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

Theorem obsrcl 20392
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 2799 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2799 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2799 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
4 eqid 2799 . . 3 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
5 eqid 2799 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
6 eqid 2799 . . 3 (ocv‘𝑊) = (ocv‘𝑊)
7 eqid 2799 . . 3 (0g𝑊) = (0g𝑊)
81, 2, 3, 4, 5, 6, 7isobs 20389 . 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 1176 1 (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089  wss 3769  ifcif 4277  {csn 4368  cfv 6101  (class class class)co 6878  Basecbs 16184  Scalarcsca 16270  ·𝑖cip 16272  0gc0g 16415  1rcur 18817  PreHilcphl 20293  ocvcocv 20329  OBasiscobs 20371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-obs 20374
This theorem is referenced by:  obsne0  20394  obs2ocv  20396  obselocv  20397  obslbs  20399
  Copyright terms: Public domain W3C validator