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Theorem lmcau 22269
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 21522 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  J  e.  Haus )
3 lmfun 20384 . . . 4  |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J )
)
4 funfvbrb 6007 . . . 4  |-  ( Fun  ( ~~> t `  J
)  ->  ( f  e.  dom  ( ~~> t `  J )  <->  f ( ~~> t `  J )
( ( ~~> t `  J ) `  f
) ) )
52, 3, 43syl 18 . . 3  |-  ( D  e.  ( *Met `  X )  ->  (
f  e.  dom  ( ~~> t `  J )  <->  f ( ~~> t `  J
) ( ( ~~> t `  J ) `  f
) ) )
6 id 23 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  D  e.  ( *Met `  X ) )
71, 6lmmbr 22215 . . . . . . 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 486 . . . . . 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 1017 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  f ( ~~> t `  J ) ( ( ~~> t `  J ) `
 f ) )  ->  f  e.  ( X  ^pm  CC )
)
10 simprr 764 . . . . . . . 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 766 . . . . . . . . 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 1018 . . . . . . . . . 10  |-  ( ( D  e.  ( *Met `  X )  /\  f ( ~~> t `  J ) ( ( ~~> t `  J ) `
 f ) )  ->  ( ( ~~> t `  J ) `  f
)  e.  X )
1312ad2antrr 730 . . . . . . . . 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 11309 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  x  e.  RR )
1514ad2antlr 731 . . . . . . . . 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 11174 . . . . . . . . . . . 12  |-  ( j  e.  ZZ  ->  j  e.  ( ZZ>= `  j )
)
1716ad2antrl 732 . . . . . . . . . . 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 5892 . . . . . . . . . . 11  |-  ( j  e.  ( ZZ>= `  j
)  ->  ( (
f  |`  ( ZZ>= `  j
) ) `  j
)  =  ( f `
 j ) )
1917, 18syl 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 ) )
2010, 17ffvelrnd 6035 . . . . . . . . . 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 2511 . . . . . . . . 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 21407 . . . . . . . . 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 1265 . . . . . . . 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 5752 . . . . . . 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 11328 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( x  /  2 )  e.  RR+ )
268simp3d 1019 . . . . . . . . . 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 6310 . . . . . . . . . . . . 13  |-  ( y  =  ( x  / 
2 )  ->  (
( ( ~~> t `  J ) `  f
) ( ball `  D
) y )  =  ( ( ( ~~> t `  J ) `  f
) ( ball `  D
) ( x  / 
2 ) ) )
2827feq3d 5731 . . . . . . . . . . . 12  |-  ( y  =  ( x  / 
2 )  ->  (
( f  |`  u
) : u --> ( ( ( ~~> t `  J
) `  f )
( ball `  D )
y )  <->  ( f  |`  u ) : u --> ( ( ( ~~> t `  J ) `  f
) ( ball `  D
) ( x  / 
2 ) ) ) )
2928rexbidv 2939 . . . . . . . . . . 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 3178 . . . . . . . . . 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 39 . . . . . . . . 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 431 . . . . . . . 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 11163 . . . . . . . . 9  |-  ZZ>= : ZZ --> ~P ZZ
34 ffn 5743 . . . . . . . . 9  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
35 reseq2 5116 . . . . . . . . . . 11  |-  ( u  =  ( ZZ>= `  j
)  ->  ( f  |`  u )  =  ( f  |`  ( ZZ>= `  j ) ) )
36 id 23 . . . . . . . . . . 11  |-  ( u  =  ( ZZ>= `  j
)  ->  u  =  ( ZZ>= `  j )
)
3735, 36feq12d 5732 . . . . . . . . . 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 6036 . . . . . . . . 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 199 . . . . . . 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 2901 . . . . . 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 2839 . . . . 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 22233 . . . . . 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 466 . . . . 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 930 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  f ( ~~> t `  J ) ( ( ~~> t `  J ) `
 f ) )  ->  f  e.  ( Cau `  D ) )
4645ex 435 . . 3  |-  ( D  e.  ( *Met `  X )  ->  (
f ( ~~> t `  J ) ( ( ~~> t `  J ) `
 f )  -> 
f  e.  ( Cau `  D ) ) )
475, 46sylbid 218 . 2  |-  ( D  e.  ( *Met `  X )  ->  (
f  e.  dom  ( ~~> t `  J )  ->  f  e.  ( Cau `  D ) ) )
4847ssrdv 3470 1  |-  ( D  e.  ( *Met `  X )  ->  dom  (
~~> t `  J ) 
C_  ( Cau `  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775   E.wrex 2776    C_ wss 3436   ~Pcpw 3979   class class class wbr 4420   dom cdm 4850   ran crn 4851    |` cres 4852   Fun wfun 5592    Fn wfn 5593   -->wf 5594   ` cfv 5598  (class class class)co 6302    ^pm cpm 7478   CCcc 9538   RRcr 9539    / cdiv 10270   2c2 10660   ZZcz 10938   ZZ>=cuz 11160   RR+crp 11303   *Metcxmt 18943   ballcbl 18945   MetOpencmopn 18948   ~~> tclm 20229   Hauscha 20311   Caucca 22210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-sup 7959  df-inf 7960  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-n0 10871  df-z 10939  df-uz 11161  df-q 11266  df-rp 11304  df-xneg 11410  df-xadd 11411  df-xmul 11412  df-icc 11643  df-topgen 15330  df-psmet 18950  df-xmet 18951  df-met 18952  df-bl 18953  df-mopn 18954  df-top 19908  df-bases 19909  df-topon 19910  df-lm 20232  df-haus 20318  df-cau 22213
This theorem is referenced by:  hlimcaui  26875
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