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Theorem nmopval 23312
Description: Value of the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
nmopval  |-  ( T : ~H --> ~H  ->  (
normop `  T )  =  sup ( { x  |  E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  x  =  ( normh `  ( T `  y
) ) ) } ,  RR* ,  <  )
)
Distinct variable group:    x, y, T

Proof of Theorem nmopval
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 xrltso 10690 . . 3  |-  <  Or  RR*
21supex 7424 . 2  |-  sup ( { x  |  E. y  e.  ~H  (
( normh `  y )  <_  1  /\  x  =  ( normh `  ( T `  y ) ) ) } ,  RR* ,  <  )  e.  _V
3 ax-hilex 22455 . 2  |-  ~H  e.  _V
4 fveq1 5686 . . . . . . . 8  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
54fveq2d 5691 . . . . . . 7  |-  ( t  =  T  ->  ( normh `  ( t `  y ) )  =  ( normh `  ( T `  y ) ) )
65eqeq2d 2415 . . . . . 6  |-  ( t  =  T  ->  (
x  =  ( normh `  ( t `  y
) )  <->  x  =  ( normh `  ( T `  y ) ) ) )
76anbi2d 685 . . . . 5  |-  ( t  =  T  ->  (
( ( normh `  y
)  <_  1  /\  x  =  ( normh `  ( t `  y
) ) )  <->  ( ( normh `  y )  <_ 
1  /\  x  =  ( normh `  ( T `  y ) ) ) ) )
87rexbidv 2687 . . . 4  |-  ( t  =  T  ->  ( E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  x  =  ( normh `  ( t `  y
) ) )  <->  E. y  e.  ~H  ( ( normh `  y )  <_  1  /\  x  =  ( normh `  ( T `  y ) ) ) ) )
98abbidv 2518 . . 3  |-  ( t  =  T  ->  { x  |  E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  x  =  ( normh `  ( t `  y
) ) ) }  =  { x  |  E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  x  =  ( normh `  ( T `  y
) ) ) } )
109supeq1d 7409 . 2  |-  ( t  =  T  ->  sup ( { x  |  E. y  e.  ~H  (
( normh `  y )  <_  1  /\  x  =  ( normh `  ( t `  y ) ) ) } ,  RR* ,  <  )  =  sup ( { x  |  E. y  e.  ~H  ( ( normh `  y )  <_  1  /\  x  =  ( normh `  ( T `  y ) ) ) } ,  RR* ,  <  ) )
11 df-nmop 23295 . 2  |-  normop  =  ( t  e.  ( ~H 
^m  ~H )  |->  sup ( { x  |  E. y  e.  ~H  (
( normh `  y )  <_  1  /\  x  =  ( normh `  ( t `  y ) ) ) } ,  RR* ,  <  ) )
122, 3, 3, 10, 11fvmptmap 7009 1  |-  ( T : ~H --> ~H  ->  (
normop `  T )  =  sup ( { x  |  E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  x  =  ( normh `  ( T `  y
) ) ) } ,  RR* ,  <  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649   {cab 2390   E.wrex 2667   class class class wbr 4172   -->wf 5409   ` cfv 5413   supcsup 7403   1c1 8947   RR*cxr 9075    < clt 9076    <_ cle 9077   ~Hchil 22375   normhcno 22379   normopcnop 22401
This theorem is referenced by:  nmopxr  23322  nmoprepnf  23323  nmoplb  23363  nmopub  23364  nmopnegi  23421  nmop0  23442  nmlnop0iALT  23451  nmopun  23470  nmcopexi  23483  pjnmopi  23604
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-hilex 22455
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-nmop 23295
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