Description: The group identity element of Hilbert space vector addition is the zero vector. (Contributed by NM, 16-Apr-2007) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | hilid | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hilablo | |
|
2 | ablogrpo | |
|
3 | 1 2 | ax-mp | |
4 | ax-hfvadd | |
|
5 | 4 | fdmi | |
6 | 3 5 | grporn | |
7 | eqid | |
|
8 | 6 7 | grpoidval | |
9 | 3 8 | ax-mp | |
10 | hvaddid2 | |
|
11 | 10 | rgen | |
12 | ax-hv0cl | |
|
13 | 6 | grpoideu | |
14 | 3 13 | ax-mp | |
15 | oveq1 | |
|
16 | 15 | eqeq1d | |
17 | 16 | ralbidv | |
18 | 17 | riota2 | |
19 | 12 14 18 | mp2an | |
20 | 11 19 | mpbi | |
21 | 9 20 | eqtri | |