Definition df-obs 21685

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 21682	. 2 class OBasis
2		vh	. . 3 setvar ℎ
3		cphl 21604	. . 3 class PreHil
4		vx	. . . . . . . . . 10 setvar 𝑥
5	4	cv 1541	. . . . . . . . 9 class 𝑥
6		vy	. . . . . . . . . 10 setvar 𝑦
7	6	cv 1541	. . . . . . . . 9 class 𝑦
8	2	cv 1541	. . . . . . . . . 10 class ℎ
9		cip 17225	. . . . . . . . . 10 class ·𝑖
10	8, 9	cfv 6498	. . . . . . . . 9 class (·𝑖‘ℎ)
11	5, 7, 10	co 7367	. . . . . . . 8 class (𝑥(·𝑖‘ℎ)𝑦)
12	4, 6	weq 1964	. . . . . . . . 9 wff 𝑥 = 𝑦
13		csca 17223	. . . . . . . . . . 11 class Scalar
14	8, 13	cfv 6498	. . . . . . . . . 10 class (Scalar‘ℎ)
15		cur 20162	. . . . . . . . . 10 class 1r
16	14, 15	cfv 6498	. . . . . . . . 9 class (1r‘(Scalar‘ℎ))
17		c0g 17402	. . . . . . . . . 10 class 0g
18	14, 17	cfv 6498	. . . . . . . . 9 class (0g‘(Scalar‘ℎ))
19	12, 16, 18	cif 4466	. . . . . . . 8 class if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
20	11, 19	wceq 1542	. . . . . . 7 wff (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
21		vb	. . . . . . . 8 setvar 𝑏
22	21	cv 1541	. . . . . . 7 class 𝑏
23	20, 6, 22	wral 3051	. . . . . 6 wff ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
24	23, 4, 22	wral 3051	. . . . 5 wff ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
25		cocv 21640	. . . . . . . 8 class ocv
26	8, 25	cfv 6498	. . . . . . 7 class (ocv‘ℎ)
27	22, 26	cfv 6498	. . . . . 6 class ((ocv‘ℎ)‘𝑏)
28	8, 17	cfv 6498	. . . . . . 7 class (0g‘ℎ)
29	28	csn 4567	. . . . . 6 class {(0g‘ℎ)}
30	27, 29	wceq 1542	. . . . 5 wff ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)}
31	24, 30	wa 395	. . . 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 17179	. . . . . 6 class Base
33	8, 32	cfv 6498	. . . . 5 class (Base‘ℎ)
34	33	cpw 4541	. . . 4 class 𝒫 (Base‘ℎ)
35	31, 21, 34	crab 3389	. . 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 5166	. 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 1542	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  21700