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Theorem nlfnval 26669
Description: Value of the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
nlfnval  |-  ( T : ~H --> CC  ->  (
null `  T )  =  ( `' T " { 0 } ) )

Proof of Theorem nlfnval
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 cnex 9573 . . 3  |-  CC  e.  _V
2 ax-hilex 25785 . . 3  |-  ~H  e.  _V
31, 2elmap 7446 . 2  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
4 cnvexg 6728 . . . 4  |-  ( T  e.  ( CC  ^m  ~H )  ->  `' T  e.  _V )
5 imaexg 6719 . . . 4  |-  ( `' T  e.  _V  ->  ( `' T " { 0 } )  e.  _V )
64, 5syl 16 . . 3  |-  ( T  e.  ( CC  ^m  ~H )  ->  ( `' T " { 0 } )  e.  _V )
7 cnveq 5163 . . . . 5  |-  ( t  =  T  ->  `' t  =  `' T
)
87imaeq1d 5323 . . . 4  |-  ( t  =  T  ->  ( `' t " {
0 } )  =  ( `' T " { 0 } ) )
9 df-nlfn 26634 . . . 4  |-  null  =  ( t  e.  ( CC  ^m  ~H )  |->  ( `' t " { 0 } ) )
108, 9fvmptg 5936 . . 3  |-  ( ( T  e.  ( CC 
^m  ~H )  /\  ( `' T " { 0 } )  e.  _V )  ->  ( null `  T
)  =  ( `' T " { 0 } ) )
116, 10mpdan 668 . 2  |-  ( T  e.  ( CC  ^m  ~H )  ->  ( null `  T )  =  ( `' T " { 0 } ) )
123, 11sylbir 213 1  |-  ( T : ~H --> CC  ->  (
null `  T )  =  ( `' T " { 0 } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381    e. wcel 1802   _Vcvv 3093   {csn 4011   `'ccnv 4985   "cima 4989   -->wf 5571   ` cfv 5575  (class class class)co 6278    ^m cmap 7419   CCcc 9490   0cc0 9492   ~Hchil 25705   nullcnl 25738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-hilex 25785
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-fv 5583  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-map 7421  df-nlfn 26634
This theorem is referenced by:  elnlfn  26716  nlelshi  26848
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