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Theorem lmcau 21624
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.
StepHypRef Expression
1 lmcau.1 . . . . 5  |-  J  =  ( MetOpen `  D )
21methaus 20896 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  J  e.  Haus )
3 lmfun 19755 . . . 4  |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J )
)
4 funfvbrb 5985 . . . 4  |-  ( Fun  ( ~~> t `  J
)  ->  ( f  e.  dom  ( ~~> t `  J )  <->  f ( ~~> t `  J )
( ( ~~> t `  J ) `  f
) ) )
52, 3, 43syl 20 . . 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 ) )
71, 6lmmbr 21570 . . . . . . 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 ) ) ) )
87biimpa 484 . . . . . 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 ) ) )
98simp1d 1009 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  f ( ~~> t `  J ) ( ( ~~> t `  J ) `
 f ) )  ->  f  e.  ( X  ^pm  CC )
)
10 simprr 757 . . . . . . . 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 759 . . . . . . . . 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
) )
128simp2d 1010 . . . . . . . . . 10  |-  ( ( D  e.  ( *Met `  X )  /\  f ( ~~> t `  J ) ( ( ~~> t `  J ) `
 f ) )  ->  ( ( ~~> t `  J ) `  f
)  e.  X )
1312ad2antrr 725 . . . . . . . . 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 11235 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  x  e.  RR )
1514ad2antlr 726 . . . . . . . . 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 11104 . . . . . . . . . . . 12  |-  ( j  e.  ZZ  ->  j  e.  ( ZZ>= `  j )
)
1716ad2antrl 727 . . . . . . . . . . 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 5870 . . . . . . . . . . 11  |-  ( j  e.  ( ZZ>= `  j
)  ->  ( (
f  |`  ( ZZ>= `  j
) ) `  j
)  =  ( f `
 j ) )
1917, 18syl 16 . . . . . . . . . 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 ) )
2010, 17ffvelrnd 6017 . . . . . . . . . 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 ) ) )
2119, 20eqeltrrd 2532 . . . . . . . . 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 20781 . . . . . . . . 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 ) )
2311, 13, 15, 21, 22syl22anc 1230 . . . . . . . 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 ) )
2410, 23fssd 5730 . . . . . . 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 11253 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( x  /  2 )  e.  RR+ )
268simp3d 1011 . . . . . . . . . 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 6289 . . . . . . . . . . . . 13  |-  ( y  =  ( x  / 
2 )  ->  (
( ( ~~> t `  J ) `  f
) ( ball `  D
) y )  =  ( ( ( ~~> t `  J ) `  f
) ( ball `  D
) ( x  / 
2 ) ) )
2827feq3d 5709 . . . . . . . . . . . 12  |-  ( y  =  ( x  / 
2 )  ->  (
( f  |`  u
) : u --> ( ( ( ~~> t `  J
) `  f )
( ball `  D )
y )  <->  ( f  |`  u ) : u --> ( ( ( ~~> t `  J ) `  f
) ( ball `  D
) ( x  / 
2 ) ) ) )
2928rexbidv 2954 . . . . . . . . . . 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 ) ) ) )
3029rspcv 3192 . . . . . . . . . 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
) ) ) )
3125, 26, 30syl2im 38 . . . . . . . . 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
) ) ) )
3231impcom 430 . . . . . . . 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 11093 . . . . . . . . 9  |-  ZZ>= : ZZ --> ~P ZZ
34 ffn 5721 . . . . . . . . 9  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
35 reseq2 5258 . . . . . . . . . . 11  |-  ( u  =  ( ZZ>= `  j
)  ->  ( f  |`  u )  =  ( f  |`  ( ZZ>= `  j ) ) )
36 id 22 . . . . . . . . . . 11  |-  ( u  =  ( ZZ>= `  j
)  ->  u  =  ( ZZ>= `  j )
)
3735, 36feq12d 5710 . . . . . . . . . 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
) ) ) )
3837rexrn 6018 . . . . . . . . 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
) ) ) )
3933, 34, 38mp2b 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
) ) )
4032, 39sylib 196 . . . . . . 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
) ) )
4124, 40reximddv 2919 . . . . . 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 ) )
4241ralrimiva 2857 . . . . 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 21588 . . . . . 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 ) ) ) )
4443adantr 465 . . . . 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 ) ) ) )
459, 42, 44mpbir2and 922 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  f ( ~~> t `  J ) ( ( ~~> t `  J ) `
 f ) )  ->  f  e.  ( Cau `  D ) )
4645ex 434 . . 3  |-  ( D  e.  ( *Met `  X )  ->  (
f ( ~~> t `  J ) ( ( ~~> t `  J ) `
 f )  -> 
f  e.  ( Cau `  D ) ) )
475, 46sylbid 215 . 2  |-  ( D  e.  ( *Met `  X )  ->  (
f  e.  dom  ( ~~> t `  J )  ->  f  e.  ( Cau `  D ) ) )
4847ssrdv 3495 1  |-  ( D  e.  ( *Met `  X )  ->  dom  (
~~> t `  J ) 
C_  ( Cau `  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794    C_ wss 3461   ~Pcpw 3997   class class class wbr 4437   dom cdm 4989   ran crn 4990    |` cres 4991   Fun wfun 5572    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    ^pm cpm 7423   CCcc 9493   RRcr 9494    / cdiv 10212   2c2 10591   ZZcz 10870   ZZ>=cuz 11090   RR+crp 11229   *Metcxmt 18277   ballcbl 18279   MetOpencmopn 18282   ~~> tclm 19600   Hauscha 19682   Caucca 21565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-n0 10802  df-z 10871  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-icc 11545  df-topgen 14718  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-top 19272  df-bases 19274  df-topon 19275  df-lm 19603  df-haus 19689  df-cau 21568
This theorem is referenced by:  hlimcaui  26026
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