Description: The set of complex numbers is an inner product (pre-Hilbert) space. (Contributed by Steve Rodriguez, 28-Apr-2007) (Revised by Mario Carneiro, 7-Nov-2013) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypothesis | cncph.6 | |
|
Assertion | cncph | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncph.6 | |
|
2 | eqid | |
|
3 | 2 | cnnv | |
4 | mulm1 | |
|
5 | 4 | adantl | |
6 | 5 | oveq2d | |
7 | negsub | |
|
8 | 6 7 | eqtrd | |
9 | 8 | fveq2d | |
10 | 9 | oveq1d | |
11 | 10 | oveq2d | |
12 | sqabsadd | |
|
13 | sqabssub | |
|
14 | 12 13 | oveq12d | |
15 | abscl | |
|
16 | 15 | recnd | |
17 | 16 | sqcld | |
18 | abscl | |
|
19 | 18 | recnd | |
20 | 19 | sqcld | |
21 | addcl | |
|
22 | 17 20 21 | syl2an | |
23 | 2cn | |
|
24 | cjcl | |
|
25 | mulcl | |
|
26 | 24 25 | sylan2 | |
27 | recl | |
|
28 | 27 | recnd | |
29 | 26 28 | syl | |
30 | mulcl | |
|
31 | 23 29 30 | sylancr | |
32 | 22 31 22 | ppncand | |
33 | 14 32 | eqtrd | |
34 | 2times | |
|
35 | 34 | eqcomd | |
36 | 22 35 | syl | |
37 | 33 36 | eqtrd | |
38 | 11 37 | eqtrd | |
39 | 38 | rgen2 | |
40 | addex | |
|
41 | mulex | |
|
42 | absf | |
|
43 | cnex | |
|
44 | fex | |
|
45 | 42 43 44 | mp2an | |
46 | cnaddabloOLD | |
|
47 | ablogrpo | |
|
48 | 46 47 | ax-mp | |
49 | ax-addf | |
|
50 | 49 | fdmi | |
51 | 48 50 | grporn | |
52 | 51 | isphg | |
53 | 40 41 45 52 | mp3an | |
54 | 3 39 53 | mpbir2an | |
55 | 1 54 | eqeltri | |