MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  obsip Structured version   Visualization version   GIF version

Theorem obsip 21648
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 2728 . . . . . 6 (ocvβ€˜π‘Š) = (ocvβ€˜π‘Š)
7 eqid 2728 . . . . . 6 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
81, 2, 3, 4, 5, 6, 7isobs 21647 . . . . 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 1145 . . . 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 7421 . . . . 5 (π‘₯ = 𝑃 β†’ (π‘₯ , 𝑦) = (𝑃 , 𝑦))
12 eqeq1 2732 . . . . . 6 (π‘₯ = 𝑃 β†’ (π‘₯ = 𝑦 ↔ 𝑃 = 𝑦))
1312ifbid 4547 . . . . 5 (π‘₯ = 𝑃 β†’ if(π‘₯ = 𝑦, 1 , 0 ) = if(𝑃 = 𝑦, 1 , 0 ))
1411, 13eqeq12d 2744 . . . 4 (π‘₯ = 𝑃 β†’ ((π‘₯ , 𝑦) = if(π‘₯ = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 )))
15 oveq2 7422 . . . . 5 (𝑦 = 𝑄 β†’ (𝑃 , 𝑦) = (𝑃 , 𝑄))
16 eqeq2 2740 . . . . . 6 (𝑦 = 𝑄 β†’ (𝑃 = 𝑦 ↔ 𝑃 = 𝑄))
1716ifbid 4547 . . . . 5 (𝑦 = 𝑄 β†’ if(𝑃 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑄, 1 , 0 ))
1815, 17eqeq12d 2744 . . . 4 (𝑦 = 𝑄 β†’ ((𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
1914, 18rspc2v 3619 . . 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 1114 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 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3057   βŠ† wss 3945  ifcif 4524  {csn 4624  β€˜cfv 6542  (class class class)co 7414  Basecbs 17173  Scalarcsca 17229  Β·π‘–cip 17231  0gc0g 17414  1rcur 20114  PreHilcphl 21549  ocvcocv 21585  OBasiscobs 21629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-obs 21632
This theorem is referenced by:  obsipid  21649  obselocv  21655
  Copyright terms: Public domain W3C validator