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Definition df-obs 21725
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 = ( ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})})
Distinct variable group:   ,𝑏,𝑥,𝑦

Detailed syntax breakdown of Definition df-obs
StepHypRef Expression
1 cobs 21722 . 2 class OBasis
2 vh . . 3 setvar
3 cphl 21642 . . 3 class PreHil
4 vx . . . . . . . . . 10 setvar 𝑥
54cv 1539 . . . . . . . . 9 class 𝑥
6 vy . . . . . . . . . 10 setvar 𝑦
76cv 1539 . . . . . . . . 9 class 𝑦
82cv 1539 . . . . . . . . . 10 class
9 cip 17302 . . . . . . . . . 10 class ·𝑖
108, 9cfv 6561 . . . . . . . . 9 class (·𝑖)
115, 7, 10co 7431 . . . . . . . 8 class (𝑥(·𝑖)𝑦)
124, 6weq 1962 . . . . . . . . 9 wff 𝑥 = 𝑦
13 csca 17300 . . . . . . . . . . 11 class Scalar
148, 13cfv 6561 . . . . . . . . . 10 class (Scalar‘)
15 cur 20178 . . . . . . . . . 10 class 1r
1614, 15cfv 6561 . . . . . . . . 9 class (1r‘(Scalar‘))
17 c0g 17484 . . . . . . . . . 10 class 0g
1814, 17cfv 6561 . . . . . . . . 9 class (0g‘(Scalar‘))
1912, 16, 18cif 4525 . . . . . . . 8 class if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘)))
2011, 19wceq 1540 . . . . . . 7 wff (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘)))
21 vb . . . . . . . 8 setvar 𝑏
2221cv 1539 . . . . . . 7 class 𝑏
2320, 6, 22wral 3061 . . . . . 6 wff 𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘)))
2423, 4, 22wral 3061 . . . . 5 wff 𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘)))
25 cocv 21678 . . . . . . . 8 class ocv
268, 25cfv 6561 . . . . . . 7 class (ocv‘)
2722, 26cfv 6561 . . . . . 6 class ((ocv‘)‘𝑏)
288, 17cfv 6561 . . . . . . 7 class (0g)
2928csn 4626 . . . . . 6 class {(0g)}
3027, 29wceq 1540 . . . . 5 wff ((ocv‘)‘𝑏) = {(0g)}
3124, 30wa 395 . . . 4 wff (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})
32 cbs 17247 . . . . . 6 class Base
338, 32cfv 6561 . . . . 5 class (Base‘)
3433cpw 4600 . . . 4 class 𝒫 (Base‘)
3531, 21, 34crab 3436 . . 3 class {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})}
362, 3, 35cmpt 5225 . 2 class ( ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})})
371, 36wceq 1540 1 wff OBasis = ( ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})})
Colors of variables: wff setvar class
This definition is referenced by:  isobs  21740
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