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Theorem h2hcau 21575
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 2565 . 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 21569 . 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 3371 . . . 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 21493 . . . . . . 7  |-  ~H  e.  _V
7 nnex 9768 . . . . . . 7  |-  NN  e.  _V
86, 7elmap 6812 . . . . . 6  |-  ( f  e.  ( ~H  ^m  NN )  <->  f : NN --> ~H )
9 nnuz 10279 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
10 h2hc.2 . . . . . . . . 9  |-  U  e.  NrmCVec
11 h2hc.4 . . . . . . . . . 10  |-  D  =  ( IndMet `  U )
125, 11imsxmet 21277 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  D  e.  ( * Met `  ~H ) )
1310, 12mp1i 11 . . . . . . . 8  |-  ( f : NN --> ~H  ->  D  e.  ( * Met `  ~H ) )
14 1z 10069 . . . . . . . . 9  |-  1  e.  ZZ
1514a1i 10 . . . . . . . 8  |-  ( f : NN --> ~H  ->  1  e.  ZZ )
16 eqidd 2297 . . . . . . . 8  |-  ( ( f : NN --> ~H  /\  k  e.  NN )  ->  ( f `  k
)  =  ( f `
 k ) )
17 eqidd 2297 . . . . . . . 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 18722 . . . . . . 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 5679 . . . . . . . . . . . . 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 10261 . . . . . . . . . . . . . 14  |-  ( ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  NN )
23 ffvelrn 5679 . . . . . . . . . . . . . 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 21574 . . . . . . . . . . . 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 4049 . . . . . . . . . 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 2572 . . . . . . . . 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 2573 . . . . . . . 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 2576 . . . . . . 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 2407 . 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 2326 1  |-  Cauchy  =  ( ( Cau `  D
)  i^i  ( ~H  ^m  NN ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   {crab 2560    i^i cin 3164   <.cop 3656   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   1c1 8754    < clt 8883   NNcn 9762   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   * Metcxmt 16385   Caucca 18695   NrmCVeccnv 21156   BaseSetcba 21158   IndMetcims 21163   ~Hchil 21515    +h cva 21516    .h csm 21517   normhcno 21519    -h cmv 21521   Cauchyccau 21522
This theorem is referenced by:  axhcompl-zf  21594  hhcau  21793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-xneg 10468  df-xadd 10469  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-xmet 16389  df-met 16390  df-bl 16391  df-cau 18698  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-vs 21171  df-nmcv 21172  df-ims 21173  df-hvsub 21567  df-hcau 21569
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