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Theorem hcau 21724
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 5457 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
2 fveq1 5457 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  z )  =  ( F `  z ) )
31, 2oveq12d 5810 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  y
)  -h  ( f `
 z ) )  =  ( ( F `
 y )  -h  ( F `  z
) ) )
43fveq2d 5462 . . . . . 6  |-  ( f  =  F  ->  ( normh `  ( ( f `
 y )  -h  ( f `  z
) ) )  =  ( normh `  ( ( F `  y )  -h  ( F `  z
) ) ) )
54breq1d 4007 . . . . 5  |-  ( f  =  F  ->  (
( normh `  ( (
f `  y )  -h  ( f `  z
) ) )  < 
x  <->  ( normh `  (
( F `  y
)  -h  ( F `
 z ) ) )  <  x ) )
65rexralbidv 2562 . . . 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 2538 . . 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 21514 . . 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 2900 . 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 21540 . . . 4  |-  ~H  e.  _V
11 nnex 9720 . . . 4  |-  NN  e.  _V
1210, 11elmap 6764 . . 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 2518   E.wrex 2519   class class class wbr 3997   -->wf 4669   ` cfv 4673  (class class class)co 5792    ^m cmap 6740    < clt 8835   NNcn 9714   ZZ>=cuz 10198   RR+crp 10322   ~Hchil 21460   normhcno 21464    -h cmv 21466   Cauchyccau 21467
This theorem is referenced by:  hcauseq  21725  hcaucvg  21726  seq1hcau  21727  chscllem2  22178
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 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-i2m1 8773  ax-1ne0 8774  ax-rrecex 8777  ax-cnre 8778  ax-hilex 21540
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-recs 6356  df-rdg 6391  df-map 6742  df-n 9715  df-hcau 21514
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