Theorem lsptpcl

Description: The span of an unordered triple is a subspace (frequently used special case of lspcl ). (Contributed by NM, 22-May-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 )
	lsptpcl.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
Assertion	lsptpcl	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 , 𝑍 } ) ∈ 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		lsptpcl.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
8		df-tp	⊢ { 𝑋 , 𝑌 , 𝑍 } = ( { 𝑋 , 𝑌 } ∪ { 𝑍 } )
9	5 6	prssd	⊢ ( 𝜑 → { 𝑋 , 𝑌 } ⊆ 𝑉 )
10	7	snssd	⊢ ( 𝜑 → { 𝑍 } ⊆ 𝑉 )
11	9 10	unssd	⊢ ( 𝜑 → ( { 𝑋 , 𝑌 } ∪ { 𝑍 } ) ⊆ 𝑉 )
12	8 11	eqsstrid	⊢ ( 𝜑 → { 𝑋 , 𝑌 , 𝑍 } ⊆ 𝑉 )
13	1 2 3	lspcl	⊢ ( ( 𝑊 ∈ LMod ∧ { 𝑋 , 𝑌 , 𝑍 } ⊆ 𝑉 ) → ( 𝑁 ‘ { 𝑋 , 𝑌 , 𝑍 } ) ∈ 𝑆 )
14	4 12 13	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 , 𝑍 } ) ∈ 𝑆 )