Definition df-obs 21647

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 21644	. 2 class OBasis
2		vh	. . 3 setvar ℎ
3		cphl 21566	. . 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 17201	. . . . . . . . . 10 class ·𝑖
10	8, 9	cfv 6499	. . . . . . . . 9 class (·𝑖‘ℎ)
11	5, 7, 10	co 7369	. . . . . . . 8 class (𝑥(·𝑖‘ℎ)𝑦)
12	4, 6	weq 1962	. . . . . . . . 9 wff 𝑥 = 𝑦
13		csca 17199	. . . . . . . . . . 11 class Scalar
14	8, 13	cfv 6499	. . . . . . . . . 10 class (Scalar‘ℎ)
15		cur 20101	. . . . . . . . . 10 class 1r
16	14, 15	cfv 6499	. . . . . . . . 9 class (1r‘(Scalar‘ℎ))
17		c0g 17378	. . . . . . . . . 10 class 0g
18	14, 17	cfv 6499	. . . . . . . . 9 class (0g‘(Scalar‘ℎ))
19	12, 16, 18	cif 4484	. . . . . . . 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 3044	. . . . . 6 wff ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
24	23, 4, 22	wral 3044	. . . . 5 wff ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ)))
25		cocv 21602	. . . . . . . 8 class ocv
26	8, 25	cfv 6499	. . . . . . 7 class (ocv‘ℎ)
27	22, 26	cfv 6499	. . . . . 6 class ((ocv‘ℎ)‘𝑏)
28	8, 17	cfv 6499	. . . . . . 7 class (0g‘ℎ)
29	28	csn 4585	. . . . . 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 17155	. . . . . 6 class Base
33	8, 32	cfv 6499	. . . . 5 class (Base‘ℎ)
34	33	cpw 4559	. . . 4 class 𝒫 (Base‘ℎ)
35	31, 21, 34	crab 3402	. . 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 5183	. 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  21662