Definition df-obs 21670

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 21667	. 2 class OBasis
2		vh	. . 3 setvar ℎ
3		cphl 21589	. . 3 class PreHil
4		vx	. . . . . . . . . 10 setvar 𝑥
5	4	cv 1539	. . . . . . . . 9 class 𝑥
6		vy	. . . . . . . . . 10 setvar 𝑦
7	6	cv 1539	. . . . . . . . 9 class 𝑦
8	2	cv 1539	. . . . . . . . . 10 class ℎ
9		cip 17281	. . . . . . . . . 10 class ·𝑖
10	8, 9	cfv 6536	. . . . . . . . 9 class (·𝑖‘ℎ)
11	5, 7, 10	co 7410	. . . . . . . 8 class (𝑥(·𝑖‘ℎ)𝑦)
12	4, 6	weq 1962	. . . . . . . . 9 wff 𝑥 = 𝑦
13		csca 17279	. . . . . . . . . . 11 class Scalar
14	8, 13	cfv 6536	. . . . . . . . . 10 class (Scalar‘ℎ)
15		cur 20146	. . . . . . . . . 10 class 1r
16	14, 15	cfv 6536	. . . . . . . . 9 class (1r‘(Scalar‘ℎ))
17		c0g 17458	. . . . . . . . . 10 class 0g
18	14, 17	cfv 6536	. . . . . . . . 9 class (0g‘(Scalar‘ℎ))
19	12, 16, 18	cif 4505	. . . . . . . 8 class if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
20	11, 19	wceq 1540	. . . . . . 7 wff (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
21		vb	. . . . . . . 8 setvar 𝑏
22	21	cv 1539	. . . . . . 7 class 𝑏
23	20, 6, 22	wral 3052	. . . . . 6 wff ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
24	23, 4, 22	wral 3052	. . . . 5 wff ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
25		cocv 21625	. . . . . . . 8 class ocv
26	8, 25	cfv 6536	. . . . . . 7 class (ocv‘ℎ)
27	22, 26	cfv 6536	. . . . . 6 class ((ocv‘ℎ)‘𝑏)
28	8, 17	cfv 6536	. . . . . . 7 class (0g‘ℎ)
29	28	csn 4606	. . . . . 6 class {(0g‘ℎ)}
30	27, 29	wceq 1540	. . . . 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 17233	. . . . . 6 class Base
33	8, 32	cfv 6536	. . . . 5 class (Base‘ℎ)
34	33	cpw 4580	. . . 4 class 𝒫 (Base‘ℎ)
35	31, 21, 34	crab 3420	. . 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 5206	. 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 1540	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  21685