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

Theorem elnlfn 22508
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 22461 . . . . . 6  |-  ( T : ~H --> CC  ->  (
null `  T )  =  ( `' T " { 0 } ) )
2 cnvimass 5033 . . . . . . 7  |-  ( `' T " { 0 } )  C_  dom  T
32a1i 10 . . . . . 6  |-  ( T : ~H --> CC  ->  ( `' T " { 0 } )  C_  dom  T )
41, 3eqsstrd 3212 . . . . 5  |-  ( T : ~H --> CC  ->  (
null `  T )  C_ 
dom  T )
5 fdm 5393 . . . . 5  |-  ( T : ~H --> CC  ->  dom 
T  =  ~H )
64, 5sseqtrd 3214 . . . 4  |-  ( T : ~H --> CC  ->  (
null `  T )  C_ 
~H )
76sseld 3179 . . 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 2350 . . . . 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 5389 . . . . 5  |-  ( T : ~H --> CC  ->  T  Fn  ~H )
12 eleq1 2343 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  ( `' T " { 0 } )  <->  A  e.  ( `' T " { 0 } ) ) )
13 fveq2 5525 . . . . . . . . 9  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
1413eqeq1d 2291 . . . . . . . 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 5563 . . . . . . . 8  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( ( T `  x )  =  0  <-> 
x T 0 ) )
18 0cn 8831 . . . . . . . . 9  |-  0  e.  CC
19 vex 2791 . . . . . . . . . 10  |-  x  e. 
_V
2019eliniseg 5042 . . . . . . . . 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 2849 . . . . 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 1623    e. wcel 1684    C_ wss 3152   {csn 3640   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255   CCcc 8735   0cc0 8737   ~Hchil 21499   nullcnl 21532
This theorem is referenced by:  elnlfn2  22509  nlelshi  22640  nlelchi  22641  riesz3i  22642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-mulcl 8799  ax-i2m1 8805  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-nlfn 22426
  Copyright terms: Public domain W3C validator