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Theorem spanval 29104
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 ⊆ ℋ → (span‘𝐴) = {𝑥S𝐴𝑥})
Distinct variable group:   𝑥,𝐴

Proof of Theorem spanval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-span 29080 . 2 span = (𝑦 ∈ 𝒫 ℋ ↦ {𝑥S𝑦𝑥})
2 sseq1 3992 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32rabbidv 3481 . . 3 (𝑦 = 𝐴 → {𝑥S𝑦𝑥} = {𝑥S𝐴𝑥})
43inteqd 4874 . 2 (𝑦 = 𝐴 {𝑥S𝑦𝑥} = {𝑥S𝐴𝑥})
5 ax-hilex 28770 . . . 4 ℋ ∈ V
65elpw2 5241 . . 3 (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
76biimpri 230 . 2 (𝐴 ⊆ ℋ → 𝐴 ∈ 𝒫 ℋ)
8 helsh 29016 . . . 4 ℋ ∈ S
9 sseq2 3993 . . . . 5 (𝑥 = ℋ → (𝐴𝑥𝐴 ⊆ ℋ))
109rspcev 3623 . . . 4 (( ℋ ∈ S𝐴 ⊆ ℋ) → ∃𝑥S 𝐴𝑥)
118, 10mpan 688 . . 3 (𝐴 ⊆ ℋ → ∃𝑥S 𝐴𝑥)
12 intexrab 5236 . . 3 (∃𝑥S 𝐴𝑥 {𝑥S𝐴𝑥} ∈ V)
1311, 12sylib 220 . 2 (𝐴 ⊆ ℋ → {𝑥S𝐴𝑥} ∈ V)
141, 4, 7, 13fvmptd3 6786 1 (𝐴 ⊆ ℋ → (span‘𝐴) = {𝑥S𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2110  wrex 3139  {crab 3142  Vcvv 3495  wss 3936  𝒫 cpw 4539   cint 4869  cfv 6350  chba 28690   S csh 28699  spancspn 28703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-1cn 10589  ax-addcl 10591  ax-hilex 28770  ax-hfvadd 28771  ax-hv0cl 28774  ax-hfvmul 28776
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-map 8402  df-nn 11633  df-hlim 28743  df-sh 28978  df-ch 28992  df-span 29080
This theorem is referenced by:  spancl  29107  spanss2  29116  spanid  29118  spanss  29119  shsval3i  29159  elspani  29314
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