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Theorem elnlfn 22524
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 22477 . . . . . 6  |-  ( T : ~H --> CC  ->  (
null `  T )  =  ( `' T " { 0 } ) )
2 cnvimass 5049 . . . . . . 7  |-  ( `' T " { 0 } )  C_  dom  T
32a1i 10 . . . . . 6  |-  ( T : ~H --> CC  ->  ( `' T " { 0 } )  C_  dom  T )
41, 3eqsstrd 3225 . . . . 5  |-  ( T : ~H --> CC  ->  (
null `  T )  C_ 
dom  T )
5 fdm 5409 . . . . 5  |-  ( T : ~H --> CC  ->  dom 
T  =  ~H )
64, 5sseqtrd 3227 . . . 4  |-  ( T : ~H --> CC  ->  (
null `  T )  C_ 
~H )
76sseld 3192 . . 3  |-  ( T : ~H --> CC  ->  ( A  e.  ( null `  T )  ->  A  e.  ~H ) )
87pm4.71rd 616 . 2  |-  ( T : ~H --> CC  ->  ( A  e.  ( null `  T )  <->  ( A  e.  ~H  /\  A  e.  ( null `  T
) ) ) )
91eleq2d 2363 . . . . 5  |-  ( T : ~H --> CC  ->  ( A  e.  ( null `  T )  <->  A  e.  ( `' T " { 0 } ) ) )
109adantr 451 . . . 4  |-  ( ( T : ~H --> CC  /\  A  e.  ~H )  ->  ( A  e.  (
null `  T )  <->  A  e.  ( `' T " { 0 } ) ) )
11 ffn 5405 . . . . 5  |-  ( T : ~H --> CC  ->  T  Fn  ~H )
12 eleq1 2356 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  ( `' T " { 0 } )  <->  A  e.  ( `' T " { 0 } ) ) )
13 fveq2 5541 . . . . . . . . 9  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
1413eqeq1d 2304 . . . . . . . 8  |-  ( x  =  A  ->  (
( T `  x
)  =  0  <->  ( T `  A )  =  0 ) )
1512, 14bibi12d 312 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  ( `' T " { 0 } )  <->  ( T `  x )  =  0 )  <->  ( A  e.  ( `' T " { 0 } )  <-> 
( T `  A
)  =  0 ) ) )
1615imbi2d 307 . . . . . 6  |-  ( x  =  A  ->  (
( T  Fn  ~H  ->  ( x  e.  ( `' T " { 0 } )  <->  ( T `  x )  =  0 ) )  <->  ( T  Fn  ~H  ->  ( A  e.  ( `' T " { 0 } )  <-> 
( T `  A
)  =  0 ) ) ) )
17 fnbrfvb 5579 . . . . . . . 8  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( ( T `  x )  =  0  <-> 
x T 0 ) )
18 0cn 8847 . . . . . . . . 9  |-  0  e.  CC
19 vex 2804 . . . . . . . . . 10  |-  x  e. 
_V
2019eliniseg 5058 . . . . . . . . 9  |-  ( 0  e.  CC  ->  (
x  e.  ( `' T " { 0 } )  <->  x T
0 ) )
2118, 20ax-mp 8 . . . . . . . 8  |-  ( x  e.  ( `' T " { 0 } )  <-> 
x T 0 )
2217, 21syl6rbbr 255 . . . . . . 7  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( x  e.  ( `' T " { 0 } )  <->  ( T `  x )  =  0 ) )
2322expcom 424 . . . . . 6  |-  ( x  e.  ~H  ->  ( T  Fn  ~H  ->  ( x  e.  ( `' T " { 0 } )  <->  ( T `  x )  =  0 ) ) )
2416, 23vtoclga 2862 . . . . 5  |-  ( A  e.  ~H  ->  ( T  Fn  ~H  ->  ( A  e.  ( `' T " { 0 } )  <->  ( T `  A )  =  0 ) ) )
2511, 24mpan9 455 . . . 4  |-  ( ( T : ~H --> CC  /\  A  e.  ~H )  ->  ( A  e.  ( `' T " { 0 } )  <->  ( T `  A )  =  0 ) )
2610, 25bitrd 244 . . 3  |-  ( ( T : ~H --> CC  /\  A  e.  ~H )  ->  ( A  e.  (
null `  T )  <->  ( T `  A )  =  0 ) )
2726pm5.32da 622 . 2  |-  ( T : ~H --> CC  ->  ( ( A  e.  ~H  /\  A  e.  ( null `  T ) )  <->  ( A  e.  ~H  /\  ( T `
 A )  =  0 ) ) )
288, 27bitrd 244 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   {csn 3653   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271   CCcc 8751   0cc0 8753   ~Hchil 21515   nullcnl 21548
This theorem is referenced by:  elnlfn2  22525  nlelshi  22656  nlelchi  22657  riesz3i  22658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-mulcl 8815  ax-i2m1 8821  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-nlfn 22442
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