Description: The norm of an operator is nonnegative. (Contributed by NM, 8-Dec-2007) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypotheses | nmoxr.1 | |
|
nmoxr.2 | |
||
nmoxr.3 | |
||
Assertion | nmooge0 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoxr.1 | |
|
2 | nmoxr.2 | |
|
3 | nmoxr.3 | |
|
4 | 0xr | |
|
5 | 4 | a1i | |
6 | simp2 | |
|
7 | eqid | |
|
8 | 1 7 | nvzcl | |
9 | ffvelrn | |
|
10 | 8 9 | sylan2 | |
11 | 10 | ancoms | |
12 | 11 | 3adant2 | |
13 | eqid | |
|
14 | 2 13 | nvcl | |
15 | 6 12 14 | syl2anc | |
16 | 15 | rexrd | |
17 | 1 2 3 | nmoxr | |
18 | 2 13 | nvge0 | |
19 | 6 12 18 | syl2anc | |
20 | 2 13 | nmosetre | |
21 | ressxr | |
|
22 | 20 21 | sstrdi | |
23 | eqid | |
|
24 | 1 7 23 | nmosetn0 | |
25 | supxrub | |
|
26 | 22 24 25 | syl2an | |
27 | 26 | 3impa | |
28 | 27 | 3comr | |
29 | 1 2 23 13 3 | nmooval | |
30 | 28 29 | breqtrrd | |
31 | 5 16 17 19 30 | xrletrd | |