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Theorem obsip 20387
Description: The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypotheses
Ref Expression
isobs.v 𝑉 = (Base‘𝑊)
isobs.h , = (·𝑖𝑊)
isobs.f 𝐹 = (Scalar‘𝑊)
isobs.u 1 = (1r𝐹)
isobs.z 0 = (0g𝐹)
Assertion
Ref Expression
obsip ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃𝐵𝑄𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))

Proof of Theorem obsip
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isobs.v . . . . . 6 𝑉 = (Base‘𝑊)
2 isobs.h . . . . . 6 , = (·𝑖𝑊)
3 isobs.f . . . . . 6 𝐹 = (Scalar‘𝑊)
4 isobs.u . . . . . 6 1 = (1r𝐹)
5 isobs.z . . . . . 6 0 = (0g𝐹)
6 eqid 2797 . . . . . 6 (ocv‘𝑊) = (ocv‘𝑊)
7 eqid 2797 . . . . . 6 (0g𝑊) = (0g𝑊)
81, 2, 3, 4, 5, 6, 7isobs 20386 . . . . 5 (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g𝑊)})))
98simp3bi 1178 . . . 4 (𝐵 ∈ (OBasis‘𝑊) → (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g𝑊)}))
109simpld 489 . . 3 (𝐵 ∈ (OBasis‘𝑊) → ∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ))
11 oveq1 6883 . . . . 5 (𝑥 = 𝑃 → (𝑥 , 𝑦) = (𝑃 , 𝑦))
12 eqeq1 2801 . . . . . 6 (𝑥 = 𝑃 → (𝑥 = 𝑦𝑃 = 𝑦))
1312ifbid 4297 . . . . 5 (𝑥 = 𝑃 → if(𝑥 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑦, 1 , 0 ))
1411, 13eqeq12d 2812 . . . 4 (𝑥 = 𝑃 → ((𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 )))
15 oveq2 6884 . . . . 5 (𝑦 = 𝑄 → (𝑃 , 𝑦) = (𝑃 , 𝑄))
16 eqeq2 2808 . . . . . 6 (𝑦 = 𝑄 → (𝑃 = 𝑦𝑃 = 𝑄))
1716ifbid 4297 . . . . 5 (𝑦 = 𝑄 → if(𝑃 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑄, 1 , 0 ))
1815, 17eqeq12d 2812 . . . 4 (𝑦 = 𝑄 → ((𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
1914, 18rspc2v 3508 . . 3 ((𝑃𝐵𝑄𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
2010, 19syl5com 31 . 2 (𝐵 ∈ (OBasis‘𝑊) → ((𝑃𝐵𝑄𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
21203impib 1145 1 ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃𝐵𝑄𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3087  wss 3767  ifcif 4275  {csn 4366  cfv 6099  (class class class)co 6876  Basecbs 16181  Scalarcsca 16267  ·𝑖cip 16269  0gc0g 16412  1rcur 18814  PreHilcphl 20290  ocvcocv 20326  OBasiscobs 20368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fv 6107  df-ov 6879  df-obs 20371
This theorem is referenced by:  obsipid  20388  obselocv  20394
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