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Theorem nmopval 25259
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 11117 . . 3  |-  <  Or  RR*
21supex 7712 . 2  |-  sup ( { x  |  E. y  e.  ~H  (
( normh `  y )  <_  1  /\  x  =  ( normh `  ( T `  y ) ) ) } ,  RR* ,  <  )  e.  _V
3 ax-hilex 24400 . 2  |-  ~H  e.  _V
4 fveq1 5689 . . . . . . . 8  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
54fveq2d 5694 . . . . . . 7  |-  ( t  =  T  ->  ( normh `  ( t `  y ) )  =  ( normh `  ( T `  y ) ) )
65eqeq2d 2453 . . . . . 6  |-  ( t  =  T  ->  (
x  =  ( normh `  ( t `  y
) )  <->  x  =  ( normh `  ( T `  y ) ) ) )
76anbi2d 703 . . . . 5  |-  ( t  =  T  ->  (
( ( normh `  y
)  <_  1  /\  x  =  ( normh `  ( t `  y
) ) )  <->  ( ( normh `  y )  <_ 
1  /\  x  =  ( normh `  ( T `  y ) ) ) ) )
87rexbidv 2735 . . . 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 2556 . . 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 7695 . 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 25242 . 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 7248 1  |-  ( T : ~H --> ~H  ->  (
normop `  T )  =  sup ( { x  |  E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  x  =  ( normh `  ( T `  y
) ) ) } ,  RR* ,  <  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   {cab 2428   E.wrex 2715   class class class wbr 4291   -->wf 5413   ` cfv 5417   supcsup 7689   1c1 9282   RR*cxr 9416    < clt 9417    <_ cle 9418   ~Hchil 24320   normhcno 24324   normopcnop 24346
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 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-hilex 24400
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-nmop 25242
This theorem is referenced by:  nmopxr  25269  nmoprepnf  25270  nmoplb  25310  nmopub  25311  nmopnegi  25368  nmop0  25389  nmlnop0iALT  25398  nmopun  25417  nmcopexi  25430  pjnmopi  25551
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