Definition df-oc 9124

Description: Define orthogonal complement of a subset (usually a subspace) of Hilbert space. The orthogonal complement is the set of all vectors orthogonal to all vectors in the subset. See ocvalt 9153 and chocval 9171 for its value. Textbooks usually denote this unary operation with the symbol _|_ as a small superscript, although Mittelstaedt uses the symbol as a prefix operation. Here we define a function (prefix operation) _|_ rather than introducing a new syntactical form. This lets us take advantage of the theorems about functions that we already have proved under set theory. Definition of [Mittelstaedt] p. 9.

Assertion

Ref	Expression
df-oc	|- _|_ = {<.x, y>. | (x (_ H~ /\ y = {z e. H~ | A.w e. x (z .ih w) = 0})}

Distinct variable group:   x,y,z,w

Detailed syntax breakdown of Definition df-oc

Step	Hyp	Ref	Expression
1		cort 8799	. 2 class _|_
2		vx	. . . . . 6 set x
3	2	cv 955	. . . . 5 class x
4		chil 8788	. . . . 5 class H~
5	3, 4	wss 2047	. . . 4 wff x (_ H~
6		vy	. . . . . 6 set y
7	6	cv 955	. . . . 5 class y
8		vz	. . . . . . . . . 10 set z
9	8	cv 955	. . . . . . . . 9 class z
10		vw	. . . . . . . . . 10 set w
11	10	cv 955	. . . . . . . . 9 class w
12		csp 8793	. . . . . . . . 9 class .ih
13	9, 11, 12	co 3963	. . . . . . . 8 class (z .ih w)
14		cc0 5234	. . . . . . . 8 class 0
15	13, 14	wceq 956	. . . . . . 7 wff (z .ih w) = 0
16	15, 10, 3	wral 1645	. . . . . 6 wff A.w e. x (z .ih w) = 0
17	16, 8, 4	crab 1648	. . . . 5 class {z e. H~ | A.w e. x (z .ih w) = 0}
18	7, 17	wceq 956	. . . 4 wff y = {z e. H~ | A.w e. x (z .ih w) = 0}
19	5, 18	wa 223	. . 3 wff (x (_ H~ /\ y = {z e. H~ | A.w e. x (z .ih w) = 0})
20	19, 2, 6	copab 2666	. 2 class {<.x, y>. | (x (_ H~ /\ y = {z e. H~ | A.w e. x (z .ih w) = 0})}
21	1, 20	wceq 956	1 wff _|_ = {<.x, y>. | (x (_ H~ /\ y = {z e. H~ | A.w e. x (z .ih w) = 0})}

This definition is referenced by:  ocvalt 9153

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