Theorem lspexchn1

Description: Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch to see if this will shorten proofs. (Contributed by NM, 20-May-2015)

	Ref	Expression
Hypotheses	lspexchn1.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lspexchn1.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
	lspexchn1.w	⊢ ( 𝜑 → 𝑊 ∈ LVec )
	lspexchn1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	lspexchn1.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
	lspexchn1.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
	lspexchn1.q	⊢ ( 𝜑 → ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 } ) )
	lspexchn1.e	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
Assertion	lspexchn1	⊢ ( 𝜑 → ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) )

Proof

Step	Hyp	Ref	Expression
1		lspexchn1.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lspexchn1.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
3		lspexchn1.w	⊢ ( 𝜑 → 𝑊 ∈ LVec )
4		lspexchn1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
5		lspexchn1.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
6		lspexchn1.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
7		lspexchn1.q	⊢ ( 𝜑 → ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 } ) )
8		lspexchn1.e	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
9		eqid	⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 )
10	3	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) → 𝑊 ∈ LVec )
11		eqid	⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
12		lveclmod	⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod )
13	3 12	syl	⊢ ( 𝜑 → 𝑊 ∈ LMod )
14	1 11 2	lspsncl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑍 } ) ∈ ( LSubSp ‘ 𝑊 ) )
15	13 6 14	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ∈ ( LSubSp ‘ 𝑊 ) )
16	9 11 13 15 5 7	lssneln0	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { ( 0g ‘ 𝑊 ) } ) )
17	16	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) → 𝑌 ∈ ( 𝑉 ∖ { ( 0g ‘ 𝑊 ) } ) )
18	4	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) → 𝑋 ∈ 𝑉 )
19	6	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) → 𝑍 ∈ 𝑉 )
20	1 2 13 5 6 7	lspsnne2	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
21	20	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
22		simpr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) → 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) )
23	1 9 2 10 17 18 19 21 22	lspexch	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
24	8 23	mtand	⊢ ( 𝜑 → ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) )