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Theorem obsss 21651
Description: An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypothesis
Ref Expression
obsss.v 𝑉 = (Baseβ€˜π‘Š)
Assertion
Ref Expression
obsss (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ 𝐡 βŠ† 𝑉)

Proof of Theorem obsss
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 obsss.v . . 3 𝑉 = (Baseβ€˜π‘Š)
2 eqid 2728 . . 3 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
3 eqid 2728 . . 3 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
4 eqid 2728 . . 3 (1rβ€˜(Scalarβ€˜π‘Š)) = (1rβ€˜(Scalarβ€˜π‘Š))
5 eqid 2728 . . 3 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
6 eqid 2728 . . 3 (ocvβ€˜π‘Š) = (ocvβ€˜π‘Š)
7 eqid 2728 . . 3 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
81, 2, 3, 4, 5, 6, 7isobs 21647 . 2 (𝐡 ∈ (OBasisβ€˜π‘Š) ↔ (π‘Š ∈ PreHil ∧ 𝐡 βŠ† 𝑉 ∧ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = if(π‘₯ = 𝑦, (1rβ€˜(Scalarβ€˜π‘Š)), (0gβ€˜(Scalarβ€˜π‘Š))) ∧ ((ocvβ€˜π‘Š)β€˜π΅) = {(0gβ€˜π‘Š)})))
98simp2bi 1144 1 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ 𝐡 βŠ† 𝑉)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = 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:  obsne0  21652  obselocv  21655  obslbs  21657
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