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 } ) )