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Theorem spanval 26978
Description: Value of the linear span of a subset of Hilbert space. The span is the intersection of all subspaces constraining the subset. Definition of span in [Schechter] p. 276. (Contributed by NM, 2-Jun-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
spanval  |-  ( A 
C_  ~H  ->  ( span `  A )  =  |^| { x  e.  SH  |  A  C_  x } )
Distinct variable group:    x, A

Proof of Theorem spanval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 26644 . . . 4  |-  ~H  e.  _V
21elpw2 4586 . . 3  |-  ( A  e.  ~P ~H  <->  A  C_  ~H )
32biimpri 210 . 2  |-  ( A 
C_  ~H  ->  A  e. 
~P ~H )
4 helsh 26890 . . . 4  |-  ~H  e.  SH
5 sseq2 3487 . . . . 5  |-  ( x  =  ~H  ->  ( A  C_  x  <->  A  C_  ~H ) )
65rspcev 3183 . . . 4  |-  ( ( ~H  e.  SH  /\  A  C_  ~H )  ->  E. x  e.  SH  A  C_  x )
74, 6mpan 675 . . 3  |-  ( A 
C_  ~H  ->  E. x  e.  SH  A  C_  x
)
8 intexrab 4581 . . 3  |-  ( E. x  e.  SH  A  C_  x  <->  |^| { x  e.  SH  |  A  C_  x }  e.  _V )
97, 8sylib 200 . 2  |-  ( A 
C_  ~H  ->  |^| { x  e.  SH  |  A  C_  x }  e.  _V )
10 sseq1 3486 . . . . 5  |-  ( y  =  A  ->  (
y  C_  x  <->  A  C_  x
) )
1110rabbidv 3073 . . . 4  |-  ( y  =  A  ->  { x  e.  SH  |  y  C_  x }  =  {
x  e.  SH  |  A  C_  x } )
1211inteqd 4258 . . 3  |-  ( y  =  A  ->  |^| { x  e.  SH  |  y  C_  x }  =  |^| { x  e.  SH  |  A  C_  x } )
13 df-span 26954 . . 3  |-  span  =  ( y  e.  ~P ~H  |->  |^| { x  e.  SH  |  y  C_  x } )
1412, 13fvmptg 5960 . 2  |-  ( ( A  e.  ~P ~H  /\ 
|^| { x  e.  SH  |  A  C_  x }  e.  _V )  ->  ( span `  A )  = 
|^| { x  e.  SH  |  A  C_  x }
)
153, 9, 14syl2anc 666 1  |-  ( A 
C_  ~H  ->  ( span `  A )  =  |^| { x  e.  SH  |  A  C_  x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1438    e. wcel 1869   E.wrex 2777   {crab 2780   _Vcvv 3082    C_ wss 3437   ~Pcpw 3980   |^|cint 4253   ` cfv 5599   ~Hchil 26564   SHcsh 26573   spancspn 26577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-i2m1 9609  ax-1ne0 9610  ax-rrecex 9613  ax-cnre 9614  ax-hilex 26644  ax-hfvadd 26645  ax-hv0cl 26648  ax-hfvmul 26650
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-map 7480  df-nn 10612  df-hlim 26617  df-sh 26852  df-ch 26866  df-span 26954
This theorem is referenced by:  spancl  26981  spanss2  26990  spanid  26992  spanss  26993  shsval3i  27033  elspani  27188
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