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Theorem hcau 26822
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
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 fveq1 5876 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
2 fveq1 5876 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  z )  =  ( F `  z ) )
31, 2oveq12d 6319 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  y
)  -h  ( f `
 z ) )  =  ( ( F `
 y )  -h  ( F `  z
) ) )
43fveq2d 5881 . . . . . 6  |-  ( f  =  F  ->  ( normh `  ( ( f `
 y )  -h  ( f `  z
) ) )  =  ( normh `  ( ( F `  y )  -h  ( F `  z
) ) ) )
54breq1d 4430 . . . . 5  |-  ( f  =  F  ->  (
( normh `  ( (
f `  y )  -h  ( f `  z
) ) )  < 
x  <->  ( normh `  (
( F `  y
)  -h  ( F `
 z ) ) )  <  x ) )
65rexralbidv 2947 . . . 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 2864 . . 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 26611 . . 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 3231 . 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 26637 . . . 4  |-  ~H  e.  _V
11 nnex 10615 . . . 4  |-  NN  e.  _V
1210, 11elmap 7504 . . 3  |-  ( F  e.  ( ~H  ^m  NN )  <->  F : NN --> ~H )
1312anbi1i 699 . 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 252 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 setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775   E.wrex 2776   class class class wbr 4420   -->wf 5593   ` cfv 5597  (class class class)co 6301    ^m cmap 7476    < clt 9675   NNcn 10609   ZZ>=cuz 11159   RR+crp 11302   ~Hchil 26557   normhcno 26561    -h cmv 26563   Cauchyccau 26564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-i2m1 9607  ax-1ne0 9608  ax-rrecex 9611  ax-cnre 9612  ax-hilex 26637
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-map 7478  df-nn 10610  df-hcau 26611
This theorem is referenced by:  hcauseq  26823  hcaucvg  26824  seq1hcau  26825  chscllem2  27276
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