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Mirrors > Home > MPE Home > Th. List > df-ph | Structured version Visualization version Unicode version |
Description: Define the class of all complex inner product spaces. An inner product space is a normed vector space whose norm satisfies the parallelogram law (a property that induces an inner product). Based on Exercise 4(b) of [ReedSimon] p. 63. The vector operation is , the scalar product is , and the norm is . An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
df-ph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ccphlo 27667 | . 2 | |
2 | cnv 27439 | . . 3 | |
3 | vx | . . . . . . . . . . . 12 | |
4 | 3 | cv 1482 | . . . . . . . . . . 11 |
5 | vy | . . . . . . . . . . . 12 | |
6 | 5 | cv 1482 | . . . . . . . . . . 11 |
7 | vg | . . . . . . . . . . . 12 | |
8 | 7 | cv 1482 | . . . . . . . . . . 11 |
9 | 4, 6, 8 | co 6650 | . . . . . . . . . 10 |
10 | vn | . . . . . . . . . . 11 | |
11 | 10 | cv 1482 | . . . . . . . . . 10 |
12 | 9, 11 | cfv 5888 | . . . . . . . . 9 |
13 | c2 11070 | . . . . . . . . 9 | |
14 | cexp 12860 | . . . . . . . . 9 | |
15 | 12, 13, 14 | co 6650 | . . . . . . . 8 |
16 | c1 9937 | . . . . . . . . . . . . 13 | |
17 | 16 | cneg 10267 | . . . . . . . . . . . 12 |
18 | vs | . . . . . . . . . . . . 13 | |
19 | 18 | cv 1482 | . . . . . . . . . . . 12 |
20 | 17, 6, 19 | co 6650 | . . . . . . . . . . 11 |
21 | 4, 20, 8 | co 6650 | . . . . . . . . . 10 |
22 | 21, 11 | cfv 5888 | . . . . . . . . 9 |
23 | 22, 13, 14 | co 6650 | . . . . . . . 8 |
24 | caddc 9939 | . . . . . . . 8 | |
25 | 15, 23, 24 | co 6650 | . . . . . . 7 |
26 | 4, 11 | cfv 5888 | . . . . . . . . . 10 |
27 | 26, 13, 14 | co 6650 | . . . . . . . . 9 |
28 | 6, 11 | cfv 5888 | . . . . . . . . . 10 |
29 | 28, 13, 14 | co 6650 | . . . . . . . . 9 |
30 | 27, 29, 24 | co 6650 | . . . . . . . 8 |
31 | cmul 9941 | . . . . . . . 8 | |
32 | 13, 30, 31 | co 6650 | . . . . . . 7 |
33 | 25, 32 | wceq 1483 | . . . . . 6 |
34 | 8 | crn 5115 | . . . . . 6 |
35 | 33, 5, 34 | wral 2912 | . . . . 5 |
36 | 35, 3, 34 | wral 2912 | . . . 4 |
37 | 36, 7, 18, 10 | coprab 6651 | . . 3 |
38 | 2, 37 | cin 3573 | . 2 |
39 | 1, 38 | wceq 1483 | 1 |
Colors of variables: wff setvar class |
This definition is referenced by: phnv 27669 isphg 27672 |
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