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Theorem h2hcau 22482
Description: The Cauchy sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
h2hc.1  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
h2hc.2  |-  U  e.  NrmCVec
h2hc.3  |-  ~H  =  ( BaseSet `  U )
h2hc.4  |-  D  =  ( IndMet `  U )
Assertion
Ref Expression
h2hcau  |-  Cauchy  =  ( ( Cau `  D
)  i^i  ( ~H  ^m  NN ) )

Proof of Theorem h2hcau
Dummy variables  f 
j  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rab 2714 . 2  |-  { f  e.  ( ~H  ^m  NN )  |  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( normh `  ( (
f `  j )  -h  ( f `  k
) ) )  < 
x }  =  {
f  |  ( f  e.  ( ~H  ^m  NN )  /\  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( normh `  ( (
f `  j )  -h  ( f `  k
) ) )  < 
x ) }
2 df-hcau 22476 . 2  |-  Cauchy  =  {
f  e.  ( ~H 
^m  NN )  | 
A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( normh `  ( ( f `  j )  -h  (
f `  k )
) )  <  x }
3 elin 3530 . . . 4  |-  ( f  e.  ( ( Cau `  D )  i^i  ( ~H  ^m  NN ) )  <-> 
( f  e.  ( Cau `  D )  /\  f  e.  ( ~H  ^m  NN ) ) )
4 ancom 438 . . . 4  |-  ( ( f  e.  ( Cau `  D )  /\  f  e.  ( ~H  ^m  NN ) )  <->  ( f  e.  ( ~H  ^m  NN )  /\  f  e.  ( Cau `  D ) ) )
5 h2hc.3 . . . . . . . 8  |-  ~H  =  ( BaseSet `  U )
65hlex 22400 . . . . . . 7  |-  ~H  e.  _V
7 nnex 10006 . . . . . . 7  |-  NN  e.  _V
86, 7elmap 7042 . . . . . 6  |-  ( f  e.  ( ~H  ^m  NN )  <->  f : NN --> ~H )
9 nnuz 10521 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
10 h2hc.2 . . . . . . . . 9  |-  U  e.  NrmCVec
11 h2hc.4 . . . . . . . . . 10  |-  D  =  ( IndMet `  U )
125, 11imsxmet 22184 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  D  e.  ( * Met `  ~H ) )
1310, 12mp1i 12 . . . . . . . 8  |-  ( f : NN --> ~H  ->  D  e.  ( * Met `  ~H ) )
14 1z 10311 . . . . . . . . 9  |-  1  e.  ZZ
1514a1i 11 . . . . . . . 8  |-  ( f : NN --> ~H  ->  1  e.  ZZ )
16 eqidd 2437 . . . . . . . 8  |-  ( ( f : NN --> ~H  /\  k  e.  NN )  ->  ( f `  k
)  =  ( f `
 k ) )
17 eqidd 2437 . . . . . . . 8  |-  ( ( f : NN --> ~H  /\  j  e.  NN )  ->  ( f `  j
)  =  ( f `
 j ) )
18 id 20 . . . . . . . 8  |-  ( f : NN --> ~H  ->  f : NN --> ~H )
199, 13, 15, 16, 17, 18iscauf 19233 . . . . . . 7  |-  ( f : NN --> ~H  ->  ( f  e.  ( Cau `  D )  <->  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( ( f `  j ) D ( f `  k ) )  <  x ) )
20 ffvelrn 5868 . . . . . . . . . . . . 13  |-  ( ( f : NN --> ~H  /\  j  e.  NN )  ->  ( f `  j
)  e.  ~H )
2120adantr 452 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ~H  /\  j  e.  NN )  /\  k  e.  (
ZZ>= `  j ) )  ->  ( f `  j )  e.  ~H )
229uztrn2 10503 . . . . . . . . . . . . . 14  |-  ( ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  NN )
23 ffvelrn 5868 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ~H  /\  k  e.  NN )  ->  ( f `  k
)  e.  ~H )
2422, 23sylan2 461 . . . . . . . . . . . . 13  |-  ( ( f : NN --> ~H  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( f `  k )  e.  ~H )
2524anassrs 630 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ~H  /\  j  e.  NN )  /\  k  e.  (
ZZ>= `  j ) )  ->  ( f `  k )  e.  ~H )
26 h2hc.1 . . . . . . . . . . . . 13  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
2726, 10, 5, 11h2hmetdval 22481 . . . . . . . . . . . 12  |-  ( ( ( f `  j
