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Theorem hlim2 22699
Description: The limit of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hlim2  |-  ( ( F : NN --> ~H  /\  A  e.  ~H )  ->  ( F  ~~>v  A  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  z )  -h  A ) )  < 
x ) )
Distinct variable groups:    x, y,
z, F    x, A, y, z

Proof of Theorem hlim2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 breq2 4219 . . . . 5  |-  ( w  =  A  ->  ( F  ~~>v  w  <->  F  ~~>v  A ) )
2 oveq2 6092 . . . . . . . . 9  |-  ( w  =  A  ->  (
( F `  z
)  -h  w )  =  ( ( F `
 z )  -h  A ) )
32fveq2d 5735 . . . . . . . 8  |-  ( w  =  A  ->  ( normh `  ( ( F `
 z )  -h  w ) )  =  ( normh `  ( ( F `  z )  -h  A ) ) )
43breq1d 4225 . . . . . . 7  |-  ( w  =  A  ->  (
( normh `  ( ( F `  z )  -h  w ) )  < 
x  <->  ( normh `  (
( F `  z
)  -h  A ) )  <  x ) )
54rexralbidv 2751 . . . . . 6  |-  ( w  =  A  ->  ( E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  w
) )  <  x  <->  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A
) )  <  x
) )
65ralbidv 2727 . . . . 5  |-  ( w  =  A  ->  ( A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  z )  -h  w ) )  < 
x  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A
) )  <  x
) )
71, 6bibi12d 314 . . . 4  |-  ( w  =  A  ->  (
( F  ~~>v  w  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  z )  -h  w ) )  < 
x )  <->  ( F  ~~>v  A 
<-> 
A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A
) )  <  x
) ) )
87imbi2d 309 . . 3  |-  ( w  =  A  ->  (
( F : NN --> ~H  ->  ( F  ~~>v  w  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  z )  -h  w ) )  < 
x ) )  <->  ( F : NN --> ~H  ->  ( F 
~~>v  A  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A
) )  <  x
) ) ) )
9 vex 2961 . . . . . 6  |-  w  e. 
_V
109hlimi 22695 . . . . 5  |-  ( F 
~~>v  w  <->  ( ( F : NN --> ~H  /\  w  e.  ~H )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  w
) )  <  x
) )
1110baib 873 . . . 4  |-  ( ( F : NN --> ~H  /\  w  e.  ~H )  ->  ( F  ~~>v  w  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  z )  -h  w ) )  < 
x ) )
1211expcom 426 . . 3  |-  ( w  e.  ~H  ->  ( F : NN --> ~H  ->  ( F  ~~>v  w  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  z )  -h  w ) )  < 
x ) ) )
138, 12vtoclga 3019 . 2  |-  ( A  e.  ~H  ->  ( F : NN --> ~H  ->  ( F  ~~>v  A  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  z )  -h  A ) )  < 
x ) ) )
1413impcom 421 1  |-  ( ( F : NN --> ~H  /\  A  e.  ~H )  ->  ( F  ~~>v  A  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  z )  -h  A ) )  < 
x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   class class class wbr 4215   -->wf 5453   ` cfv 5457  (class class class)co 6084    < clt 9125   NNcn 10005   ZZ>=cuz 10493   RR+crp 10617   ~Hchil 22427   normhcno 22431    -h cmv 22433    ~~>v chli 22435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-i2m1 9063  ax-1ne0 9064  ax-rrecex 9067  ax-cnre 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-recs 6636  df-rdg 6671  df-nn 10006  df-hlim 22480
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