MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  obsrcl Structured version   Visualization version   GIF version

Theorem obsrcl 21761
Description: Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
Assertion
Ref Expression
obsrcl (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)

Proof of Theorem obsrcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2735 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2735 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
4 eqid 2735 . . 3 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
5 eqid 2735 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
6 eqid 2735 . . 3 (ocv‘𝑊) = (ocv‘𝑊)
7 eqid 2735 . . 3 (0g𝑊) = (0g𝑊)
81, 2, 3, 4, 5, 6, 7isobs 21758 . 2 (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵 ⊆ (Base‘𝑊) ∧ (∀𝑥𝐵𝑦𝐵 (𝑥(·𝑖𝑊)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘𝑊)), (0g‘(Scalar‘𝑊))) ∧ ((ocv‘𝑊)‘𝐵) = {(0g𝑊)})))
98simp1bi 1144 1 (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  wss 3963  ifcif 4531  {csn 4631  cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301  ·𝑖cip 17303  0gc0g 17486  1rcur 20199  PreHilcphl 21660  ocvcocv 21696  OBasiscobs 21740
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-obs 21743
This theorem is referenced by:  obsne0  21763  obs2ocv  21765  obselocv  21766  obslbs  21768
  Copyright terms: Public domain W3C validator