Description: The algebraic span of a set of vectors is a vector subspace. (Contributed by Mario Carneiro, 7-Jan-2015)
Ref | Expression | ||
---|---|---|---|
Hypotheses | aspval.a | |
|
aspval.v | |
||
aspval.l | |
||
Assertion | asplss | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aspval.a | |
|
2 | aspval.v | |
|
3 | aspval.l | |
|
4 | 1 2 3 | aspval | |
5 | assalmod | |
|
6 | 5 | adantr | |
7 | ssrab2 | |
|
8 | inss2 | |
|
9 | 7 8 | sstri | |
10 | 9 | a1i | |
11 | fvex | |
|
12 | 4 11 | eqeltrrdi | |
13 | intex | |
|
14 | 12 13 | sylibr | |
15 | 3 | lssintcl | |
16 | 6 10 14 15 | syl3anc | |
17 | 4 16 | eqeltrd | |