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Theorem obsss 20796
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 2818 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2818 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
4 eqid 2818 . . 3 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
5 eqid 2818 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
6 eqid 2818 . . 3 (ocv‘𝑊) = (ocv‘𝑊)
7 eqid 2818 . . 3 (0g𝑊) = (0g𝑊)
81, 2, 3, 4, 5, 6, 7isobs 20792 . 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 1138 1 (𝐵 ∈ (OBasis‘𝑊) → 𝐵𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wss 3933  ifcif 4463  {csn 4557  cfv 6348  (class class class)co 7145  Basecbs 16471  Scalarcsca 16556  ·𝑖cip 16558  0gc0g 16701  1rcur 19180  PreHilcphl 20696  ocvcocv 20732  OBasiscobs 20774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-obs 20777
This theorem is referenced by:  obsne0  20797  obselocv  20800  obslbs  20802
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