)  e.  ~H  /\  ( f `  k
)  e.  ~H )  ->  ( ( f `  j ) D ( f `  k ) )  =  ( normh `  ( ( f `  j )  -h  (
f `  k )
) ) )
2821, 25, 27syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( f : NN --> ~H  /\  j  e.  NN )  /\  k  e.  (
ZZ>= `  j ) )  ->  ( ( f `
 j ) D ( f `  k
) )  =  (
normh `  ( ( f `
 j )  -h  ( f `  k
) ) ) )
2928breq1d 4222 . . . . . . . . . 10  |-  ( ( ( f : NN --> ~H  /\  j  e.  NN )  /\  k  e.  (
ZZ>= `  j ) )  ->  ( ( ( f `  j ) D ( f `  k ) )  < 
x  <->  ( normh `  (
( f `  j
)  -h  ( f `
 k ) ) )  <  x ) )
3029ralbidva 2721 . . . . . . . . 9  |-  ( ( f : NN --> ~H  /\  j  e.  NN )  ->  ( A. k  e.  ( ZZ>= `  j )
( ( f `  j ) D ( f `  k ) )  <  x  <->  A. k  e.  ( ZZ>= `  j )
( normh `  ( (
f `  j )  -h  ( f `  k
) ) )  < 
x ) )
3130rexbidva 2722 . . . . . . . 8  |-  ( f : NN --> ~H  ->  ( E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( ( f `  j ) D ( f `  k ) )  < 
x  <->  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( normh `  ( ( f `  j )  -h  (
f `  k )
) )  <  x
) )
3231ralbidv 2725 . . . . . . 7  |-  ( f : NN --> ~H  ->  ( A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( ( f `  j ) D ( f `  k ) )  < 
x  <->  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( normh `  ( ( f `  j )  -h  (
f `  k )
) )  <  x
) )
3319, 32bitrd 245 . . . . . 6  |-  ( f : NN --> ~H  ->  ( f  e.  ( Cau `  D )  <->  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( normh `  ( (
f `  j )  -h  ( f `  k
) ) )  < 
x ) )
348, 33sylbi 188 . . . . 5  |-  ( f  e.  ( ~H  ^m  NN )  ->  ( f  e.  ( Cau `  D
)  <->  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( normh `  ( ( f `  j )  -h  (
f `  k )
) )  <  x
) )
3534pm5.32i 619 . . . 4  |-  ( ( f  e.  ( ~H 
^m  NN )  /\  f  e.  ( Cau `  D ) )  <->  ( f  e.  ( ~H  ^m  NN )  /\  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( normh `  ( ( f `  j )  -h  (
f `  k )
) )  <  x
) )
363, 4, 353bitri 263 . . 3  |-  ( f  e.  ( ( Cau `  D )  i^i  ( ~H  ^m  NN ) )  <-> 
( f  e.  ( ~H  ^m  NN )  /\  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( normh `  ( ( f `  j )  -h  (
f `  k )
) )  <  x
) )
3736abbi2i 2547 . 2  |-  ( ( Cau `  D )  i^i  ( ~H  ^m  NN ) )  =  {
f  |  ( f  e.  ( ~H  ^m  NN )  /\  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( normh `  ( (
f `  j )  -h  ( f `  k
) ) )  < 
x ) }
381, 2, 373eqtr4i 2466 1  |-  Cauchy  =  ( ( Cau `  D
)  i^i  ( ~H  ^m  NN ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705   E.wrex 2706   {crab 2709    i^i cin 3319   <.cop 3817   class class class wbr 4212   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   1c1 8991    < clt 9120   NNcn 10000   ZZcz 10282   ZZ>=cuz 10488   RR+crp 10612   * Metcxmt 16686   Caucca 19206   NrmCVeccnv 22063   BaseSetcba 22065   IndMetcims 22070   ~Hchil 22422    +h cva 22423    .h csm 22424   normhcno 22426    -h cmv 22428   Cauchyccau 22429
This theorem is referenced by:  axhcompl-zf  22501  hhcau  22700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-xneg 10710  df-xadd 10711  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-cau 19209  df-grpo 21779  df-gid 21780  df-ginv 21781  df-gdiv 21782  df-ablo 21870  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-vs 22078  df-nmcv 22079  df-ims 22080  df-hvsub 22474  df-hcau 22476
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