Theorem lspsnne2

Description: Two ways to express that vectors have different spans. (Contributed by NM, 20-May-2015)

	Ref	Expression
Hypotheses	lspsnne2.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lspsnne2.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
	lspsnne2.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
	lspsnne2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	lspsnne2.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
	lspsnne2.e	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) )
Assertion	lspsnne2	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )

Proof

Step	Hyp	Ref	Expression
1		lspsnne2.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lspsnne2.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
3		lspsnne2.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
4		lspsnne2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
5		lspsnne2.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
6		lspsnne2.e	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) )
7		eqimss	⊢ ( ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) → ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 } ) )
8		eqid	⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
9	1 8 2	lspsncl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) )
10	3 5 9	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) )
11	1 8 2 3 10 4	lspsnel5	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ↔ ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 } ) ) )
12	7 11	syl5ibr	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) )
13	12	necon3bd	⊢ ( 𝜑 → ( ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) )
14	6 13	mpd	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )