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Theorem elnlfn 23423
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 23376 . . . . . 6  |-  ( T : ~H --> CC  ->  (
null `  T )  =  ( `' T " { 0 } ) )
2 cnvimass 5216 . . . . . 6  |-  ( `' T " { 0 } )  C_  dom  T
31, 2syl6eqss 3390 . . . . 5  |-  ( T : ~H --> CC  ->  (
null `  T )  C_ 
dom  T )
4 fdm 5587 . . . . 5  |-  ( T : ~H --> CC  ->  dom 
T  =  ~H )
53, 4sseqtrd 3376 . . . 4  |-  ( T : ~H --> CC  ->  (
null `  T )  C_ 
~H )
65sseld 3339 . . 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 2502 . . . . 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 5583 . . . . 5  |-  ( T : ~H --> CC  ->  T  Fn  ~H )
11 eleq1 2495 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  ( `' T " { 0 } )  <->  A  e.  ( `' T " { 0 } ) ) )
12 fveq2 5720 . . . . . . . . 9  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
1312eqeq1d 2443 . . . . . . . 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 5759 . . . . . . . 8  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( ( T `  x )  =  0  <-> 
x T 0 ) )
17 0cn 9076 . . . . . . . . 9  |-  0  e.  CC
18 vex 2951 . . . . . . . . . 10  |-  x  e. 
_V
1918eliniseg 5225 . . . . . . . . 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 3009 . . . . 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 1652    e. wcel 1725   {csn 3806   class class class wbr 4204   `'ccnv 4869   dom cdm 4870   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446   CCcc 8980   0cc0 8982   ~Hchil 22414   nullcnl 22447
This theorem is referenced by:  elnlfn2  23424  nlelshi  23555  nlelchi  23556  riesz3i  23557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-mulcl 9044  ax-i2m1 9050  ax-hilex 22494
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-nlfn 23341
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