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Theorem hcau 24585
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 5689 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
2 fveq1 5689 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  z )  =  ( F `  z ) )
31, 2oveq12d 6108 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  y
)  -h  ( f `
 z ) )  =  ( ( F `
 y )  -h  ( F `  z
) ) )
43fveq2d 5694 . . . . . 6  |-  ( f  =  F  ->  ( normh `  ( ( f `
 y )  -h  ( f `  z
) ) )  =  ( normh `  ( ( F `  y )  -h  ( F `  z
) ) ) )
54breq1d 4301 . . . . 5  |-  ( f  =  F  ->  (
( normh `  ( (
f `  y )  -h  ( f `  z
) ) )  < 
x  <->  ( normh `  (
( F `  y
)  -h  ( F `
 z ) ) )  <  x ) )
65rexralbidv 2758 . . . 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 2734 . . 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 24374 . . 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 3118 . 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 24400 . . . 4  |-  ~H  e.  _V
11 nnex 10327 . . . 4  |-  NN  e.  _V
1210, 11elmap 7240 . . 3  |-  ( F  e.  ( ~H  ^m  NN )  <->  F : NN --> ~H )
1312anbi1i 695 . 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 249 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 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2714   E.wrex 2715   class class class wbr 4291   -->wf 5413   ` cfv 5417  (class class class)co 6090    ^m cmap 7213    < clt 9417   NNcn 10321   ZZ>=cuz 10860   RR+crp 10990   ~Hchil 24320   normhcno 24324    -h cmv 24326   Cauchyccau 24327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-i2m1 9349  ax-1ne0 9350  ax-rrecex 9353  ax-cnre 9354  ax-hilex 24400
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6831  df-rdg 6865  df-map 7215  df-nn 10322  df-hcau 24374
This theorem is referenced by:  hcauseq  24586  hcaucvg  24587  seq1hcau  24588  chscllem2  25040
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