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Theorem nmfnval 23380
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 10735 . . 3  |-  <  Or  RR*
21supex 7469 . 2  |-  sup ( { x  |  E. y  e.  ~H  (
( normh `  y )  <_  1  /\  x  =  ( abs `  ( T `  y )
) ) } ,  RR* ,  <  )  e. 
_V
3 ax-hilex 22503 . 2  |-  ~H  e.  _V
4 cnex 9072 . 2  |-  CC  e.  _V
5 fveq1 5728 . . . . . . . 8  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
65fveq2d 5733 . . . . . . 7  |-  ( t  =  T  ->  ( abs `  ( t `  y ) )  =  ( abs `  ( T `  y )
) )
76eqeq2d 2448 . . . . . 6  |-  ( t  =  T  ->  (
x  =  ( abs `  ( t `  y
) )  <->  x  =  ( abs `  ( T `
 y ) ) ) )
87anbi2d 686 . . . . 5  |-  ( t  =  T  ->  (
( ( normh `  y
)  <_  1  /\  x  =  ( abs `  ( t `  y
) ) )  <->  ( ( normh `  y )  <_ 
1  /\  x  =  ( abs `  ( T `
 y ) ) ) ) )
98rexbidv 2727 . . . 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 2551 . . 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 7452 . 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 23349 . 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 7051 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 360    = wceq 1653   {cab 2423   E.wrex 2707   class class class wbr 4213   -->wf 5451   ` cfv 5455   supcsup 7446   CCcc 8989   1c1 8992   RR*cxr 9120    < clt 9121    <_ cle 9122   abscabs 12040   ~Hchil 22423   normhcno 22427   normfncnmf 22455
This theorem is referenced by:  nmfnxr  23383  nmfnrepnf  23384  nmfnlb  23428  nmfnleub  23429  nmfn0  23491  nmcfnexi  23555  branmfn  23609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-hilex 22503
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-po 4504  df-so 4505  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-nmfn 23349
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