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Theorem nmfnval 22456
Description: Value of the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
nmfnval  |-  ( T : ~H --> CC  ->  (
normfn `  T )  =  sup ( { x  |  E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  x  =  ( abs `  ( T `  y
) ) ) } ,  RR* ,  <  )
)
Distinct variable group:    x, y, T

Proof of Theorem nmfnval
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 xrltso 10475 . . 3  |-  <  Or  RR*
21supex 7214 . 2  |-  sup ( { x  |  E. y  e.  ~H  (
( normh `  y )  <_  1  /\  x  =  ( abs `  ( T `  y )
) ) } ,  RR* ,  <  )  e. 
_V
3 ax-hilex 21579 . 2  |-  ~H  e.  _V
4 cnex 8818 . 2  |-  CC  e.  _V
5 fveq1 5524 . . . . . . . 8  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
65fveq2d 5529 . . . . . . 7  |-  ( t  =  T  ->  ( abs `  ( t `  y ) )  =  ( abs `  ( T `  y )
) )
76eqeq2d 2294 . . . . . 6  |-  ( t  =  T  ->  (
x  =  ( abs `  ( t `  y
) )  <->  x  =  ( abs `  ( T `
 y ) ) ) )
87anbi2d 684 . . . . 5  |-  ( t  =  T  ->  (
( ( normh `  y
)  <_  1  /\  x  =  ( abs `  ( t `  y
) ) )  <->  ( ( normh `  y )  <_ 
1  /\  x  =  ( abs `  ( T `
 y ) ) ) ) )
98rexbidv 2564 . . . 4  |-  ( t  =  T  ->  ( E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  x  =  ( abs `  ( t `  y
) ) )  <->  E. y  e.  ~H  ( ( normh `  y )  <_  1  /\  x  =  ( abs `  ( T `  y ) ) ) ) )
109abbidv 2397 . . 3  |-  ( t  =  T  ->  { x  |  E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  x  =  ( abs `  ( t `  y
) ) ) }  =  { x  |  E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  x  =  ( abs `  ( T `  y
) ) ) } )
1110supeq1d 7199 . 2  |-  ( t  =  T  ->  sup ( { x  |  E. y  e.  ~H  (
( normh `  y )  <_  1  /\  x  =  ( abs `  (
t `  y )
) ) } ,  RR* ,  <  )  =  sup ( { x  |  E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  x  =  ( abs `  ( T `  y
) ) ) } ,  RR* ,  <  )
)
12 df-nmfn 22425 . 2  |-  normfn  =  ( t  e.  ( CC 
^m  ~H )  |->  sup ( { x  |  E. y  e.  ~H  (
( normh `  y )  <_  1  /\  x  =  ( abs `  (
t `  y )
) ) } ,  RR* ,  <  ) )
132, 3, 4, 11, 12fvmptmap 6804 1  |-  ( T : ~H --> CC  ->  (
normfn `  T )  =  sup ( { x  |  E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  x  =  ( abs `  ( T `  y
) ) ) } ,  RR* ,  <  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623   {cab 2269   E.wrex 2544   class class class wbr 4023   -->wf 5251   ` cfv 5255   supcsup 7193   CCcc 8735   1c1 8738   RR*cxr 8866    < clt 8867    <_ cle 8868   abscabs 11719   ~Hchil 21499   normhcno 21503   normfncnmf 21531
This theorem is referenced by:  nmfnxr  22459  nmfnrepnf  22460  nmfnlb  22504  nmfnleub  22505  nmfn0  22567  nmcfnexi  22631  branmfn  22685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-nmfn 22425
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