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Theorem obsocv 21148
Description: An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypotheses
Ref Expression
obsocv.z 0 = (0gβ€˜π‘Š)
obsocv.o βŠ₯ = (ocvβ€˜π‘Š)
Assertion
Ref Expression
obsocv (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ ( βŠ₯ β€˜π΅) = { 0 })

Proof of Theorem obsocv
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 eqid 2733 . . . 4 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
3 eqid 2733 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
4 eqid 2733 . . . 4 (1rβ€˜(Scalarβ€˜π‘Š)) = (1rβ€˜(Scalarβ€˜π‘Š))
5 eqid 2733 . . . 4 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
6 obsocv.o . . . 4 βŠ₯ = (ocvβ€˜π‘Š)
7 obsocv.z . . . 4 0 = (0gβ€˜π‘Š)
81, 2, 3, 4, 5, 6, 7isobs 21142 . . 3 (𝐡 ∈ (OBasisβ€˜π‘Š) ↔ (π‘Š ∈ PreHil ∧ 𝐡 βŠ† (Baseβ€˜π‘Š) ∧ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = if(π‘₯ = 𝑦, (1rβ€˜(Scalarβ€˜π‘Š)), (0gβ€˜(Scalarβ€˜π‘Š))) ∧ ( βŠ₯ β€˜π΅) = { 0 })))
98simp3bi 1148 . 2 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = if(π‘₯ = 𝑦, (1rβ€˜(Scalarβ€˜π‘Š)), (0gβ€˜(Scalarβ€˜π‘Š))) ∧ ( βŠ₯ β€˜π΅) = { 0 }))
109simprd 497 1 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ ( βŠ₯ β€˜π΅) = { 0 })
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   βŠ† wss 3911  ifcif 4487  {csn 4587  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  Scalarcsca 17141  Β·π‘–cip 17143  0gc0g 17326  1rcur 19918  PreHilcphl 21044  ocvcocv 21080  OBasiscobs 21124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-obs 21127
This theorem is referenced by:  obs2ocv  21149  obs2ss  21151
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