HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  spanval Structured version   Visualization version   GIF version

Theorem spanval 29268
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 29244 . 2 span = (𝑦 ∈ 𝒫 ℋ ↦ {𝑥S𝑦𝑥})
2 sseq1 3902 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32rabbidv 3381 . . 3 (𝑦 = 𝐴 → {𝑥S𝑦𝑥} = {𝑥S𝐴𝑥})
43inteqd 4841 . 2 (𝑦 = 𝐴 {𝑥S𝑦𝑥} = {𝑥S𝐴𝑥})
5 ax-hilex 28934 . . . 4 ℋ ∈ V
65elpw2 5213 . . 3 (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
76biimpri 231 . 2 (𝐴 ⊆ ℋ → 𝐴 ∈ 𝒫 ℋ)
8 helsh 29180 . . . 4 ℋ ∈ S
9 sseq2 3903 . . . . 5 (𝑥 = ℋ → (𝐴𝑥𝐴 ⊆ ℋ))
109rspcev 3526 . . . 4 (( ℋ ∈ S𝐴 ⊆ ℋ) → ∃𝑥S 𝐴𝑥)
118, 10mpan 690 . . 3 (𝐴 ⊆ ℋ → ∃𝑥S 𝐴𝑥)
12 intexrab 5208 . . 3 (∃𝑥S 𝐴𝑥 {𝑥S𝐴𝑥} ∈ V)
1311, 12sylib 221 . 2 (𝐴 ⊆ ℋ → {𝑥S𝐴𝑥} ∈ V)
141, 4, 7, 13fvmptd3 6798 1 (𝐴 ⊆ ℋ → (span‘𝐴) = {𝑥S𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  wrex 3054  {crab 3057  Vcvv 3398  wss 3843  𝒫 cpw 4488   cint 4836  cfv 6339  chba 28854   S csh 28863  spancspn 28867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-1cn 10673  ax-addcl 10675  ax-hilex 28934  ax-hfvadd 28935  ax-hv0cl 28938  ax-hfvmul 28940
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-map 8439  df-nn 11717  df-hlim 28907  df-sh 29142  df-ch 29156  df-span 29244
This theorem is referenced by:  spancl  29271  spanss2  29280  spanid  29282  spanss  29283  shsval3i  29323  elspani  29478
  Copyright terms: Public domain W3C validator