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Theorem elnlfn 23280
Description: Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elnlfn  |-  ( T : ~H --> CC  ->  ( A  e.  ( null `  T )  <->  ( A  e.  ~H  /\  ( T `
 A )  =  0 ) ) )

Proof of Theorem elnlfn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nlfnval 23233 . . . . . 6  |-  ( T : ~H --> CC  ->  (
null `  T )  =  ( `' T " { 0 } ) )
2 cnvimass 5165 . . . . . 6  |-  ( `' T " { 0 } )  C_  dom  T
31, 2syl6eqss 3342 . . . . 5  |-  ( T : ~H --> CC  ->  (
null `  T )  C_ 
dom  T )
4 fdm 5536 . . . . 5  |-  ( T : ~H --> CC  ->  dom 
T  =  ~H )
53, 4sseqtrd 3328 . . . 4  |-  ( T : ~H --> CC  ->  (
null `  T )  C_ 
~H )
65sseld 3291 . . 3  |-  ( T : ~H --> CC  ->  ( A  e.  ( null `  T )  ->  A  e.  ~H ) )
76pm4.71rd 617 . 2  |-  ( T : ~H --> CC  ->  ( A  e.  ( null `  T )  <->  ( A  e.  ~H  /\  A  e.  ( null `  T
) ) ) )
81eleq2d 2455 . . . . 5  |-  ( T : ~H --> CC  ->  ( A  e.  ( null `  T )  <->  A  e.  ( `' T " { 0 } ) ) )
98adantr 452 . . . 4  |-  ( ( T : ~H --> CC  /\  A  e.  ~H )  ->  ( A  e.  (
null `  T )  <->  A  e.  ( `' T " { 0 } ) ) )
10 ffn 5532 . . . . 5  |-  ( T : ~H --> CC  ->  T  Fn  ~H )
11 eleq1 2448 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  ( `' T " { 0 } )  <->  A  e.  ( `' T " { 0 } ) ) )
12 fveq2 5669 . . . . . . . . 9  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
1312eqeq1d 2396 . . . . . . . 8  |-  ( x  =  A  ->  (
( T `  x
)  =  0  <->  ( T `  A )  =  0 ) )
1411, 13bibi12d 313 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  ( `' T " { 0 } )  <->  ( T `  x )  =  0 )  <->  ( A  e.  ( `' T " { 0 } )  <-> 
( T `  A
)  =  0 ) ) )
1514imbi2d 308 . . . . . 6  |-  ( x  =  A  ->  (
( T  Fn  ~H  ->  ( x  e.  ( `' T " { 0 } )  <->  ( T `  x )  =  0 ) )  <->  ( T  Fn  ~H  ->  ( A  e.  ( `' T " { 0 } )  <-> 
( T `  A
)  =  0 ) ) ) )
16 fnbrfvb 5707 . . . . . . . 8  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( ( T `  x )  =  0  <-> 
x T 0 ) )
17 0cn 9018 . . . . . . . . 9  |-  0  e.  CC
18 vex 2903 . . . . . . . . . 10  |-  x  e. 
_V
1918eliniseg 5174 . . . . . . . . 9  |-  ( 0  e.  CC  ->  (
x  e.  ( `' T " { 0 } )  <->  x T
0 ) )
2017, 19ax-mp 8 . . . . . . . 8  |-  ( x  e.  ( `' T " { 0 } )  <-> 
x T 0 )
2116, 20syl6rbbr 256 . . . . . . 7  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( x  e.  ( `' T " { 0 } )  <->  ( T `  x )  =  0 ) )
2221expcom 425 . . . . . 6  |-  ( x  e.  ~H  ->  ( T  Fn  ~H  ->  ( x  e.  ( `' T " { 0 } )  <->  ( T `  x )  =  0 ) ) )
2315, 22vtoclga 2961 . . . . 5  |-  ( A  e.  ~H  ->  ( T  Fn  ~H  ->  ( A  e.  ( `' T " { 0 } )  <->  ( T `  A )  =  0 ) ) )
2410, 23mpan9 456 . . . 4  |-  ( ( T : ~H --> CC  /\  A  e.  ~H )  ->  ( A  e.  ( `' T " { 0 } )  <->  ( T `  A )  =  0 ) )
259, 24bitrd 245 . . 3  |-  ( ( T : ~H --> CC  /\  A  e.  ~H )  ->  ( A  e.  (
null `  T )  <->  ( T `  A )  =  0 ) )
2625pm5.32da 623 . 2  |-  ( T : ~H --> CC  ->  ( ( A  e.  ~H  /\  A  e.  ( null `  T ) )  <->  ( A  e.  ~H  /\  ( T `
 A )  =  0 ) ) )
277, 26bitrd 245 1  |-  ( T : ~H --> CC  ->  ( A  e.  ( null `  T )  <->  ( A  e.  ~H  /\  ( T `
 A )  =  0 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {csn 3758   class class class wbr 4154   `'ccnv 4818   dom cdm 4819   "cima 4822    Fn wfn 5390   -->wf 5391   ` cfv 5395   CCcc 8922   0cc0 8924   ~Hchil 22271   nullcnl 22304
This theorem is referenced by:  elnlfn2  23281  nlelshi  23412  nlelchi  23413  riesz3i  23414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-mulcl 8986  ax-i2m1 8992  ax-hilex 22351
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-map 6957  df-nlfn 23198
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