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Theorem elcnop 25261
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 6109 . . . . . . . 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 4302 . . . . . 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 2759 . . . 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 2757 . . 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 25244 . . 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 3119 . 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 24401 . . . 4  |-  ~H  e.  _V
1211, 11elmap 7241 . . 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 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   class class class wbr 4292   -->wf 5414   ` cfv 5418  (class class class)co 6091    ^m cmap 7214    < clt 9418   RR+crp 10991   ~Hchil 24321   normhcno 24325    -h cmv 24327   ConOpccop 24348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-hilex 24401
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-map 7216  df-cnop 25244
This theorem is referenced by:  cnopc  25317  0cnop  25383  idcnop  25385  lnopconi  25438
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