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Theorem spanval 25913
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 25578 . . . 4  |-  ~H  e.  _V
21elpw2 4604 . . 3  |-  ( A  e.  ~P ~H  <->  A  C_  ~H )
32biimpri 206 . 2  |-  ( A 
C_  ~H  ->  A  e. 
~P ~H )
4 helsh 25825 . . . 4  |-  ~H  e.  SH
5 sseq2 3519 . . . . 5  |-  ( x  =  ~H  ->  ( A  C_  x  <->  A  C_  ~H ) )
65rspcev 3207 . . . 4  |-  ( ( ~H  e.  SH  /\  A  C_  ~H )  ->  E. x  e.  SH  A  C_  x )
74, 6mpan 670 . . 3  |-  ( A 
C_  ~H  ->  E. x  e.  SH  A  C_  x
)
8 intexrab 4599 . . 3  |-  ( E. x  e.  SH  A  C_  x  <->  |^| { x  e.  SH  |  A  C_  x }  e.  _V )
97, 8sylib 196 . 2  |-  ( A 
C_  ~H  ->  |^| { x  e.  SH  |  A  C_  x }  e.  _V )
10 sseq1 3518 . . . . 5  |-  ( y  =  A  ->  (
y  C_  x  <->  A  C_  x
) )
1110rabbidv 3098 . . . 4  |-  ( y  =  A  ->  { x  e.  SH  |  y  C_  x }  =  {
x  e.  SH  |  A  C_  x } )
1211inteqd 4280 . . 3  |-  ( y  =  A  ->  |^| { x  e.  SH  |  y  C_  x }  =  |^| { x  e.  SH  |  A  C_  x } )
13 df-span 25889 . . 3  |-  span  =  ( y  e.  ~P ~H  |->  |^| { x  e.  SH  |  y  C_  x } )
1412, 13fvmptg 5939 . 2  |-  ( ( A  e.  ~P ~H  /\ 
|^| { x  e.  SH  |  A  C_  x }  e.  _V )  ->  ( span `  A )  = 
|^| { x  e.  SH  |  A  C_  x }
)
153, 9, 14syl2anc 661 1  |-  ( A 
C_  ~H  ->  ( span `  A )  =  |^| { x  e.  SH  |  A  C_  x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   E.wrex 2808   {crab 2811   _Vcvv 3106    C_ wss 3469   ~Pcpw 4003   |^|cint 4275   ` cfv 5579   ~Hchil 25498   SHcsh 25507   spancspn 25511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rrecex 9553  ax-cnre 9554  ax-hilex 25578  ax-hfvadd 25579  ax-hv0cl 25582  ax-hfvmul 25584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-map 7412  df-nn 10526  df-hlim 25551  df-sh 25786  df-ch 25801  df-span 25889
This theorem is referenced by:  spancl  25916  spanss2  25925  spanid  25927  spanss  25928  shsval3i  25968  elspani  26123
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