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Mirrors > Home > HSE Home > Th. List > df-oc | Structured version Visualization version GIF version |
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 ocval 30533 and chocvali 30552 for its value. Textbooks usually denote this unary operation with the symbol โฅ as a small superscript, although Mittelstaedt uses the symbol as a prefix operation. Here we define a function (prefix operation) โฅ rather than introducing a new syntactic form. This lets us take advantage of the theorems about functions that we already have proved under set theory. Definition of [Mittelstaedt] p. 9. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
df-oc | โข โฅ = (๐ฅ โ ๐ซ โ โฆ {๐ฆ โ โ โฃ โ๐ง โ ๐ฅ (๐ฆ ยทih ๐ง) = 0}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cort 30183 | . 2 class โฅ | |
2 | vx | . . 3 setvar ๐ฅ | |
3 | chba 30172 | . . . 4 class โ | |
4 | 3 | cpw 4603 | . . 3 class ๐ซ โ |
5 | vy | . . . . . . . 8 setvar ๐ฆ | |
6 | 5 | cv 1541 | . . . . . . 7 class ๐ฆ |
7 | vz | . . . . . . . 8 setvar ๐ง | |
8 | 7 | cv 1541 | . . . . . . 7 class ๐ง |
9 | csp 30175 | . . . . . . 7 class ยทih | |
10 | 6, 8, 9 | co 7409 | . . . . . 6 class (๐ฆ ยทih ๐ง) |
11 | cc0 11110 | . . . . . 6 class 0 | |
12 | 10, 11 | wceq 1542 | . . . . 5 wff (๐ฆ ยทih ๐ง) = 0 |
13 | 2 | cv 1541 | . . . . 5 class ๐ฅ |
14 | 12, 7, 13 | wral 3062 | . . . 4 wff โ๐ง โ ๐ฅ (๐ฆ ยทih ๐ง) = 0 |
15 | 14, 5, 3 | crab 3433 | . . 3 class {๐ฆ โ โ โฃ โ๐ง โ ๐ฅ (๐ฆ ยทih ๐ง) = 0} |
16 | 2, 4, 15 | cmpt 5232 | . 2 class (๐ฅ โ ๐ซ โ โฆ {๐ฆ โ โ โฃ โ๐ง โ ๐ฅ (๐ฆ ยทih ๐ง) = 0}) |
17 | 1, 16 | wceq 1542 | 1 wff โฅ = (๐ฅ โ ๐ซ โ โฆ {๐ฆ โ โ โฃ โ๐ง โ ๐ฅ (๐ฆ ยทih ๐ง) = 0}) |
Colors of variables: wff setvar class |
This definition is referenced by: ocval 30533 |
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