Definition df-obs 21260

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 21257	. 2 class OBasis
2		vh	. . 3 setvar ℎ
3		cphl 21177	. . 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 17202	. . . . . . . . . 10 class ·𝑖
10	8, 9	cfv 6544	. . . . . . . . 9 class (·𝑖‘ℎ)
11	5, 7, 10	co 7409	. . . . . . . 8 class (𝑥(·𝑖‘ℎ)𝑦)
12	4, 6	weq 1967	. . . . . . . . 9 wff 𝑥 = 𝑦
13		csca 17200	. . . . . . . . . . 11 class Scalar
14	8, 13	cfv 6544	. . . . . . . . . 10 class (Scalar‘ℎ)
15		cur 20004	. . . . . . . . . 10 class 1r
16	14, 15	cfv 6544	. . . . . . . . 9 class (1r‘(Scalar‘ℎ))
17		c0g 17385	. . . . . . . . . 10 class 0g
18	14, 17	cfv 6544	. . . . . . . . 9 class (0g‘(Scalar‘ℎ))
19	12, 16, 18	cif 4529	. . . . . . . 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 3062	. . . . . 6 wff ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
24	23, 4, 22	wral 3062	. . . . 5 wff ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
25		cocv 21213	. . . . . . . 8 class ocv
26	8, 25	cfv 6544	. . . . . . 7 class (ocv‘ℎ)
27	22, 26	cfv 6544	. . . . . 6 class ((ocv‘ℎ)‘𝑏)
28	8, 17	cfv 6544	. . . . . . 7 class (0g‘ℎ)
29	28	csn 4629	. . . . . 6 class {(0g‘ℎ)}
30	27, 29	wceq 1542	. . . . 5 wff ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)}
31	24, 30	wa 397	. . . 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 17144	. . . . . 6 class Base
33	8, 32	cfv 6544	. . . . 5 class (Base‘ℎ)
34	33	cpw 4603	. . . 4 class 𝒫 (Base‘ℎ)
35	31, 21, 34	crab 3433	. . 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 5232	. 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  21275