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Theorem spanval 31163
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 31139 . 2 span = (𝑦 ∈ 𝒫 β„‹ ↦ ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝑦 βŠ† π‘₯})
2 sseq1 4007 . . . 4 (𝑦 = 𝐴 β†’ (𝑦 βŠ† π‘₯ ↔ 𝐴 βŠ† π‘₯))
32rabbidv 3438 . . 3 (𝑦 = 𝐴 β†’ {π‘₯ ∈ Sβ„‹ ∣ 𝑦 βŠ† π‘₯} = {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯})
43inteqd 4958 . 2 (𝑦 = 𝐴 β†’ ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝑦 βŠ† π‘₯} = ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯})
5 ax-hilex 30829 . . . 4 β„‹ ∈ V
65elpw2 5351 . . 3 (𝐴 ∈ 𝒫 β„‹ ↔ 𝐴 βŠ† β„‹)
76biimpri 227 . 2 (𝐴 βŠ† β„‹ β†’ 𝐴 ∈ 𝒫 β„‹)
8 helsh 31075 . . . 4 β„‹ ∈ Sβ„‹
9 sseq2 4008 . . . . 5 (π‘₯ = β„‹ β†’ (𝐴 βŠ† π‘₯ ↔ 𝐴 βŠ† β„‹))
109rspcev 3611 . . . 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 5346 . . 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 7033 1 (𝐴 βŠ† β„‹ β†’ (spanβ€˜π΄) = ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3067  {crab 3430  Vcvv 3473   βŠ† wss 3949  π’« cpw 4606  βˆ© cint 4953  β€˜cfv 6553   β„‹chba 30749   Sβ„‹ csh 30758  spancspn 30762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-1cn 11204  ax-addcl 11206  ax-hilex 30829  ax-hfvadd 30830  ax-hv0cl 30833  ax-hfvmul 30835
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-map 8853  df-nn 12251  df-hlim 30802  df-sh 31037  df-ch 31051  df-span 31139
This theorem is referenced by:  spancl  31166  spanss2  31175  spanid  31177  spanss  31178  shsval3i  31218  elspani  31373
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