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Theorem h2hcau 21559
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 2552 . 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 21553 . 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 3358 . . . 4  |-  ( f  e.  ( ( Cau `  D )  i^i  ( ~H  ^m  NN ) )  <-> 
( f  e.  ( Cau `  D )  /\  f  e.  ( ~H  ^m  NN ) ) )
4 ancom 437 . . . 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 21477 . . . . . . 7  |-  ~H  e.  _V
7 nnex 9752 . . . . . . 7  |-  NN  e.  _V
86, 7elmap 6796 . . . . . 6  |-  ( f  e.  ( ~H  ^m  NN )  <->  f : NN --> ~H )
9 nnuz 10263 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
10 h2hc.2 . . . . . . . . 9  |-  U  e.  NrmCVec
11 h2hc.4 . . . . . . . . . 10  |-  D  =  ( IndMet `  U )
125, 11imsxmet 21261 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  D  e.  ( * Met `  ~H ) )
1310, 12mp1i 11 . . . . . . . 8  |-  ( f : NN --> ~H  ->  D  e.  ( * Met `  ~H ) )
14 1z 10053 . . . . . . . . 9  |-  1  e.  ZZ
1514a1i 10 . . . . . . . 8  |-  ( f : NN --> ~H  ->  1  e.  ZZ )
16 eqidd 2284 . . . . . . . 8  |-  ( ( f : NN --> ~H  /\  k  e.  NN )  ->  ( f `  k
)  =  ( f `
 k ) )
17 eqidd 2284 . . . . . . . 8  |-  ( ( f : NN --> ~H  /\  j  e.  NN )  ->  ( f `  j
)  =  ( f `
 j ) )
18 id 19 . . . . . . . 8  |-  ( f : NN --> ~H  ->  f : NN --> ~H )
199, 13, 15, 16, 17, 18iscauf 18706 . . . . . . 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 5663 . . . . . . . . . . . . 13  |-  ( ( f : NN --> ~H  /\  j  e.  NN )  ->  ( f `  j
)  e.  ~H )
2120adantr 451 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ~H  /\  j  e.  NN )  /\  k  e.  (
ZZ>= `  j ) )  ->  ( f `  j )  e.  ~H )
229uztrn2 10245 . . . . . . . . . . . . . 14  |-  ( ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  NN )
23 ffvelrn 5663 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ~H  /\  k  e.  NN )  ->  ( f `  k
)  e.  ~H )
2422, 23sylan2 460 . . . . . . . . . . . . 13  |-  ( ( f : NN --> ~H  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( f `  k )  e.  ~H )
2524anassrs 629 . . . . . . . . . . . 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 21558 . . . . . . . . . . . 12  |-  ( ( ( f `  j
)  e.  ~H  /\  ( f `  k
)  e.  ~H )  ->  ( ( f `  j ) D ( f `  k ) )  =  ( normh `  ( ( f `  j )  -h  (
f `  k )
) ) )
2821, 25, 27syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( f : NN --> ~H  /\  j  e.  NN )  /\  k  e.  (
ZZ>= `  j ) )  ->  ( ( f `
 j ) D ( f `  k
) )  =  (
normh `  ( ( f `
 j )  -h  ( f `  k
) ) ) )
2928breq1d 4033 . . . . . . . . . 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 2559 . . . . . . . . 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 2560 . . . . . . . 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 2563 . . . . . . 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 244 . . . . . 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 187 . . . . 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 618 . . . 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 262 . . 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 2394 . 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 2313 1  |-  Cauchy  =  ( ( Cau `  D
)  i^i  ( ~H  ^m  NN ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151   <.cop 3643   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   1c1 8738    < clt 8867   NNcn 9746   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   * Metcxmt 16369   Caucca 18679   NrmCVeccnv 21140   BaseSetcba 21142   IndMetcims 21147   ~Hchil 21499    +h cva 21500    .h csm 21501   normhcno 21503    -h cmv 21505   Cauchyccau 21506
This theorem is referenced by:  axhcompl-zf  21578  hhcau  21777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-xneg 10452  df-xadd 10453  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-xmet 16373  df-met 16374  df-bl 16375  df-cau 18682  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-hvsub 21551  df-hcau 21553
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