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Theorem elcnop 26607
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 5871 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  w )  =  ( T `  w ) )
2 fveq1 5871 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
31, 2oveq12d 6313 . . . . . . . 8  |-  ( t  =  T  ->  (
( t `  w
)  -h  ( t `
 x ) )  =  ( ( T `
 w )  -h  ( T `  x
) ) )
43fveq2d 5876 . . . . . . 7  |-  ( t  =  T  ->  ( normh `  ( ( t `
 w )  -h  ( t `  x
) ) )  =  ( normh `  ( ( T `  w )  -h  ( T `  x
) ) ) )
54breq1d 4463 . . . . . 6  |-  ( t  =  T  ->  (
( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y  <->  ( normh `  (
( T `  w
)  -h  ( T `
 x ) ) )  <  y ) )
65imbi2d 316 . . . . 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 2986 . . . 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 2911 . . 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 26590 . . 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 3268 . 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 25747 . . . 4  |-  ~H  e.  _V
1211, 11elmap 7459 . . 3  |-  ( T  e.  ( ~H  ^m  ~H )  <->  T : ~H --> ~H )
1312anbi1i 695 . 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 249 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   class class class wbr 4453   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432    < clt 9640   RR+crp 11232   ~Hchil 25667   normhcno 25671    -h cmv 25673   ConOpccop 25694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-hilex 25747
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-cnop 26590
This theorem is referenced by:  cnopc  26663  0cnop  26729  idcnop  26731  lnopconi  26784
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