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Definition df-obs 16934
 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 Scalar Scalar
Distinct variable group:   ,,,

Detailed syntax breakdown of Definition df-obs
StepHypRef Expression
1 cobs 16931 . 2 OBasis
2 vh . . 3
3 cphl 16857 . . 3
4 vx . . . . . . . . . 10
54cv 1652 . . . . . . . . 9
6 vy . . . . . . . . . 10
76cv 1652 . . . . . . . . 9
82cv 1652 . . . . . . . . . 10
9 cip 13536 . . . . . . . . . 10
108, 9cfv 5456 . . . . . . . . 9
115, 7, 10co 6083 . . . . . . . 8
124, 6weq 1654 . . . . . . . . 9
13 csca 13534 . . . . . . . . . . 11 Scalar
148, 13cfv 5456 . . . . . . . . . 10 Scalar
15 cur 15664 . . . . . . . . . 10
1614, 15cfv 5456 . . . . . . . . 9 Scalar
17 c0g 13725 . . . . . . . . . 10
1814, 17cfv 5456 . . . . . . . . 9 Scalar
1912, 16, 18cif 3741 . . . . . . . 8 Scalar Scalar
2011, 19wceq 1653 . . . . . . 7 Scalar Scalar
21 vb . . . . . . . 8
2221cv 1652 . . . . . . 7
2320, 6, 22wral 2707 . . . . . 6 Scalar Scalar
2423, 4, 22wral 2707 . . . . 5 Scalar Scalar
25 cocv 16889 . . . . . . . 8
268, 25cfv 5456 . . . . . . 7
2722, 26cfv 5456 . . . . . 6
288, 17cfv 5456 . . . . . . 7
2928csn 3816 . . . . . 6
3027, 29wceq 1653 . . . . 5
3124, 30wa 360 . . . 4 Scalar Scalar
32 cbs 13471 . . . . . 6
338, 32cfv 5456 . . . . 5
3433cpw 3801 . . . 4
3531, 21, 34crab 2711 . . 3 Scalar Scalar
362, 3, 35cmpt 4268 . 2 Scalar Scalar
371, 36wceq 1653 1 OBasis Scalar Scalar
 Colors of variables: wff set class This definition is referenced by:  isobs  16949
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