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Theorem nlfnval 23376
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 9063 . . 3  |-  CC  e.  _V
2 ax-hilex 22494 . . 3  |-  ~H  e.  _V
31, 2elmap 7034 . 2  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
4 cnvexg 5397 . . . 4  |-  ( T  e.  ( CC  ^m  ~H )  ->  `' T  e.  _V )
5 imaexg 5209 . . . 4  |-  ( `' T  e.  _V  ->  ( `' T " { 0 } )  e.  _V )
64, 5syl 16 . . 3  |-  ( T  e.  ( CC  ^m  ~H )  ->  ( `' T " { 0 } )  e.  _V )
7 cnveq 5038 . . . . 5  |-  ( t  =  T  ->  `' t  =  `' T
)
87imaeq1d 5194 . . . 4  |-  ( t  =  T  ->  ( `' t " {
0 } )  =  ( `' T " { 0 } ) )
9 df-nlfn 23341 . . . 4  |-  null  =  ( t  e.  ( CC  ^m  ~H )  |->  ( `' t " { 0 } ) )
108, 9fvmptg 5796 . . 3  |-  ( ( T  e.  ( CC 
^m  ~H )  /\  ( `' T " { 0 } )  e.  _V )  ->  ( null `  T
)  =  ( `' T " { 0 } ) )
116, 10mpdan 650 . 2  |-  ( T  e.  ( CC  ^m  ~H )  ->  ( null `  T )  =  ( `' T " { 0 } ) )
123, 11sylbir 205 1  |-  ( T : ~H --> CC  ->  (
null `  T )  =  ( `' T " { 0 } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806   `'ccnv 4869   "cima 4873   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^m cmap 7010   CCcc 8980   0cc0 8982   ~Hchil 22414   nullcnl 22447
This theorem is referenced by:  elnlfn  23423  nlelshi  23555
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-hilex 22494
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-nlfn 23341
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