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Theorem nmfnval 25285
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 11123 . . 3  |-  <  Or  RR*
21supex 7718 . 2  |-  sup ( { x  |  E. y  e.  ~H  (
( normh `  y )  <_  1  /\  x  =  ( abs `  ( T `  y )
) ) } ,  RR* ,  <  )  e. 
_V
3 ax-hilex 24406 . 2  |-  ~H  e.  _V
4 cnex 9368 . 2  |-  CC  e.  _V
5 fveq1 5695 . . . . . . . 8  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
65fveq2d 5700 . . . . . . 7  |-  ( t  =  T  ->  ( abs `  ( t `  y ) )  =  ( abs `  ( T `  y )
) )
76eqeq2d 2454 . . . . . 6  |-  ( t  =  T  ->  (
x  =  ( abs `  ( t `  y
) )  <->  x  =  ( abs `  ( T `
 y ) ) ) )
87anbi2d 703 . . . . 5  |-  ( t  =  T  ->  (
( ( normh `  y
)  <_  1  /\  x  =  ( abs `  ( t `  y
) ) )  <->  ( ( normh `  y )  <_ 
1  /\  x  =  ( abs `  ( T `
 y ) ) ) ) )
98rexbidv 2741 . . . 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 2562 . . 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 7701 . 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 25254 . 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 7254 1  |-  ( T : ~H --> CC  ->  (
normfn `  T )  =  sup ( { x  |  E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  x  =  ( abs `  ( T `  y
) ) ) } ,  RR* ,  <  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   {cab 2429   E.wrex 2721   class class class wbr 4297   -->wf 5419   ` cfv 5423   supcsup 7695   CCcc 9285   1c1 9288   RR*cxr 9422    < clt 9423    <_ cle 9424   abscabs 12728   ~Hchil 24326   normhcno 24330   normfncnmf 24358
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 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-hilex 24406
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-nmfn 25254
This theorem is referenced by:  nmfnxr  25288  nmfnrepnf  25289  nmfnlb  25333  nmfnleub  25334  nmfn0  25396  nmcfnexi  25460  branmfn  25514
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