Definition df-obs 20851

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 = (ℎ ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘ℎ) ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) ∧ ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)})})

Distinct variable group:   ℎ,𝑏,𝑥,𝑦

Detailed syntax breakdown of Definition df-obs

Step	Hyp	Ref	Expression
1		cobs 20848	. 2 class OBasis
2		vh	. . 3 setvar ℎ
3		cphl 20770	. . 3 class PreHil
4		vx	. . . . . . . . . 10 setvar 𝑥
5	4	cv 1536	. . . . . . . . 9 class 𝑥
6		vy	. . . . . . . . . 10 setvar 𝑦
7	6	cv 1536	. . . . . . . . 9 class 𝑦
8	2	cv 1536	. . . . . . . . . 10 class ℎ
9		cip 16572	. . . . . . . . . 10 class ·𝑖
10	8, 9	cfv 6357	. . . . . . . . 9 class (·𝑖‘ℎ)
11	5, 7, 10	co 7158	. . . . . . . 8 class (𝑥(·𝑖‘ℎ)𝑦)
12	4, 6	weq 1964	. . . . . . . . 9 wff 𝑥 = 𝑦
13		csca 16570	. . . . . . . . . . 11 class Scalar
14	8, 13	cfv 6357	. . . . . . . . . 10 class (Scalar‘ℎ)
15		cur 19253	. . . . . . . . . 10 class 1r
16	14, 15	cfv 6357	. . . . . . . . 9 class (1r‘(Scalar‘ℎ))
17		c0g 16715	. . . . . . . . . 10 class 0g
18	14, 17	cfv 6357	. . . . . . . . 9 class (0g‘(Scalar‘ℎ))
19	12, 16, 18	cif 4469	. . . . . . . 8 class if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
20	11, 19	wceq 1537	. . . . . . 7 wff (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
21		vb	. . . . . . . 8 setvar 𝑏
22	21	cv 1536	. . . . . . 7 class 𝑏
23	20, 6, 22	wral 3140	. . . . . 6 wff ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
24	23, 4, 22	wral 3140	. . . . 5 wff ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
25		cocv 20806	. . . . . . . 8 class ocv
26	8, 25	cfv 6357	. . . . . . 7 class (ocv‘ℎ)
27	22, 26	cfv 6357	. . . . . 6 class ((ocv‘ℎ)‘𝑏)
28	8, 17	cfv 6357	. . . . . . 7 class (0g‘ℎ)
29	28	csn 4569	. . . . . 6 class {(0g‘ℎ)}
30	27, 29	wceq 1537	. . . . 5 wff ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)}
31	24, 30	wa 398	. . . 4 wff (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) ∧ ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)})
32		cbs 16485	. . . . . 6 class Base
33	8, 32	cfv 6357	. . . . 5 class (Base‘ℎ)
34	33	cpw 4541	. . . 4 class 𝒫 (Base‘ℎ)
35	31, 21, 34	crab 3144	. . 3 class {𝑏 ∈ 𝒫 (Base‘ℎ) ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) ∧ ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)})}
36	2, 3, 35	cmpt 5148	. 2 class (ℎ ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘ℎ) ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) ∧ ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)})})
37	1, 36	wceq 1537	1 wff OBasis = (ℎ ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘ℎ) ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) ∧ ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)})})

This definition is referenced by:  isobs  20866