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Theorem obsip 21586
Description: The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypotheses
Ref Expression
isobs.v 𝑉 = (Baseβ€˜π‘Š)
isobs.h , = (Β·π‘–β€˜π‘Š)
isobs.f 𝐹 = (Scalarβ€˜π‘Š)
isobs.u 1 = (1rβ€˜πΉ)
isobs.z 0 = (0gβ€˜πΉ)
Assertion
Ref Expression
obsip ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))

Proof of Theorem obsip
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isobs.v . . . . . 6 𝑉 = (Baseβ€˜π‘Š)
2 isobs.h . . . . . 6 , = (Β·π‘–β€˜π‘Š)
3 isobs.f . . . . . 6 𝐹 = (Scalarβ€˜π‘Š)
4 isobs.u . . . . . 6 1 = (1rβ€˜πΉ)
5 isobs.z . . . . . 6 0 = (0gβ€˜πΉ)
6 eqid 2724 . . . . . 6 (ocvβ€˜π‘Š) = (ocvβ€˜π‘Š)
7 eqid 2724 . . . . . 6 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
81, 2, 3, 4, 5, 6, 7isobs 21585 . . . . 5 (𝐡 ∈ (OBasisβ€˜π‘Š) ↔ (π‘Š ∈ PreHil ∧ 𝐡 βŠ† 𝑉 ∧ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ , 𝑦) = if(π‘₯ = 𝑦, 1 , 0 ) ∧ ((ocvβ€˜π‘Š)β€˜π΅) = {(0gβ€˜π‘Š)})))
98simp3bi 1144 . . . 4 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ , 𝑦) = if(π‘₯ = 𝑦, 1 , 0 ) ∧ ((ocvβ€˜π‘Š)β€˜π΅) = {(0gβ€˜π‘Š)}))
109simpld 494 . . 3 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ , 𝑦) = if(π‘₯ = 𝑦, 1 , 0 ))
11 oveq1 7409 . . . . 5 (π‘₯ = 𝑃 β†’ (π‘₯ , 𝑦) = (𝑃 , 𝑦))
12 eqeq1 2728 . . . . . 6 (π‘₯ = 𝑃 β†’ (π‘₯ = 𝑦 ↔ 𝑃 = 𝑦))
1312ifbid 4544 . . . . 5 (π‘₯ = 𝑃 β†’ if(π‘₯ = 𝑦, 1 , 0 ) = if(𝑃 = 𝑦, 1 , 0 ))
1411, 13eqeq12d 2740 . . . 4 (π‘₯ = 𝑃 β†’ ((π‘₯ , 𝑦) = if(π‘₯ = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 )))
15 oveq2 7410 . . . . 5 (𝑦 = 𝑄 β†’ (𝑃 , 𝑦) = (𝑃 , 𝑄))
16 eqeq2 2736 . . . . . 6 (𝑦 = 𝑄 β†’ (𝑃 = 𝑦 ↔ 𝑃 = 𝑄))
1716ifbid 4544 . . . . 5 (𝑦 = 𝑄 β†’ if(𝑃 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑄, 1 , 0 ))
1815, 17eqeq12d 2740 . . . 4 (𝑦 = 𝑄 β†’ ((𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
1914, 18rspc2v 3615 . . 3 ((𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ , 𝑦) = if(π‘₯ = 𝑦, 1 , 0 ) β†’ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
2010, 19syl5com 31 . 2 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ ((𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
21203impib 1113 1 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   βŠ† wss 3941  ifcif 4521  {csn 4621  β€˜cfv 6534  (class class class)co 7402  Basecbs 17145  Scalarcsca 17201  Β·π‘–cip 17203  0gc0g 17386  1rcur 20078  PreHilcphl 21487  ocvcocv 21523  OBasiscobs 21567
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 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-obs 21570
This theorem is referenced by:  obsipid  21587  obselocv  21593
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