Description: The value of a linear Hilbert space operator at zero is zero. Remark in Beran p. 99. (Contributed by NM, 13-Aug-2006) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | lnop0 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn | |
|
2 | ax-hv0cl | |
|
3 | 1 2 | hvmulcli | |
4 | ax-hvaddid | |
|
5 | 3 4 | ax-mp | |
6 | ax-hvmulid | |
|
7 | 2 6 | ax-mp | |
8 | 5 7 | eqtri | |
9 | 8 | fveq2i | |
10 | lnopl | |
|
11 | 2 2 10 | mpanr12 | |
12 | 1 11 | mpan2 | |
13 | 9 12 | eqtr3id | |
14 | lnopf | |
|
15 | ffvelrn | |
|
16 | 2 15 | mpan2 | |
17 | 14 16 | syl | |
18 | ax-hvmulid | |
|
19 | 17 18 | syl | |
20 | 19 | oveq1d | |
21 | 13 20 | eqtrd | |
22 | 21 | oveq1d | |
23 | hvsubid | |
|
24 | 17 23 | syl | |
25 | hvpncan | |
|
26 | 25 | anidms | |
27 | 17 26 | syl | |
28 | 22 24 27 | 3eqtr3rd | |