HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  spanval Structured version   Visualization version   GIF version
Theorem spanval 30573
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 βŠ† β„‹ β†’ (spanβ€˜π΄) = ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯})
Distinct variable group:   π‘₯,𝐴
Proof of Theorem spanval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-span 30549 . 2 span = (𝑦 ∈ 𝒫 β„‹ ↦ ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝑦 βŠ† π‘₯})
2 sseq1 4006 . . . 4 (𝑦 = 𝐴 β†’ (𝑦 βŠ† π‘₯ ↔ 𝐴 βŠ† π‘₯))
32rabbidv 3440 . . 3 (𝑦 = 𝐴 β†’ {π‘₯ ∈ Sβ„‹ ∣ 𝑦 βŠ† π‘₯} = {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯})
43inteqd 4954 . 2 (𝑦 = 𝐴 β†’ ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝑦 βŠ† π‘₯} = ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯})
5 ax-hilex 30239 . . . 4 β„‹ ∈ V
65elpw2 5344 . . 3 (𝐴 ∈ 𝒫 β„‹ ↔ 𝐴 βŠ† β„‹)
76biimpri 227 . 2 (𝐴 βŠ† β„‹ β†’ 𝐴 ∈ 𝒫 β„‹)
8 helsh 30485 . . . 4 β„‹ ∈ Sβ„‹
9 sseq2 4007 . . . . 5 (π‘₯ = β„‹ β†’ (𝐴 βŠ† π‘₯ ↔ 𝐴 βŠ† β„‹))
109rspcev 3612 . . . 4 (( β„‹ ∈ Sβ„‹ ∧ 𝐴 βŠ† β„‹) β†’ βˆƒπ‘₯ ∈ Sβ„‹ 𝐴 βŠ† π‘₯)
118, 10mpan 688 . . 3 (𝐴 βŠ† β„‹ β†’ βˆƒπ‘₯ ∈ Sβ„‹ 𝐴 βŠ† π‘₯)
12 intexrab 5339 . . 3 (βˆƒπ‘₯ ∈ Sβ„‹ 𝐴 βŠ† π‘₯ ↔ ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯} ∈ V)
1311, 12sylib 217 . 2 (𝐴 βŠ† β„‹ β†’ ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯} ∈ V)
141, 4, 7, 13fvmptd3 7018 1 (𝐴 βŠ† β„‹ β†’ (spanβ€˜π΄) = ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βŠ† wss 3947  π’« cpw 4601  βˆ© cint 4949  β€˜cfv 6540   β„‹chba 30159   Sβ„‹ csh 30168  spancspn 30172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-1cn 11164  ax-addcl 11166  ax-hilex 30239  ax-hfvadd 30240  ax-hv0cl 30243  ax-hfvmul 30245
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-map 8818  df-nn 12209  df-hlim 30212  df-sh 30447  df-ch 30461  df-span 30549
This theorem is referenced by:  spancl  30576  spanss2  30585  spanid  30587  spanss  30588  shsval3i  30628  elspani  30783
  Copyright terms: Public domain W3C validator