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Theorem obsip 21759
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 2735 . . . . . 6 (ocv‘𝑊) = (ocv‘𝑊)
7 eqid 2735 . . . . . 6 (0g𝑊) = (0g𝑊)
81, 2, 3, 4, 5, 6, 7isobs 21758 . . . . 5 (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g𝑊)})))
98simp3bi 1146 . . . 4 (𝐵 ∈ (OBasis‘𝑊) → (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g𝑊)}))
109simpld 494 . . 3 (𝐵 ∈ (OBasis‘𝑊) → ∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ))
11 oveq1 7438 . . . . 5 (𝑥 = 𝑃 → (𝑥 , 𝑦) = (𝑃 , 𝑦))
12 eqeq1 2739 . . . . . 6 (𝑥 = 𝑃 → (𝑥 = 𝑦𝑃 = 𝑦))
1312ifbid 4554 . . . . 5 (𝑥 = 𝑃 → if(𝑥 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑦, 1 , 0 ))
1411, 13eqeq12d 2751 . . . 4 (𝑥 = 𝑃 → ((𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 )))
15 oveq2 7439 . . . . 5 (𝑦 = 𝑄 → (𝑃 , 𝑦) = (𝑃 , 𝑄))
16 eqeq2 2747 . . . . . 6 (𝑦 = 𝑄 → (𝑃 = 𝑦𝑃 = 𝑄))
1716ifbid 4554 . . . . 5 (𝑦 = 𝑄 → if(𝑃 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑄, 1 , 0 ))
1815, 17eqeq12d 2751 . . . 4 (𝑦 = 𝑄 → ((𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
1914, 18rspc2v 3633 . . 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 1115 1 ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃𝐵𝑄𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wss 3963  ifcif 4531  {csn 4631  cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301  ·𝑖cip 17303  0gc0g 17486  1rcur 20199  PreHilcphl 21660  ocvcocv 21696  OBasiscobs 21740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-obs 21743
This theorem is referenced by:  obsipid  21760  obselocv  21766
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