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Theorem elcnop 22871
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 5631 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  w )  =  ( T `  w ) )
2 fveq1 5631 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
31, 2oveq12d 5999 . . . . . . . 8  |-  ( t  =  T  ->  (
( t `  w
)  -h  ( t `
 x ) )  =  ( ( T `
 w )  -h  ( T `  x
) ) )
43fveq2d 5636 . . . . . . 7  |-  ( t  =  T  ->  ( normh `  ( ( t `
 w )  -h  ( t `  x
) ) )  =  ( normh `  ( ( T `  w )  -h  ( T `  x
) ) ) )
54breq1d 4135 . . . . . 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 2672 . . . 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 2670 . . 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 22854 . . 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 3011 . 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 22013 . . . 4  |-  ~H  e.  _V
1211, 11elmap 6939 . . 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 1647    e. wcel 1715   A.wral 2628   E.wrex 2629   class class class wbr 4125   -->wf 5354   ` cfv 5358  (class class class)co 5981    ^m cmap 6915    < clt 9014   RR+crp 10505   ~Hchil 21933   normhcno 21937    -h cmv 21939   ConOpccop 21960
This theorem is referenced by:  cnopc  22927  0cnop  22993  idcnop  22995  lnopconi  23048
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-hilex 22013
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-map 6917  df-cnop 22854
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