MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-obs Structured version   Visualization version   GIF version

Definition df-obs 21252
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 = ( ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})})
Distinct variable group:   ,𝑏,𝑥,𝑦

Detailed syntax breakdown of Definition df-obs
StepHypRef Expression
1 cobs 21249 . 2 class OBasis
2 vh . . 3 setvar
3 cphl 21169 . . 3 class PreHil
4 vx . . . . . . . . . 10 setvar 𝑥
54cv 1541 . . . . . . . . 9 class 𝑥
6 vy . . . . . . . . . 10 setvar 𝑦
76cv 1541 . . . . . . . . 9 class 𝑦
82cv 1541 . . . . . . . . . 10 class
9 cip 17199 . . . . . . . . . 10 class ·𝑖
108, 9cfv 6541 . . . . . . . . 9 class (·𝑖)
115, 7, 10co 7406 . . . . . . . 8 class (𝑥(·𝑖)𝑦)
124, 6weq 1967 . . . . . . . . 9 wff 𝑥 = 𝑦
13 csca 17197 . . . . . . . . . . 11 class Scalar
148, 13cfv 6541 . . . . . . . . . 10 class (Scalar‘)
15 cur 19999 . . . . . . . . . 10 class 1r
1614, 15cfv 6541 . . . . . . . . 9 class (1r‘(Scalar‘))
17 c0g 17382 . . . . . . . . . 10 class 0g
1814, 17cfv 6541 . . . . . . . . 9 class (0g‘(Scalar‘))
1912, 16, 18cif 4528 . . . . . . . 8 class if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘)))
2011, 19wceq 1542 . . . . . . 7 wff (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘)))
21 vb . . . . . . . 8 setvar 𝑏
2221cv 1541 . . . . . . 7 class 𝑏
2320, 6, 22wral 3062 . . . . . 6 wff 𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘)))
2423, 4, 22wral 3062 . . . . 5 wff 𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘)))
25 cocv 21205 . . . . . . . 8 class ocv
268, 25cfv 6541 . . . . . . 7 class (ocv‘)
2722, 26cfv 6541 . . . . . 6 class ((ocv‘)‘𝑏)
288, 17cfv 6541 . . . . . . 7 class (0g)
2928csn 4628 . . . . . 6 class {(0g)}
3027, 29wceq 1542 . . . . 5 wff ((ocv‘)‘𝑏) = {(0g)}
3124, 30wa 397 . . . 4 wff (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})
32 cbs 17141 . . . . . 6 class Base
338, 32cfv 6541 . . . . 5 class (Base‘)
3433cpw 4602 . . . 4 class 𝒫 (Base‘)
3531, 21, 34crab 3433 . . 3 class {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})}
362, 3, 35cmpt 5231 . 2 class ( ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})})
371, 36wceq 1542 1 wff OBasis = ( ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})})
Colors of variables: wff setvar class
This definition is referenced by:  isobs  21267
  Copyright terms: Public domain W3C validator