Metamath Proof Explorer


Theorem lsmssspx

Description: Subspace sum (in its extended domain) is a subset of the span of the union of its arguments. (Contributed by NM, 6-Aug-2014)

Ref Expression
Hypotheses lsmsp2.v V = Base W
lsmsp2.n N = LSpan W
lsmsp2.p ˙ = LSSum W
lsmssspx.t φ T V
lsmssspx.u φ U V
lsmssspx.w φ W LMod
Assertion lsmssspx φ T ˙ U N T U

Proof

Step Hyp Ref Expression
1 lsmsp2.v V = Base W
2 lsmsp2.n N = LSpan W
3 lsmsp2.p ˙ = LSSum W
4 lsmssspx.t φ T V
5 lsmssspx.u φ U V
6 lsmssspx.w φ W LMod
7 1 2 lspssv W LMod T V N T V
8 6 4 7 syl2anc φ N T V
9 1 2 lspssid W LMod T V T N T
10 6 4 9 syl2anc φ T N T
11 1 3 lsmless1x W LMod N T V U V T N T T ˙ U N T ˙ U
12 6 8 5 10 11 syl31anc φ T ˙ U N T ˙ U
13 1 2 lspssv W LMod U V N U V
14 6 5 13 syl2anc φ N U V
15 1 2 lspssid W LMod U V U N U
16 6 5 15 syl2anc φ U N U
17 1 3 lsmless2x W LMod N T V N U V U N U N T ˙ U N T ˙ N U
18 6 8 14 16 17 syl31anc φ N T ˙ U N T ˙ N U
19 12 18 sstrd φ T ˙ U N T ˙ N U
20 1 2 3 lsmsp2 W LMod T V U V N T ˙ N U = N T U
21 6 4 5 20 syl3anc φ N T ˙ N U = N T U
22 19 21 sseqtrd φ T ˙ U N T U