HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  nlfnval Structured version   Unicode version

Theorem nlfnval 27520
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 9621 . . 3  |-  CC  e.  _V
2 ax-hilex 26638 . . 3  |-  ~H  e.  _V
31, 2elmap 7505 . 2  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
4 cnvexg 6750 . . . 4  |-  ( T  e.  ( CC  ^m  ~H )  ->  `' T  e.  _V )
5 imaexg 6741 . . . 4  |-  ( `' T  e.  _V  ->  ( `' T " { 0 } )  e.  _V )
64, 5syl 17 . . 3  |-  ( T  e.  ( CC  ^m  ~H )  ->  ( `' T " { 0 } )  e.  _V )
7 cnveq 5024 . . . . 5  |-  ( t  =  T  ->  `' t  =  `' T
)
87imaeq1d 5183 . . . 4  |-  ( t  =  T  ->  ( `' t " {
0 } )  =  ( `' T " { 0 } ) )
9 df-nlfn 27485 . . . 4  |-  null  =  ( t  e.  ( CC  ^m  ~H )  |->  ( `' t " { 0 } ) )
108, 9fvmptg 5959 . . 3  |-  ( ( T  e.  ( CC 
^m  ~H )  /\  ( `' T " { 0 } )  e.  _V )  ->  ( null `  T
)  =  ( `' T " { 0 } ) )
116, 10mpdan 672 . 2  |-  ( T  e.  ( CC  ^m  ~H )  ->  ( null `  T )  =  ( `' T " { 0 } ) )
123, 11sylbir 216 1  |-  ( T : ~H --> CC  ->  (
null `  T )  =  ( `' T " { 0 } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1868   _Vcvv 3081   {csn 3996   `'ccnv 4849   "cima 4853   -->wf 5594   ` cfv 5598  (class class class)co 6302    ^m cmap 7477   CCcc 9538   0cc0 9540   ~Hchil 26558   nullcnl 26591
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 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-hilex 26638
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-fv 5606  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-map 7479  df-nlfn 27485
This theorem is referenced by:  elnlfn  27567  nlelshi  27699
  Copyright terms: Public domain W3C validator