Theorem lbsextlem1

Description: Lemma for lbsext . The set S is the set of all linearly independent sets containing C ; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014)

	Ref	Expression
Hypotheses	lbsext.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lbsext.j	⊢ 𝐽 = ( LBasis ‘ 𝑊 )
	lbsext.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
	lbsext.w	⊢ ( 𝜑 → 𝑊 ∈ LVec )
	lbsext.c	⊢ ( 𝜑 → 𝐶 ⊆ 𝑉 )
	lbsext.x	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐶 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝐶 ∖ { 𝑥 } ) ) )
	lbsext.s	⊢ 𝑆 = { 𝑧 ∈ 𝒫 𝑉 ∣ ( 𝐶 ⊆ 𝑧 ∧ ∀ 𝑥 ∈ 𝑧 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑧 ∖ { 𝑥 } ) ) ) }
Assertion	lbsextlem1	⊢ ( 𝜑 → 𝑆 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		lbsext.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lbsext.j	⊢ 𝐽 = ( LBasis ‘ 𝑊 )
3		lbsext.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
4		lbsext.w	⊢ ( 𝜑 → 𝑊 ∈ LVec )
5		lbsext.c	⊢ ( 𝜑 → 𝐶 ⊆ 𝑉 )
6		lbsext.x	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐶 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝐶 ∖ { 𝑥 } ) ) )
7		lbsext.s	⊢ 𝑆 = { 𝑧 ∈ 𝒫 𝑉 ∣ ( 𝐶 ⊆ 𝑧 ∧ ∀ 𝑥 ∈ 𝑧 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑧 ∖ { 𝑥 } ) ) ) }
8	1	fvexi	⊢ 𝑉 ∈ V
9	8	elpw2	⊢ ( 𝐶 ∈ 𝒫 𝑉 ↔ 𝐶 ⊆ 𝑉 )
10	5 9	sylibr	⊢ ( 𝜑 → 𝐶 ∈ 𝒫 𝑉 )
11		ssid	⊢ 𝐶 ⊆ 𝐶
12	6 11	jctil	⊢ ( 𝜑 → ( 𝐶 ⊆ 𝐶 ∧ ∀ 𝑥 ∈ 𝐶 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝐶 ∖ { 𝑥 } ) ) ) )
13		sseq2	⊢ ( 𝑧 = 𝐶 → ( 𝐶 ⊆ 𝑧 ↔ 𝐶 ⊆ 𝐶 ) )
14		difeq1	⊢ ( 𝑧 = 𝐶 → ( 𝑧 ∖ { 𝑥 } ) = ( 𝐶 ∖ { 𝑥 } ) )
15	14	fveq2d	⊢ ( 𝑧 = 𝐶 → ( 𝑁 ‘ ( 𝑧 ∖ { 𝑥 } ) ) = ( 𝑁 ‘ ( 𝐶 ∖ { 𝑥 } ) ) )
16	15	eleq2d	⊢ ( 𝑧 = 𝐶 → ( 𝑥 ∈ ( 𝑁 ‘ ( 𝑧 ∖ { 𝑥 } ) ) ↔ 𝑥 ∈ ( 𝑁 ‘ ( 𝐶 ∖ { 𝑥 } ) ) ) )
17	16	notbid	⊢ ( 𝑧 = 𝐶 → ( ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑧 ∖ { 𝑥 } ) ) ↔ ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝐶 ∖ { 𝑥 } ) ) ) )
18	17	raleqbi1dv	⊢ ( 𝑧 = 𝐶 → ( ∀ 𝑥 ∈ 𝑧 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑧 ∖ { 𝑥 } ) ) ↔ ∀ 𝑥 ∈ 𝐶 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝐶 ∖ { 𝑥 } ) ) ) )
19	13 18	anbi12d	⊢ ( 𝑧 = 𝐶 → ( ( 𝐶 ⊆ 𝑧 ∧ ∀ 𝑥 ∈ 𝑧 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑧 ∖ { 𝑥 } ) ) ) ↔ ( 𝐶 ⊆ 𝐶 ∧ ∀ 𝑥 ∈ 𝐶 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝐶 ∖ { 𝑥 } ) ) ) ) )
20	19 7	elrab2	⊢ ( 𝐶 ∈ 𝑆 ↔ ( 𝐶 ∈ 𝒫 𝑉 ∧ ( 𝐶 ⊆ 𝐶 ∧ ∀ 𝑥 ∈ 𝐶 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝐶 ∖ { 𝑥 } ) ) ) ) )
21	10 12 20	sylanbrc	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
22	21	ne0d	⊢ ( 𝜑 → 𝑆 ≠ ∅ )