Description: A necessary and sufficient condition for the product of two projectors to be a projector is that the projectors commute. Part 1 of Theorem 1 of AkhiezerGlazman p. 65. (Contributed by NM, 3-Jun-2006) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypotheses | pjclem1.1 | |
|
pjclem1.2 | |
||
Assertion | pjcmul1i | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjclem1.1 | |
|
2 | pjclem1.2 | |
|
3 | 1 2 | pjclem4 | |
4 | pjmfn | |
|
5 | 1 2 | chincli | |
6 | fnfvelrn | |
|
7 | 4 5 6 | mp2an | |
8 | 3 7 | eqeltrdi | |
9 | pjadj2 | |
|
10 | 1 | pjbdlni | |
11 | 2 | pjbdlni | |
12 | 10 11 | adjcoi | |
13 | pjadj3 | |
|
14 | 2 13 | ax-mp | |
15 | pjadj3 | |
|
16 | 1 15 | ax-mp | |
17 | 14 16 | coeq12i | |
18 | 12 17 | eqtri | |
19 | 9 18 | eqtr3di | |
20 | 8 19 | impbii | |