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Theorem elnlfn 26670
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 26623 . . . . . 6  |-  ( T : ~H --> CC  ->  (
null `  T )  =  ( `' T " { 0 } ) )
2 cnvimass 5363 . . . . . 6  |-  ( `' T " { 0 } )  C_  dom  T
31, 2syl6eqss 3559 . . . . 5  |-  ( T : ~H --> CC  ->  (
null `  T )  C_ 
dom  T )
4 fdm 5741 . . . . 5  |-  ( T : ~H --> CC  ->  dom 
T  =  ~H )
53, 4sseqtrd 3545 . . . 4  |-  ( T : ~H --> CC  ->  (
null `  T )  C_ 
~H )
65sseld 3508 . . 3  |-  ( T : ~H --> CC  ->  ( A  e.  ( null `  T )  ->  A  e.  ~H ) )
76pm4.71rd 635 . 2  |-  ( T : ~H --> CC  ->  ( A  e.  ( null `  T )  <->  ( A  e.  ~H  /\  A  e.  ( null `  T
) ) ) )
81eleq2d 2537 . . . . 5  |-  ( T : ~H --> CC  ->  ( A  e.  ( null `  T )  <->  A  e.  ( `' T " { 0 } ) ) )
98adantr 465 . . . 4  |-  ( ( T : ~H --> CC  /\  A  e.  ~H )  ->  ( A  e.  (
null `  T )  <->  A  e.  ( `' T " { 0 } ) ) )
10 ffn 5737 . . . . 5  |-  ( T : ~H --> CC  ->  T  Fn  ~H )
11 eleq1 2539 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  ( `' T " { 0 } )  <->  A  e.  ( `' T " { 0 } ) ) )
12 fveq2 5872 . . . . . . . . 9  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
1312eqeq1d 2469 . . . . . . . 8  |-  ( x  =  A  ->  (
( T `  x
)  =  0  <->  ( T `  A )  =  0 ) )
1411, 13bibi12d 321 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  ( `' T " { 0 } )  <->  ( T `  x )  =  0 )  <->  ( A  e.  ( `' T " { 0 } )  <-> 
( T `  A
)  =  0 ) ) )
1514imbi2d 316 . . . . . 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 5914 . . . . . . . 8  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( ( T `  x )  =  0  <-> 
x T 0 ) )
17 0cn 9600 . . . . . . . . 9  |-  0  e.  CC
18 vex 3121 . . . . . . . . . 10  |-  x  e. 
_V
1918eliniseg 5372 . . . . . . . . 9  |-  ( 0  e.  CC  ->  (
x  e.  ( `' T " { 0 } )  <->  x T
0 ) )
2017, 19ax-mp 5 . . . . . . . 8  |-  ( x  e.  ( `' T " { 0 } )  <-> 
x T 0 )
2116, 20syl6rbbr 264 . . . . . . 7  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( x  e.  ( `' T " { 0 } )  <->  ( T `  x )  =  0 ) )
2221expcom 435 . . . . . 6  |-  ( x  e.  ~H  ->  ( T  Fn  ~H  ->  ( x  e.  ( `' T " { 0 } )  <->  ( T `  x )  =  0 ) ) )
2315, 22vtoclga 3182 . . . . 5  |-  ( A  e.  ~H  ->  ( T  Fn  ~H  ->  ( A  e.  ( `' T " { 0 } )  <->  ( T `  A )  =  0 ) ) )
2410, 23mpan9 469 . . . 4  |-  ( ( T : ~H --> CC  /\  A  e.  ~H )  ->  ( A  e.  ( `' T " { 0 } )  <->  ( T `  A )  =  0 ) )
259, 24bitrd 253 . . 3  |-  ( ( T : ~H --> CC  /\  A  e.  ~H )  ->  ( A  e.  (
null `  T )  <->  ( T `  A )  =  0 ) )
2625pm5.32da 641 . 2  |-  ( T : ~H --> CC  ->  ( ( A  e.  ~H  /\  A  e.  ( null `  T ) )  <->  ( A  e.  ~H  /\  ( T `
 A )  =  0 ) ) )
277, 26bitrd 253 1  |-  ( T : ~H --> CC  ->  ( A  e.  ( null `  T )  <->  ( A  e.  ~H  /\  ( T `
 A )  =  0 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {csn 4033   class class class wbr 4453   `'ccnv 5004   dom cdm 5005   "cima 5008    Fn wfn 5589   -->wf 5590   ` cfv 5594   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-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-mulcl 9566  ax-i2m1 9572  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:  elnlfn2  26671  nlelshi  26802  nlelchi  26803  riesz3i  26804
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