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Theorem nlfnval 25285
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 9363 . . 3  |-  CC  e.  _V
2 ax-hilex 24401 . . 3  |-  ~H  e.  _V
31, 2elmap 7241 . 2  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
4 cnvexg 6524 . . . 4  |-  ( T  e.  ( CC  ^m  ~H )  ->  `' T  e.  _V )
5 imaexg 6515 . . . 4  |-  ( `' T  e.  _V  ->  ( `' T " { 0 } )  e.  _V )
64, 5syl 16 . . 3  |-  ( T  e.  ( CC  ^m  ~H )  ->  ( `' T " { 0 } )  e.  _V )
7 cnveq 5013 . . . . 5  |-  ( t  =  T  ->  `' t  =  `' T
)
87imaeq1d 5168 . . . 4  |-  ( t  =  T  ->  ( `' t " {
0 } )  =  ( `' T " { 0 } ) )
9 df-nlfn 25250 . . . 4  |-  null  =  ( t  e.  ( CC  ^m  ~H )  |->  ( `' t " { 0 } ) )
108, 9fvmptg 5772 . . 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 1369    e. wcel 1756   _Vcvv 2972   {csn 3877   `'ccnv 4839   "cima 4843   -->wf 5414   ` cfv 5418  (class class class)co 6091    ^m cmap 7214   CCcc 9280   0cc0 9282   ~Hchil 24321   nullcnl 24354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-hilex 24401
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-map 7216  df-nlfn 25250
This theorem is referenced by:  elnlfn  25332  nlelshi  25464
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