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Theorem obsip 21127
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 2736 . . . . . 6 (ocvβ€˜π‘Š) = (ocvβ€˜π‘Š)
7 eqid 2736 . . . . . 6 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
81, 2, 3, 4, 5, 6, 7isobs 21126 . . . . 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 1147 . . . 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 495 . . 3 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ , 𝑦) = if(π‘₯ = 𝑦, 1 , 0 ))
11 oveq1 7364 . . . . 5 (π‘₯ = 𝑃 β†’ (π‘₯ , 𝑦) = (𝑃 , 𝑦))
12 eqeq1 2740 . . . . . 6 (π‘₯ = 𝑃 β†’ (π‘₯ = 𝑦 ↔ 𝑃 = 𝑦))
1312ifbid 4509 . . . . 5 (π‘₯ = 𝑃 β†’ if(π‘₯ = 𝑦, 1 , 0 ) = if(𝑃 = 𝑦, 1 , 0 ))
1411, 13eqeq12d 2752 . . . 4 (π‘₯ = 𝑃 β†’ ((π‘₯ , 𝑦) = if(π‘₯ = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 )))
15 oveq2 7365 . . . . 5 (𝑦 = 𝑄 β†’ (𝑃 , 𝑦) = (𝑃 , 𝑄))
16 eqeq2 2748 . . . . . 6 (𝑦 = 𝑄 β†’ (𝑃 = 𝑦 ↔ 𝑃 = 𝑄))
1716ifbid 4509 . . . . 5 (𝑦 = 𝑄 β†’ if(𝑃 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑄, 1 , 0 ))
1815, 17eqeq12d 2752 . . . 4 (𝑦 = 𝑄 β†’ ((𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
1914, 18rspc2v 3590 . . 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 1116 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3064   βŠ† wss 3910  ifcif 4486  {csn 4586  β€˜cfv 6496  (class class class)co 7357  Basecbs 17083  Scalarcsca 17136  Β·π‘–cip 17138  0gc0g 17321  1rcur 19913  PreHilcphl 21028  ocvcocv 21064  OBasiscobs 21108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-obs 21111
This theorem is referenced by:  obsipid  21128  obselocv  21134
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