ( Cau ` D ) ) )
47	5, 46	sylbid 230	. 2 |- ( D e. ( *Met ` X ) -> ( f e. dom ( ~~> t ` J ) -> f e. ( Cau ` D ) ) )
48	47	ssrdv 3609	1 |- ( D e. ( *Met ` X ) -> dom ( ~~> t ` J ) C_ ( Cau ` D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   RRcr 9935   / cdiv 10684   2c2 11070   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   *Metcxmt 19731   ballcbl 19733   MetOpencmopn 19736   ~~> tclm 21030   Hauscha 21112   Caucca 23051

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-pm 7860  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-icc 12182  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-lm 21033  df-haus 21119  df-cau 23054

This theorem is referenced by:  hlimcaui  28093