Description: The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014) (Proof shortened by Mario Carneiro, 19-Jun-2014)
Ref | Expression | ||
---|---|---|---|
Hypotheses | lss0cl.z | |
|
lss0cl.s | |
||
Assertion | lss0cl | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lss0cl.z | |
|
2 | lss0cl.s | |
|
3 | 2 | lssn0 | |
4 | n0 | |
|
5 | 3 4 | sylib | |
6 | 5 | adantl | |
7 | simp1 | |
|
8 | eqid | |
|
9 | 8 2 | lssel | |
10 | 9 | 3adant1 | |
11 | eqid | |
|
12 | 8 1 11 | lmodsubid | |
13 | 7 10 12 | syl2anc | |
14 | 11 2 | lssvsubcl | |
15 | 14 | anabsan2 | |
16 | 15 | 3impa | |
17 | 13 16 | eqeltrrd | |
18 | 17 | 3expia | |
19 | 18 | exlimdv | |
20 | 6 19 | mpd | |