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Theorem obsss 20868
Description: An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypothesis
Ref Expression
obsss.v 𝑉 = (Base‘𝑊)
Assertion
Ref Expression
obsss (𝐵 ∈ (OBasis‘𝑊) → 𝐵𝑉)

Proof of Theorem obsss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 obsss.v . . 3 𝑉 = (Base‘𝑊)
2 eqid 2824 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2824 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
4 eqid 2824 . . 3 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
5 eqid 2824 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
6 eqid 2824 . . 3 (ocv‘𝑊) = (ocv‘𝑊)
7 eqid 2824 . . 3 (0g𝑊) = (0g𝑊)
81, 2, 3, 4, 5, 6, 7isobs 20864 . 2 (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥(·𝑖𝑊)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘𝑊)), (0g‘(Scalar‘𝑊))) ∧ ((ocv‘𝑊)‘𝐵) = {(0g𝑊)})))
98simp2bi 1143 1 (𝐵 ∈ (OBasis‘𝑊) → 𝐵𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  wss 3919  ifcif 4450  {csn 4550  cfv 6343  (class class class)co 7149  Basecbs 16483  Scalarcsca 16568  ·𝑖cip 16570  0gc0g 16713  1rcur 19251  PreHilcphl 20768  ocvcocv 20804  OBasiscobs 20846
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-obs 20849
This theorem is referenced by:  obsne0  20869  obselocv  20872  obslbs  20874
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