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Theorem hlim2 21787
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 4043 . . . . 5  |-  ( w  =  A  ->  ( F  ~~>v  w  <->  F  ~~>v  A ) )
2 oveq2 5882 . . . . . . . . 9  |-  ( w  =  A  ->  (
( F `  z
)  -h  w )  =  ( ( F `
 z )  -h  A ) )
32fveq2d 5545 . . . . . . . 8  |-  ( w  =  A  ->  ( normh `  ( ( F `
 z )  -h  w ) )  =  ( normh `  ( ( F `  z )  -h  A ) ) )
43breq1d 4049 . . . . . . 7  |-  ( w  =  A  ->  (
( normh `  ( ( F `  z )  -h  w ) )  < 
x  <->  ( normh `  (
( F `  z
)  -h  A ) )  <  x ) )
54rexralbidv 2600 . . . . . 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 2576 . . . . 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 312 . . . 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 307 . . 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 2804 . . . . . 6  |-  w  e. 
_V
109hlimi 21783 . . . . 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 871 . . . 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 424 . . 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 2862 . 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 419 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874    < clt 8883   NNcn 9762   ZZ>=cuz 10246   RR+crp 10370   ~Hchil 21515   normhcno 21519    -h cmv 21521    ~~>v chli 21523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-recs 6404  df-rdg 6439  df-nn 9763  df-hlim 21568
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