Theorem lspsnss2

Description: Comparable spans of singletons must have proportional vectors. See lspsneq for equal span version. (Contributed by NM, 7-Jun-2015)

	Ref	Expression
Hypotheses	lspsnss2.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lspsnss2.s	⊢ 𝑆 = ( Scalar ‘ 𝑊 )
	lspsnss2.k	⊢ 𝐾 = ( Base ‘ 𝑆 )
	lspsnss2.t	⊢ · = ( ·𝑠 ‘ 𝑊 )
	lspsnss2.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
	lspsnss2.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
	lspsnss2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	lspsnss2.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
Assertion	lspsnss2	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 } ) ↔ ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lspsnss2.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lspsnss2.s	⊢ 𝑆 = ( Scalar ‘ 𝑊 )
3		lspsnss2.k	⊢ 𝐾 = ( Base ‘ 𝑆 )
4		lspsnss2.t	⊢ · = ( ·𝑠 ‘ 𝑊 )
5		lspsnss2.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
6		lspsnss2.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
7		lspsnss2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
8		lspsnss2.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
9		eqid	⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
10	1 9 5	lspsncl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) )
11	6 8 10	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) )
12	1 9 5 6 11 7	ellspsn5b	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ↔ ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 } ) ) )
13	2 3 1 4 5	ellspsn	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ↔ ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑌 ) ) )
14	6 8 13	syl2anc	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ↔ ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑌 ) ) )
15	12 14	bitr3d	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 } ) ↔ ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑌 ) ) )