Theorem lspprcl

Description: The span of a pair is a subspace (frequently used special case of lspcl ). (Contributed by NM, 11-Apr-2015)

	Ref	Expression
Hypotheses	lspval.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lspval.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
	lspval.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
	lspprcl.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
	lspprcl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	lspprcl.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
Assertion	lspprcl	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		lspval.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lspval.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
3		lspval.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
4		lspprcl.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
5		lspprcl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
6		lspprcl.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
7	5 6	prssd	⊢ ( 𝜑 → { 𝑋 , 𝑌 } ⊆ 𝑉 )
8	1 2 3	lspcl	⊢ ( ( 𝑊 ∈ LMod ∧ { 𝑋 , 𝑌 } ⊆ 𝑉 ) → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ 𝑆 )
9	4 7 8	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ 𝑆 )