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Theorem h2hcau 26022
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 2816 . 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 26016 . 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 3683 . . . 4  |-  ( f  e.  ( ( Cau `  D )  i^i  ( ~H  ^m  NN ) )  <-> 
( f  e.  ( Cau `  D )  /\  f  e.  ( ~H  ^m  NN ) ) )
4 ancom 450 . . . 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 25940 . . . . . . 7  |-  ~H  e.  _V
7 nnex 10562 . . . . . . 7  |-  NN  e.  _V
86, 7elmap 7466 . . . . . 6  |-  ( f  e.  ( ~H  ^m  NN )  <->  f : NN --> ~H )
9 nnuz 11141 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
10 h2hc.2 . . . . . . . . 9  |-  U  e.  NrmCVec
11 h2hc.4 . . . . . . . . . 10  |-  D  =  ( IndMet `  U )
125, 11imsxmet 25724 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  D  e.  ( *Met `  ~H ) )
1310, 12mp1i 12 . . . . . . . 8  |-  ( f : NN --> ~H  ->  D  e.  ( *Met `  ~H ) )
14 1zzd 10916 . . . . . . . 8  |-  ( f : NN --> ~H  ->  1  e.  ZZ )
15 eqidd 2458 . . . . . . . 8  |-  ( ( f : NN --> ~H  /\  k  e.  NN )  ->  ( f `  k
)  =  ( f `
 k ) )
16 eqidd 2458 . . . . . . . 8  |-  ( ( f : NN --> ~H  /\  j  e.  NN )  ->  ( f `  j
)  =  ( f `
 j ) )
17 id 22 . . . . . . . 8  |-  ( f : NN --> ~H  ->  f : NN --> ~H )
189, 13, 14, 15, 16, 17iscauf 21844 . . . . . . 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 ) )
19 ffvelrn 6030 . . . . . . . . . . . . 13  |-  ( ( f : NN --> ~H  /\  j  e.  NN )  ->  ( f `  j
)  e.  ~H )
2019adantr 465 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ~H  /\  j  e.  NN )  /\  k  e.  (
ZZ>= `  j ) )  ->  ( f `  j )  e.  ~H )
21 eluznn 11177 . . . . . . . . . . . . . 14  |-  ( ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  NN )
22 ffvelrn 6030 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ~H  /\  k  e.  NN )  ->  ( f `  k
)  e.  ~H )
2321, 22sylan2 474 . . . . . . . . . . . . 13  |-  ( ( f : NN --> ~H  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( f `  k )  e.  ~H )
2423anassrs 648 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ~H  /\  j  e.  NN )  /\  k  e.  (
ZZ>= `  j ) )  ->  ( f `  k )  e.  ~H )
25 h2hc.1 . . . . . . . . . . . . 13  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
2625, 10, 5, 11h2hmetdval 26021 . . . . . . . . . . . 12  |-  ( ( ( f `  j
)  e.  ~H  /\  ( f `  k
)  e.  ~H )  ->  ( ( f `  j ) D ( f `  k ) )  =  ( normh `  ( ( f `  j )  -h  (
f `  k )
) ) )
2720, 24, 26syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( f : NN --> ~H  /\  j  e.  NN )  /\  k  e.  (
ZZ>= `  j ) )  ->  ( ( f `
 j ) D ( f `  k
) )  =  (
normh `  ( ( f `
 j )  -h  ( f `  k
) ) ) )
2827breq1d 4466 . . . . . . . . . 10  |-  ( ( ( f : NN --> ~H  /\  j  e.  NN )  /\  k  e.  (
ZZ>= `  j ) )  ->  ( ( ( f `  j ) D ( f `  k ) )  < 
x  <->  ( normh `  (
( f `  j
)  -h  ( f `
 k ) ) )  <  x ) )
2928ralbidva 2893 . . . . . . . . 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 ) )
3029rexbidva 2965 . . . . . . . 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
) )
3130ralbidv 2896 . . . . . . 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
) )
3218, 31bitrd 253 . . . . . 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 ) )
338, 32sylbi 195 . . . . 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
) )
3433pm5.32i 637 . . . 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
) )
353, 4, 343bitri 271 . . 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
) )
3635abbi2i 2590 . 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 ) }
371, 2, 363eqtr4i 2496 1  |-  Cauchy  =  ( ( Cau `  D
)  i^i  ( ~H  ^m  NN ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   A.wral 2807   E.wrex 2808   {crab 2811    i^i cin 3470   <.cop 4038   class class class wbr 4456   -->wf 5590   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   1c1 9510    < clt 9645   NNcn 10556   ZZ>=cuz 11106   RR+crp 11245   *Metcxmt 18529   Caucca 21817   NrmCVeccnv 25603   BaseSetcba 25605   IndMetcims 25610   ~Hchil 25962    +h cva 25963    .h csm 25964   normhcno 25966    -h cmv 25968   Cauchyccau 25969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-xneg 11343  df-xadd 11344  df-seq 12110  df-exp 12169  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-cau 21820  df-grpo 25319  df-gid 25320  df-ginv 25321  df-gdiv 25322  df-ablo 25410  df-vc 25565  df-nv 25611  df-va 25614  df-ba 25615  df-sm 25616  df-0v 25617  df-vs 25618  df-nmcv 25619  df-ims 25620  df-hvsub 26014  df-hcau 26016
This theorem is referenced by:  axhcompl-zf  26041  hhcau  26241
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