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Theorem nmfnval 27364
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 11440 . . 3  |-  <  Or  RR*
21supex 7983 . 2  |-  sup ( { x  |  E. y  e.  ~H  (
( normh `  y )  <_  1  /\  x  =  ( abs `  ( T `  y )
) ) } ,  RR* ,  <  )  e. 
_V
3 ax-hilex 26487 . 2  |-  ~H  e.  _V
4 cnex 9619 . 2  |-  CC  e.  _V
5 fveq1 5880 . . . . . . . 8  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
65fveq2d 5885 . . . . . . 7  |-  ( t  =  T  ->  ( abs `  ( t `  y ) )  =  ( abs `  ( T `  y )
) )
76eqeq2d 2443 . . . . . 6  |-  ( t  =  T  ->  (
x  =  ( abs `  ( t `  y
) )  <->  x  =  ( abs `  ( T `
 y ) ) ) )
87anbi2d 708 . . . . 5  |-  ( t  =  T  ->  (
( ( normh `  y
)  <_  1  /\  x  =  ( abs `  ( t `  y
) ) )  <->  ( ( normh `  y )  <_ 
1  /\  x  =  ( abs `  ( T `
 y ) ) ) ) )
98rexbidv 2946 . . . 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 2565 . . 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 7966 . 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 27333 . 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 7516 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 370    = wceq 1437   {cab 2414   E.wrex 2783   class class class wbr 4426   -->wf 5597   ` cfv 5601   supcsup 7960   CCcc 9536   1c1 9539   RR*cxr 9673    < clt 9674    <_ cle 9675   abscabs 13276   ~Hchil 26407   normhcno 26411   normfncnmf 26439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-hilex 26487
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-nmfn 27333
This theorem is referenced by:  nmfnxr  27367  nmfnrepnf  27368  nmfnlb  27412  nmfnleub  27413  nmfn0  27475  nmcfnexi  27539  branmfn  27593
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