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Theorem hcau 21688
Description: Member of the set of Cauchy sequences on a Hilbert space. Definition for Cauchy sequence in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hcau  |-  ( F  e.  Cauchy 
<->  ( F : NN --> ~H  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  y )  -h  ( F `  z )
) )  <  x
) )
Distinct variable group:    x, y, z, F

Proof of Theorem hcau
StepHypRef Expression
1 fveq1 5422 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
2 fveq1 5422 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  z )  =  ( F `  z ) )
31, 2oveq12d 5775 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  y
)  -h  ( f `
 z ) )  =  ( ( F `
 y )  -h  ( F `  z
) ) )
43fveq2d 5427 . . . . . 6  |-  ( f  =  F  ->  ( normh `  ( ( f `
 y )  -h  ( f `  z
) ) )  =  ( normh `  ( ( F `  y )  -h  ( F `  z
) ) ) )
54breq1d 3973 . . . . 5  |-  ( f  =  F  ->  (
( normh `  ( (
f `  y )  -h  ( f `  z
) ) )  < 
x  <->  ( normh `  (
( F `  y
)  -h  ( F `
 z ) ) )  <  x ) )
65rexralbidv 2558 . . . 4  |-  ( f  =  F  ->  ( E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( f `  y )  -h  (
f `  z )
) )  <  x  <->  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  y )  -h  ( F `  z )
) )  <  x
) )
76ralbidv 2534 . . 3  |-  ( f  =  F  ->  ( A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( (
f `  y )  -h  ( f `  z
) ) )  < 
x  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  y )  -h  ( F `  z )
) )  <  x
) )
8 df-hcau 21478 . . 3  |-  Cauchy  =  {
f  e.  ( ~H 
^m  NN )  | 
A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( f `  y )  -h  (
f `  z )
) )  <  x }
97, 8elrab2 2876 . 2  |-  ( F  e.  Cauchy 
<->  ( F  e.  ( ~H  ^m  NN )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  y )  -h  ( F `  z )
) )  <  x
) )
10 ax-hilex 21504 . . . 4  |-  ~H  e.  _V
11 nnex 9685 . . . 4  |-  NN  e.  _V
1210, 11elmap 6729 . . 3  |-  ( F  e.  ( ~H  ^m  NN )  <->  F : NN --> ~H )
1312anbi1i 679 . 2  |-  ( ( F  e.  ( ~H 
^m  NN )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  y )  -h  ( F `  z
) ) )  < 
x )  <->  ( F : NN --> ~H  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  y )  -h  ( F `  z
) ) )  < 
x ) )
149, 13bitri 242 1  |-  ( F  e.  Cauchy 
<->  ( F : NN --> ~H  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  y )  -h  ( F `  z )
) )  <  x
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516   E.wrex 2517   class class class wbr 3963   -->wf 4634   ` cfv 4638  (class class class)co 5757    ^m cmap 6705    < clt 8800   NNcn 9679   ZZ>=cuz 10162   RR+crp 10286   ~Hchil 21424   normhcno 21428    -h cmv 21430   Cauchyccau 21431
This theorem is referenced by:  hcauseq  21689  hcaucvg  21690  seq1hcau  21691  chscllem2  22160
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-i2m1 8738  ax-1ne0 8739  ax-rrecex 8742  ax-cnre 8743  ax-hilex 21504
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-recs 6321  df-rdg 6356  df-map 6707  df-n 9680  df-hcau 21478
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