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Theorem spanval 22823
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 22490 . . . 4  |-  ~H  e.  _V
21elpw2 4356 . . 3  |-  ( A  e.  ~P ~H  <->  A  C_  ~H )
32biimpri 198 . 2  |-  ( A 
C_  ~H  ->  A  e. 
~P ~H )
4 helsh 22735 . . . 4  |-  ~H  e.  SH
5 sseq2 3362 . . . . 5  |-  ( x  =  ~H  ->  ( A  C_  x  <->  A  C_  ~H ) )
65rspcev 3044 . . . 4  |-  ( ( ~H  e.  SH  /\  A  C_  ~H )  ->  E. x  e.  SH  A  C_  x )
74, 6mpan 652 . . 3  |-  ( A 
C_  ~H  ->  E. x  e.  SH  A  C_  x
)
8 intexrab 4351 . . 3  |-  ( E. x  e.  SH  A  C_  x  <->  |^| { x  e.  SH  |  A  C_  x }  e.  _V )
97, 8sylib 189 . 2  |-  ( A 
C_  ~H  ->  |^| { x  e.  SH  |  A  C_  x }  e.  _V )
10 sseq1 3361 . . . . 5  |-  ( y  =  A  ->  (
y  C_  x  <->  A  C_  x
) )
1110rabbidv 2940 . . . 4  |-  ( y  =  A  ->  { x  e.  SH  |  y  C_  x }  =  {
x  e.  SH  |  A  C_  x } )
1211inteqd 4047 . . 3  |-  ( y  =  A  ->  |^| { x  e.  SH  |  y  C_  x }  =  |^| { x  e.  SH  |  A  C_  x } )
13 df-span 22799 . . 3  |-  span  =  ( y  e.  ~P ~H  |->  |^| { x  e.  SH  |  y  C_  x } )
1412, 13fvmptg 5795 . 2  |-  ( ( A  e.  ~P ~H  /\ 
|^| { x  e.  SH  |  A  C_  x }  e.  _V )  ->  ( span `  A )  = 
|^| { x  e.  SH  |  A  C_  x }
)
153, 9, 14syl2anc 643 1  |-  ( A 
C_  ~H  ->  ( span `  A )  =  |^| { x  e.  SH  |  A  C_  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   E.wrex 2698   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   |^|cint 4042   ` cfv 5445   ~Hchil 22410   SHcsh 22419   spancspn 22423
This theorem is referenced by:  spancl  22826  spanss2  22835  spanid  22837  spanss  22838  shsval3i  22878  elspani  23033
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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-i2m1 9047  ax-1ne0 9048  ax-rrecex 9051  ax-cnre 9052  ax-hilex 22490  ax-hfvadd 22491  ax-hv0cl 22494  ax-hfvmul 22496
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-recs 6624  df-rdg 6659  df-map 7011  df-nn 9990  df-hlim 22463  df-sh 22697  df-ch 22712  df-span 22799
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