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Theorem obsrcl 20416
 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 2801 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2801 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2801 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
4 eqid 2801 . . 3 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
5 eqid 2801 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
6 eqid 2801 . . 3 (ocv‘𝑊) = (ocv‘𝑊)
7 eqid 2801 . . 3 (0g𝑊) = (0g𝑊)
81, 2, 3, 4, 5, 6, 7isobs 20413 . 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 1142 1 (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∀wral 3109   ⊆ wss 3884  ifcif 4428  {csn 4528  ‘cfv 6328  (class class class)co 7139  Basecbs 16479  Scalarcsca 16564  ·𝑖cip 16566  0gc0g 16709  1rcur 19248  PreHilcphl 20317  ocvcocv 20353  OBasiscobs 20395 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-obs 20398 This theorem is referenced by:  obsne0  20418  obs2ocv  20420  obselocv  20421  obslbs  20423
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