Definition df-oc 9275

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 9283 and chocval 9301 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 8979	. 2 class _|_
2		vx	. . . . . 6 set x
3	2	cv 1098	. . . . 5 class x
4		chil 8968	. . . . 5 class H~
5	3, 4	wss 2018	. . . 4 wff x (_ H~
6		vy	. . . . . 6 set y
7	6	cv 1098	. . . . 5 class y
8		vz	. . . . . . . . . 10 set z
9	8	cv 1098	. . . . . . . . 9 class z
10		vw	. . . . . . . . . 10 set w
11	10	cv 1098	. . . . . . . . 9 class w
12		csp 8973	. . . . . . . . 9 class .ih
13	9, 11, 12	co 3902	. . . . . . . 8 class (z .ih w)
14		cc0 5157	. . . . . . . 8 class 0
15	13, 14	wceq 1099	. . . . . . 7 wff (z .ih w) = 0
16	15, 10, 3	wral 1621	. . . . . 6 wff A.w e. x (z .ih w) = 0
17	16, 8, 4	crab 1624	. . . . 5 class {z e. H~ | A.w e. x (z .ih w) = 0}
18	7, 17	wceq 1099	. . . 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 2634	. 2 class {<.x, y>. | (x (_ H~ /\ y = {z e. H~ | A.w e. x (z .ih w) = 0})}
21	1, 20	wceq 1099	1 wff _|_ = {<.x, y>. | (x (_ H~ /\ y = {z e. H~ | A.w e. x (z .ih w) = 0})}

This definition is referenced by:  ocvalt 9283

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