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Theorem obsocv 20287
 Description: An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypotheses
Ref Expression
obsocv.z 0 = (0g𝑊)
obsocv.o = (ocv‘𝑊)
Assertion
Ref Expression
obsocv (𝐵 ∈ (OBasis‘𝑊) → ( 𝐵) = { 0 })

Proof of Theorem obsocv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2771 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2771 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2771 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
4 eqid 2771 . . . 4 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
5 eqid 2771 . . . 4 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
6 obsocv.o . . . 4 = (ocv‘𝑊)
7 obsocv.z . . . 4 0 = (0g𝑊)
81, 2, 3, 4, 5, 6, 7isobs 20281 . . 3 (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵 ⊆ (Base‘𝑊) ∧ (∀𝑥𝐵𝑦𝐵 (𝑥(·𝑖𝑊)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘𝑊)), (0g‘(Scalar‘𝑊))) ∧ ( 𝐵) = { 0 })))
98simp3bi 1141 . 2 (𝐵 ∈ (OBasis‘𝑊) → (∀𝑥𝐵𝑦𝐵 (𝑥(·𝑖𝑊)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘𝑊)), (0g‘(Scalar‘𝑊))) ∧ ( 𝐵) = { 0 }))
109simprd 483 1 (𝐵 ∈ (OBasis‘𝑊) → ( 𝐵) = { 0 })
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061   ⊆ wss 3723  ifcif 4225  {csn 4316  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  Scalarcsca 16152  ·𝑖cip 16154  0gc0g 16308  1rcur 18709  PreHilcphl 20186  ocvcocv 20221  OBasiscobs 20263 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-obs 20266 This theorem is referenced by:  obs2ocv  20288  obs2ss  20290
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