Theorem lbsext

Description: For any linearly independent subset C of V , there is a basis containing the vectors in C . (Contributed by Mario Carneiro, 25-Jun-2014) (Revised by Mario Carneiro, 17-May-2015)

	Ref	Expression
Hypotheses	lbsex.j	⊢ 𝐽 = ( LBasis ‘ 𝑊 )
	lbsex.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lbsex.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
Assertion	lbsext	⊢ ( ( 𝑊 ∈ LVec ∧ 𝐶 ⊆ 𝑉 ∧ ∀ 𝑥 ∈ 𝐶 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝐶 ∖ { 𝑥 } ) ) ) → ∃ 𝑠 ∈ 𝐽 𝐶 ⊆ 𝑠 )

Proof

Step	Hyp	Ref	Expression
1		lbsex.j	⊢ 𝐽 = ( LBasis ‘ 𝑊 )
2		lbsex.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
3		lbsex.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
4	2	fvexi	⊢ 𝑉 ∈ V
5	4	pwex	⊢ 𝒫 𝑉 ∈ V
6		numth3	⊢ ( 𝒫 𝑉 ∈ V → 𝒫 𝑉 ∈ dom card )
7	5 6	ax-mp	⊢ 𝒫 𝑉 ∈ dom card
8	7	jctr	⊢ ( 𝑊 ∈ LVec → ( 𝑊 ∈ LVec ∧ 𝒫 𝑉 ∈ dom card ) )
9	1 2 3	lbsextg	⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝒫 𝑉 ∈ dom card ) ∧ 𝐶 ⊆ 𝑉 ∧ ∀ 𝑥 ∈ 𝐶 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝐶 ∖ { 𝑥 } ) ) ) → ∃ 𝑠 ∈ 𝐽 𝐶 ⊆ 𝑠 )
10	8 9	syl3an1	⊢ ( ( 𝑊 ∈ LVec ∧ 𝐶 ⊆ 𝑉 ∧ ∀ 𝑥 ∈ 𝐶 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝐶 ∖ { 𝑥 } ) ) ) → ∃ 𝑠 ∈ 𝐽 𝐶 ⊆ 𝑠 )