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Metamath Proof Explorer

Theorem lspsnvsi

Description: Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014) (Proof shortened by Mario Carneiro, 4-Sep-2014)

Ref Expression
Hypotheses lspsn.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
lspsn.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
lspsn.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
lspsn.t ⊢ · = ( ·𝑠 ‘ 𝑊 )
lspsn.n ⊢ 𝑁 = ( LSpan ‘ 𝑊 )
Assertion lspsnvsi ( ( 𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { ( 𝑅 · 𝑋 ) } ) ⊆ ( 𝑁 ‘ { 𝑋 } ) )

Proof

Step Hyp Ref Expression
1 lspsn.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2 lspsn.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
3 lspsn.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
4 lspsn.t ⊢ · = ( ·𝑠 ‘ 𝑊 )
5 lspsn.n ⊢ 𝑁 = ( LSpan ‘ 𝑊 )
6 eqid ⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
7 simp1 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → 𝑊 ∈ LMod )
8 simp3 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ 𝑉 )
9 8 snssd ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → { 𝑋 } ⊆ 𝑉 )
10 3 6 5 lspcl ⊢ ( ( 𝑊 ∈ LMod ∧ { 𝑋 } ⊆ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) )
11 7 9 10 syl2anc ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) )
12 simp2 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → 𝑅 ∈ 𝐾 )
13 3 4 1 2 5 7 12 8 ellspsni ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → ( 𝑅 · 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) )
14 6 5 7 11 13 ellspsn5 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { ( 𝑅 · 𝑋 ) } ) ⊆ ( 𝑁 ‘ { 𝑋 } ) )