Description: The bra function maps vectors one-to-one onto the set of continuous linear functionals. (Contributed by NM, 26-May-2006) (Proof shortened by Mario Carneiro, 16-Nov-2013) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | bra11 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hilex | |
|
2 | 1 | mptex | |
3 | df-bra | |
|
4 | 2 3 | fnmpti | |
5 | rnbra | |
|
6 | fveq1 | |
|
7 | braval | |
|
8 | 7 | adantlr | |
9 | braval | |
|
10 | 9 | adantll | |
11 | 8 10 | eqeq12d | |
12 | 6 11 | syl5ib | |
13 | 12 | ralrimdva | |
14 | hial2eq2 | |
|
15 | 13 14 | sylibd | |
16 | 15 | rgen2 | |
17 | dff1o6 | |
|
18 | 4 5 16 17 | mpbir3an | |