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Theorem obsip 21672
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 2725 . . . . . 6 (ocv‘𝑊) = (ocv‘𝑊)
7 eqid 2725 . . . . . 6 (0g𝑊) = (0g𝑊)
81, 2, 3, 4, 5, 6, 7isobs 21671 . . . . 5 (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g𝑊)})))
98simp3bi 1144 . . . 4 (𝐵 ∈ (OBasis‘𝑊) → (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g𝑊)}))
109simpld 493 . . 3 (𝐵 ∈ (OBasis‘𝑊) → ∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ))
11 oveq1 7426 . . . . 5 (𝑥 = 𝑃 → (𝑥 , 𝑦) = (𝑃 , 𝑦))
12 eqeq1 2729 . . . . . 6 (𝑥 = 𝑃 → (𝑥 = 𝑦𝑃 = 𝑦))
1312ifbid 4553 . . . . 5 (𝑥 = 𝑃 → if(𝑥 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑦, 1 , 0 ))
1411, 13eqeq12d 2741 . . . 4 (𝑥 = 𝑃 → ((𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 )))
15 oveq2 7427 . . . . 5 (𝑦 = 𝑄 → (𝑃 , 𝑦) = (𝑃 , 𝑄))
16 eqeq2 2737 . . . . . 6 (𝑦 = 𝑄 → (𝑃 = 𝑦𝑃 = 𝑄))
1716ifbid 4553 . . . . 5 (𝑦 = 𝑄 → if(𝑃 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑄, 1 , 0 ))
1815, 17eqeq12d 2741 . . . 4 (𝑦 = 𝑄 → ((𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
1914, 18rspc2v 3617 . . 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 1113 1 ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃𝐵𝑄𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  wss 3944  ifcif 4530  {csn 4630  cfv 6549  (class class class)co 7419  Basecbs 17183  Scalarcsca 17239  ·𝑖cip 17241  0gc0g 17424  1rcur 20133  PreHilcphl 21573  ocvcocv 21609  OBasiscobs 21653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-obs 21656
This theorem is referenced by:  obsipid  21673  obselocv  21679
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