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Theorem nmfnval 22472
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 10491 . . 3  |-  <  Or  RR*
21supex 7230 . 2  |-  sup ( { x  |  E. y  e.  ~H  (
( normh `  y )  <_  1  /\  x  =  ( abs `  ( T `  y )
) ) } ,  RR* ,  <  )  e. 
_V
3 ax-hilex 21595 . 2  |-  ~H  e.  _V
4 cnex 8834 . 2  |-  CC  e.  _V
5 fveq1 5540 . . . . . . . 8  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
65fveq2d 5545 . . . . . . 7  |-  ( t  =  T  ->  ( abs `  ( t `  y ) )  =  ( abs `  ( T `  y )
) )
76eqeq2d 2307 . . . . . 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 2577 . . . 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 2410 . . 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 7215 . 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 22441 . 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 6820 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 1632   {cab 2282   E.wrex 2557   class class class wbr 4039   -->wf 5267   ` cfv 5271   supcsup 7209   CCcc 8751   1c1 8754   RR*cxr 8882    < clt 8883    <_ cle 8884   abscabs 11735   ~Hchil 21515   normhcno 21519   normfncnmf 21547
This theorem is referenced by:  nmfnxr  22475  nmfnrepnf  22476  nmfnlb  22520  nmfnleub  22521  nmfn0  22583  nmcfnexi  22647  branmfn  22701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-hilex 21595
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-nmfn 22441
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