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Theorem spanval 31090
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 31066 . 2 span = (𝑦 ∈ 𝒫 β„‹ ↦ ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝑦 βŠ† π‘₯})
2 sseq1 4002 . . . 4 (𝑦 = 𝐴 β†’ (𝑦 βŠ† π‘₯ ↔ 𝐴 βŠ† π‘₯))
32rabbidv 3434 . . 3 (𝑦 = 𝐴 β†’ {π‘₯ ∈ Sβ„‹ ∣ 𝑦 βŠ† π‘₯} = {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯})
43inteqd 4948 . 2 (𝑦 = 𝐴 β†’ ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝑦 βŠ† π‘₯} = ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯})
5 ax-hilex 30756 . . . 4 β„‹ ∈ V
65elpw2 5338 . . 3 (𝐴 ∈ 𝒫 β„‹ ↔ 𝐴 βŠ† β„‹)
76biimpri 227 . 2 (𝐴 βŠ† β„‹ β†’ 𝐴 ∈ 𝒫 β„‹)
8 helsh 31002 . . . 4 β„‹ ∈ Sβ„‹
9 sseq2 4003 . . . . 5 (π‘₯ = β„‹ β†’ (𝐴 βŠ† π‘₯ ↔ 𝐴 βŠ† β„‹))
109rspcev 3606 . . . 4 (( β„‹ ∈ Sβ„‹ ∧ 𝐴 βŠ† β„‹) β†’ βˆƒπ‘₯ ∈ Sβ„‹ 𝐴 βŠ† π‘₯)
118, 10mpan 687 . . 3 (𝐴 βŠ† β„‹ β†’ βˆƒπ‘₯ ∈ Sβ„‹ 𝐴 βŠ† π‘₯)
12 intexrab 5333 . . 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 7014 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 3064  {crab 3426  Vcvv 3468   βŠ† wss 3943  π’« cpw 4597  βˆ© cint 4943  β€˜cfv 6536   β„‹chba 30676   Sβ„‹ csh 30685  spancspn 30689
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-1cn 11167  ax-addcl 11169  ax-hilex 30756  ax-hfvadd 30757  ax-hv0cl 30760  ax-hfvmul 30762
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-map 8821  df-nn 12214  df-hlim 30729  df-sh 30964  df-ch 30978  df-span 31066
This theorem is referenced by:  spancl  31093  spanss2  31102  spanid  31104  spanss  31105  shsval3i  31145  elspani  31300
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