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Theorem obsip 20267
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 2760 . . . . . 6 (ocv‘𝑊) = (ocv‘𝑊)
7 eqid 2760 . . . . . 6 (0g𝑊) = (0g𝑊)
81, 2, 3, 4, 5, 6, 7isobs 20266 . . . . 5 (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g𝑊)})))
98simp3bi 1142 . . . 4 (𝐵 ∈ (OBasis‘𝑊) → (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g𝑊)}))
109simpld 477 . . 3 (𝐵 ∈ (OBasis‘𝑊) → ∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ))
11 oveq1 6820 . . . . 5 (𝑥 = 𝑃 → (𝑥 , 𝑦) = (𝑃 , 𝑦))
12 eqeq1 2764 . . . . . 6 (𝑥 = 𝑃 → (𝑥 = 𝑦𝑃 = 𝑦))
1312ifbid 4252 . . . . 5 (𝑥 = 𝑃 → if(𝑥 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑦, 1 , 0 ))
1411, 13eqeq12d 2775 . . . 4 (𝑥 = 𝑃 → ((𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 )))
15 oveq2 6821 . . . . 5 (𝑦 = 𝑄 → (𝑃 , 𝑦) = (𝑃 , 𝑄))
16 eqeq2 2771 . . . . . 6 (𝑦 = 𝑄 → (𝑃 = 𝑦𝑃 = 𝑄))
1716ifbid 4252 . . . . 5 (𝑦 = 𝑄 → if(𝑃 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑄, 1 , 0 ))
1815, 17eqeq12d 2775 . . . 4 (𝑦 = 𝑄 → ((𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
1914, 18rspc2v 3461 . . 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 1109 1 ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃𝐵𝑄𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wss 3715  ifcif 4230  {csn 4321  cfv 6049  (class class class)co 6813  Basecbs 16059  Scalarcsca 16146  ·𝑖cip 16148  0gc0g 16302  1rcur 18701  PreHilcphl 20171  ocvcocv 20206  OBasiscobs 20248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-obs 20251
This theorem is referenced by:  obsipid  20268  obselocv  20274
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