Description: The composition of two bounded linear operators is bounded. (Contributed by NM, 9-Mar-2006) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypotheses | nmoptri.1 | |
|
nmoptri.2 | |
||
Assertion | bdopcoi | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoptri.1 | |
|
2 | nmoptri.2 | |
|
3 | bdopln | |
|
4 | 1 3 | ax-mp | |
5 | bdopln | |
|
6 | 2 5 | ax-mp | |
7 | 4 6 | lnopcoi | |
8 | 4 | lnopfi | |
9 | 6 | lnopfi | |
10 | 8 9 | hocofi | |
11 | nmopxr | |
|
12 | 10 11 | ax-mp | |
13 | nmopre | |
|
14 | 1 13 | ax-mp | |
15 | nmopre | |
|
16 | 2 15 | ax-mp | |
17 | 14 16 | remulcli | |
18 | nmopgtmnf | |
|
19 | 10 18 | ax-mp | |
20 | 1 2 | nmopcoi | |
21 | xrre | |
|
22 | 12 17 19 20 21 | mp4an | |
23 | elbdop2 | |
|
24 | 7 22 23 | mpbir2an | |