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Theorem elcnop 23317
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 5690 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  w )  =  ( T `  w ) )
2 fveq1 5690 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
31, 2oveq12d 6062 . . . . . . . 8  |-  ( t  =  T  ->  (
( t `  w
)  -h  ( t `
 x ) )  =  ( ( T `
 w )  -h  ( T `  x
) ) )
43fveq2d 5695 . . . . . . 7  |-  ( t  =  T  ->  ( normh `  ( ( t `
 w )  -h  ( t `  x
) ) )  =  ( normh `  ( ( T `  w )  -h  ( T `  x
) ) ) )
54breq1d 4186 . . . . . 6  |-  ( t  =  T  ->  (
( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y  <->  ( normh `  (
( T `  w
)  -h  ( T `
 x ) ) )  <  y ) )
65imbi2d 308 . . . . 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 2714 . . . 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 2712 . . 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 23300 . . 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 3058 . 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 22459 . . . 4  |-  ~H  e.  _V
1211, 11elmap 7005 . . 3  |-  ( T  e.  ( ~H  ^m  ~H )  <->  T : ~H --> ~H )
1312anbi1i 677 . 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 241 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   E.wrex 2671   class class class wbr 4176   -->wf 5413   ` cfv 5417  (class class class)co 6044    ^m cmap 6981    < clt 9080   RR+crp 10572   ~Hchil 22379   normhcno 22383    -h cmv 22385   ConOpccop 22406
This theorem is referenced by:  cnopc  23373  0cnop  23439  idcnop  23441  lnopconi  23494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-hilex 22459
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-map 6983  df-cnop 23300
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