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Theorem elcnop 22437
Description: Property defining a continuous Hilbert space operator. (Contributed by NM, 28-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elcnop  |-  ( T  e.  ConOp 
<->  ( T : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( normh `  ( ( T `  w )  -h  ( T `  x
) ) )  < 
y ) ) )
Distinct variable group:    x, w, y, z, T

Proof of Theorem elcnop
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 fveq1 5524 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  w )  =  ( T `  w ) )
2 fveq1 5524 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
31, 2oveq12d 5876 . . . . . . . 8  |-  ( t  =  T  ->  (
( t `  w
)  -h  ( t `
 x ) )  =  ( ( T `
 w )  -h  ( T `  x
) ) )
43fveq2d 5529 . . . . . . 7  |-  ( t  =  T  ->  ( normh `  ( ( t `
 w )  -h  ( t `  x
) ) )  =  ( normh `  ( ( T `  w )  -h  ( T `  x
) ) ) )
54breq1d 4033 . . . . . 6  |-  ( t  =  T  ->  (
( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y  <->  ( normh `  (
( T `  w
)  -h  ( T `
 x ) ) )  <  y ) )
65imbi2d 307 . . . . 5  |-  ( t  =  T  ->  (
( ( normh `  (
w  -h  x ) )  <  z  -> 
( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y )  <->  ( ( normh `  ( w  -h  x ) )  < 
z  ->  ( normh `  ( ( T `  w )  -h  ( T `  x )
) )  <  y
) ) )
76rexralbidv 2587 . . . 4  |-  ( t  =  T  ->  ( E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y )  <->  E. z  e.  RR+  A. w  e. 
~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( normh `  ( ( T `  w )  -h  ( T `  x
) ) )  < 
y ) ) )
872ralbidv 2585 . . 3  |-  ( t  =  T  ->  ( A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e. 
~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y )  <->  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
( normh `  ( w  -h  x ) )  < 
z  ->  ( normh `  ( ( T `  w )  -h  ( T `  x )
) )  <  y
) ) )
9 df-cnop 22420 . . 3  |-  ConOp  =  {
t  e.  ( ~H 
^m  ~H )  |  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y ) }
108, 9elrab2 2925 . 2  |-  ( T  e.  ConOp 
<->  ( T  e.  ( ~H  ^m  ~H )  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e. 
~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( normh `  ( ( T `  w )  -h  ( T `  x
) ) )  < 
y ) ) )
11 ax-hilex 21579 . . . 4  |-  ~H  e.  _V
1211, 11elmap 6796 . . 3  |-  ( T  e.  ( ~H  ^m  ~H )  <->  T : ~H --> ~H )
1312anbi1i 676 . 2  |-  ( ( T  e.  ( ~H 
^m  ~H )  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( normh `  ( ( T `  w )  -h  ( T `  x
) ) )  < 
y ) )  <->  ( T : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
( normh `  ( w  -h  x ) )  < 
z  ->  ( normh `  ( ( T `  w )  -h  ( T `  x )
) )  <  y
) ) )
1410, 13bitri 240 1  |-  ( T  e.  ConOp 
<->  ( T : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( normh `  ( ( T `  w )  -h  ( T `  x
) ) )  < 
y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772    < clt 8867   RR+crp 10354   ~Hchil 21499   normhcno 21503    -h cmv 21505   ConOpccop 21526
This theorem is referenced by:  cnopc  22493  0cnop  22559  idcnop  22561  lnopconi  22614
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-cnop 22420
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