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Theorem nlfnval 26623
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 9585 . . 3  |-  CC  e.  _V
2 ax-hilex 25739 . . 3  |-  ~H  e.  _V
31, 2elmap 7459 . 2  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
4 cnvexg 6741 . . . 4  |-  ( T  e.  ( CC  ^m  ~H )  ->  `' T  e.  _V )
5 imaexg 6732 . . . 4  |-  ( `' T  e.  _V  ->  ( `' T " { 0 } )  e.  _V )
64, 5syl 16 . . 3  |-  ( T  e.  ( CC  ^m  ~H )  ->  ( `' T " { 0 } )  e.  _V )
7 cnveq 5182 . . . . 5  |-  ( t  =  T  ->  `' t  =  `' T
)
87imaeq1d 5342 . . . 4  |-  ( t  =  T  ->  ( `' t " {
0 } )  =  ( `' T " { 0 } ) )
9 df-nlfn 26588 . . . 4  |-  null  =  ( t  e.  ( CC  ^m  ~H )  |->  ( `' t " { 0 } ) )
108, 9fvmptg 5955 . . 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 1379    e. wcel 1767   _Vcvv 3118   {csn 4033   `'ccnv 5004   "cima 5008   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   CCcc 9502   0cc0 9504   ~Hchil 25659   nullcnl 25692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-hilex 25739
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-nlfn 26588
This theorem is referenced by:  elnlfn  26670  nlelshi  26802
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