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Theorem nmopval 22429
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
Dummy variable  t is distinct from all other variables.

Proof of Theorem nmopval
StepHypRef Expression
1 xrltso 10471 . . 3  |-  <  Or  RR*
21supex 7210 . 2  |-  sup ( { x  |  E. y  e.  ~H  (
( normh `  y )  <_  1  /\  x  =  ( normh `  ( T `  y ) ) ) } ,  RR* ,  <  )  e.  _V
3 ax-hilex 21572 . 2  |-  ~H  e.  _V
4 fveq1 5485 . . . . . . . 8  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
54fveq2d 5490 . . . . . . 7  |-  ( t  =  T  ->  ( normh `  ( t `  y ) )  =  ( normh `  ( T `  y ) ) )
65eqeq2d 2296 . . . . . 6  |-  ( t  =  T  ->  (
x  =  ( normh `  ( t `  y
) )  <->  x  =  ( normh `  ( T `  y ) ) ) )
76anbi2d 686 . . . . 5  |-  ( t  =  T  ->  (
( ( normh `  y
)  <_  1  /\  x  =  ( normh `  ( t `  y
) ) )  <->  ( ( normh `  y )  <_ 
1  /\  x  =  ( normh `  ( T `  y ) ) ) ) )
87rexbidv 2566 . . . 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 2399 . . 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 7195 . 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 22412 . 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 6800 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 6    /\ wa 360    = wceq 1624   {cab 2271   E.wrex 2546   class class class wbr 4025   -->wf 5218   ` cfv 5222   supcsup 7189   1c1 8734   RR*cxr 8862    < clt 8863    <_ cle 8864   ~Hchil 21492   normhcno 21496   normopcnop 21518
This theorem is referenced by:  nmopxr  22439  nmoprepnf  22440  nmoplb  22480  nmopub  22481  nmopnegi  22538  nmop0  22559  nmlnop0iALT  22568  nmopun  22587  nmcopexi  22600  pjnmopi  22721
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8789  ax-resscn 8790  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-hilex 21572
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-sup 7190  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-nmop 22412
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