Theorem lspsn0

Description: Span of the singleton of the zero vector. ( spansn0 analog.) (Contributed by NM, 15-Jan-2014) (Proof shortened by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lspsn0.z	⊢ 0 = ( 0g ‘ 𝑊 )
	lspsn0.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
Assertion	lspsn0	⊢ ( 𝑊 ∈ LMod → ( 𝑁 ‘ { 0 } ) = { 0 } )

Proof

Step	Hyp	Ref	Expression
1		lspsn0.z	⊢ 0 = ( 0g ‘ 𝑊 )
2		lspsn0.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
3		eqid	⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
4	1 3	lsssn0	⊢ ( 𝑊 ∈ LMod → { 0 } ∈ ( LSubSp ‘ 𝑊 ) )
5	3 2	lspid	⊢ ( ( 𝑊 ∈ LMod ∧ { 0 } ∈ ( LSubSp ‘ 𝑊 ) ) → ( 𝑁 ‘ { 0 } ) = { 0 } )
6	4 5	mpdan	⊢ ( 𝑊 ∈ LMod → ( 𝑁 ‘ { 0 } ) = { 0 } )