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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 ) ) )