Description: A projection is self-adjoint. Property (i) of Beran p. 109. (Contributed by NM, 30-Oct-1999) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypotheses | pjidm.1 | |
|
pjidm.2 | |
||
pjadj.3 | |
||
Assertion | pjadjii | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjidm.1 | |
|
2 | pjidm.2 | |
|
3 | pjadj.3 | |
|
4 | 3 2 | pjorthi | |
5 | 1 4 | ax-mp | |
6 | 5 | fveq2i | |
7 | cj0 | |
|
8 | 6 7 | eqtri | |
9 | 1 | choccli | |
10 | 9 2 | pjhclii | |
11 | 1 3 | pjhclii | |
12 | 10 11 | his1i | |
13 | 2 3 | pjorthi | |
14 | 1 13 | ax-mp | |
15 | 8 12 14 | 3eqtr4ri | |
16 | 15 | oveq2i | |
17 | 1 2 | pjhclii | |
18 | 9 3 | pjhclii | |
19 | his7 | |
|
20 | 17 11 18 19 | mp3an | |
21 | ax-his2 | |
|
22 | 17 10 11 21 | mp3an | |
23 | 16 20 22 | 3eqtr4i | |
24 | 1 3 | pjpji | |
25 | 24 | oveq2i | |
26 | 1 2 | pjpji | |
27 | 26 | oveq1i | |
28 | 23 25 27 | 3eqtr4i | |