Metamath Proof Explorer

Theorem lsmsp2

Description: Subspace sum of spans of subsets is the span of their union. ( spanuni analog.) (Contributed by NM, 22-Feb-2014) (Revised by Mario Carneiro, 21-Jun-2014)

Ref Expression
Hypotheses lsmsp2.v ⊢ V = Base W
lsmsp2.n ⊢ N = LSpan ⁡ W
lsmsp2.p ⊢ ⊕ ˙ = LSSum ⁡ W
Assertion lsmsp2 ⊢ W ∈ LMod ∧ T ⊆ V ∧ U ⊆ V → N ⁡ T ⊕ ˙ N ⁡ 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 simp1 ⊢ W ∈ LMod ∧ T ⊆ V ∧ U ⊆ V → W ∈ LMod
5 eqid ⊢ LSubSp ⁡ W = LSubSp ⁡ W
6 1 5 2 lspcl ⊢ W ∈ LMod ∧ T ⊆ V → N ⁡ T ∈ LSubSp ⁡ W
7 6 3adant3 ⊢ W ∈ LMod ∧ T ⊆ V ∧ U ⊆ V → N ⁡ T ∈ LSubSp ⁡ W
8 1 5 2 lspcl ⊢ W ∈ LMod ∧ U ⊆ V → N ⁡ U ∈ LSubSp ⁡ W
9 8 3adant2 ⊢ W ∈ LMod ∧ T ⊆ V ∧ U ⊆ V → N ⁡ U ∈ LSubSp ⁡ W
10 5 2 3 lsmsp ⊢ W ∈ LMod ∧ N ⁡ T ∈ LSubSp ⁡ W ∧ N ⁡ U ∈ LSubSp ⁡ W → N ⁡ T ⊕ ˙ N ⁡ U = N ⁡ N ⁡ T ∪ N ⁡ U
11 4 7 9 10 syl3anc ⊢ W ∈ LMod ∧ T ⊆ V ∧ U ⊆ V → N ⁡ T ⊕ ˙ N ⁡ U = N ⁡ N ⁡ T ∪ N ⁡ U
12 1 2 lspun ⊢ W ∈ LMod ∧ T ⊆ V ∧ U ⊆ V → N ⁡ T ∪ U = N ⁡ N ⁡ T ∪ N ⁡ U
13 11 12 eqtr4d ⊢ W ∈ LMod ∧ T ⊆ V ∧ U ⊆ V → N ⁡ T ⊕ ˙ N ⁡ U = N ⁡ T ∪ U