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Definition df-obs 21127
Description: Define the set of all orthonormal bases for a pre-Hilbert space. An orthonormal basis is a set of mutually orthogonal vectors with norm 1 and such that the linear span is dense in the whole space. (As this is an "algebraic" definition, before we have topology available, we express this denseness by saying that the double orthocomplement is the whole space, or equivalently, the single orthocomplement is trivial.) (Contributed by Mario Carneiro, 23-Oct-2015.)
Assertion
Ref Expression
df-obs OBasis = (β„Ž ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Baseβ€˜β„Ž) ∣ (βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯(Β·π‘–β€˜β„Ž)𝑦) = if(π‘₯ = 𝑦, (1rβ€˜(Scalarβ€˜β„Ž)), (0gβ€˜(Scalarβ€˜β„Ž))) ∧ ((ocvβ€˜β„Ž)β€˜π‘) = {(0gβ€˜β„Ž)})})
Distinct variable group:   β„Ž,𝑏,π‘₯,𝑦

Detailed syntax breakdown of Definition df-obs
StepHypRef Expression
1 cobs 21124 . 2 class OBasis
2 vh . . 3 setvar β„Ž
3 cphl 21044 . . 3 class PreHil
4 vx . . . . . . . . . 10 setvar π‘₯
54cv 1541 . . . . . . . . 9 class π‘₯
6 vy . . . . . . . . . 10 setvar 𝑦
76cv 1541 . . . . . . . . 9 class 𝑦
82cv 1541 . . . . . . . . . 10 class β„Ž
9 cip 17143 . . . . . . . . . 10 class ·𝑖
108, 9cfv 6497 . . . . . . . . 9 class (Β·π‘–β€˜β„Ž)
115, 7, 10co 7358 . . . . . . . 8 class (π‘₯(Β·π‘–β€˜β„Ž)𝑦)
124, 6weq 1967 . . . . . . . . 9 wff π‘₯ = 𝑦
13 csca 17141 . . . . . . . . . . 11 class Scalar
148, 13cfv 6497 . . . . . . . . . 10 class (Scalarβ€˜β„Ž)
15 cur 19918 . . . . . . . . . 10 class 1r
1614, 15cfv 6497 . . . . . . . . 9 class (1rβ€˜(Scalarβ€˜β„Ž))
17 c0g 17326 . . . . . . . . . 10 class 0g
1814, 17cfv 6497 . . . . . . . . 9 class (0gβ€˜(Scalarβ€˜β„Ž))
1912, 16, 18cif 4487 . . . . . . . 8 class if(π‘₯ = 𝑦, (1rβ€˜(Scalarβ€˜β„Ž)), (0gβ€˜(Scalarβ€˜β„Ž)))
2011, 19wceq 1542 . . . . . . 7 wff (π‘₯(Β·π‘–β€˜β„Ž)𝑦) = if(π‘₯ = 𝑦, (1rβ€˜(Scalarβ€˜β„Ž)), (0gβ€˜(Scalarβ€˜β„Ž)))
21 vb . . . . . . . 8 setvar 𝑏
2221cv 1541 . . . . . . 7 class 𝑏
2320, 6, 22wral 3061 . . . . . 6 wff βˆ€π‘¦ ∈ 𝑏 (π‘₯(Β·π‘–β€˜β„Ž)𝑦) = if(π‘₯ = 𝑦, (1rβ€˜(Scalarβ€˜β„Ž)), (0gβ€˜(Scalarβ€˜β„Ž)))
2423, 4, 22wral 3061 . . . . 5 wff βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯(Β·π‘–β€˜β„Ž)𝑦) = if(π‘₯ = 𝑦, (1rβ€˜(Scalarβ€˜β„Ž)), (0gβ€˜(Scalarβ€˜β„Ž)))
25 cocv 21080 . . . . . . . 8 class ocv
268, 25cfv 6497 . . . . . . 7 class (ocvβ€˜β„Ž)
2722, 26cfv 6497 . . . . . 6 class ((ocvβ€˜β„Ž)β€˜π‘)
288, 17cfv 6497 . . . . . . 7 class (0gβ€˜β„Ž)
2928csn 4587 . . . . . 6 class {(0gβ€˜β„Ž)}
3027, 29wceq 1542 . . . . 5 wff ((ocvβ€˜β„Ž)β€˜π‘) = {(0gβ€˜β„Ž)}
3124, 30wa 397 . . . 4 wff (βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯(Β·π‘–β€˜β„Ž)𝑦) = if(π‘₯ = 𝑦, (1rβ€˜(Scalarβ€˜β„Ž)), (0gβ€˜(Scalarβ€˜β„Ž))) ∧ ((ocvβ€˜β„Ž)β€˜π‘) = {(0gβ€˜β„Ž)})
32 cbs 17088 . . . . . 6 class Base
338, 32cfv 6497 . . . . 5 class (Baseβ€˜β„Ž)
3433cpw 4561 . . . 4 class 𝒫 (Baseβ€˜β„Ž)
3531, 21, 34crab 3406 . . 3 class {𝑏 ∈ 𝒫 (Baseβ€˜β„Ž) ∣ (βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯(Β·π‘–β€˜β„Ž)𝑦) = if(π‘₯ = 𝑦, (1rβ€˜(Scalarβ€˜β„Ž)), (0gβ€˜(Scalarβ€˜β„Ž))) ∧ ((ocvβ€˜β„Ž)β€˜π‘) = {(0gβ€˜β„Ž)})}
362, 3, 35cmpt 5189 . 2 class (β„Ž ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Baseβ€˜β„Ž) ∣ (βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯(Β·π‘–β€˜β„Ž)𝑦) = if(π‘₯ = 𝑦, (1rβ€˜(Scalarβ€˜β„Ž)), (0gβ€˜(Scalarβ€˜β„Ž))) ∧ ((ocvβ€˜β„Ž)β€˜π‘) = {(0gβ€˜β„Ž)})})
371, 36wceq 1542 1 wff OBasis = (β„Ž ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Baseβ€˜β„Ž) ∣ (βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯(Β·π‘–β€˜β„Ž)𝑦) = if(π‘₯ = 𝑦, (1rβ€˜(Scalarβ€˜β„Ž)), (0gβ€˜(Scalarβ€˜β„Ž))) ∧ ((ocvβ€˜β„Ž)β€˜π‘) = {(0gβ€˜β„Ž)})})
Colors of variables: wff setvar class
This definition is referenced by:  isobs  21142
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