Theorem lbssp

Description: The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014)

	Ref	Expression
Hypotheses	lbsss.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lbsss.j	⊢ 𝐽 = ( LBasis ‘ 𝑊 )
	lbssp.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
Assertion	lbssp	⊢ ( 𝐵 ∈ 𝐽 → ( 𝑁 ‘ 𝐵 ) = 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		lbsss.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lbsss.j	⊢ 𝐽 = ( LBasis ‘ 𝑊 )
3		lbssp.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
4		elfvdm	⊢ ( 𝐵 ∈ ( LBasis ‘ 𝑊 ) → 𝑊 ∈ dom LBasis )
5	4 2	eleq2s	⊢ ( 𝐵 ∈ 𝐽 → 𝑊 ∈ dom LBasis )
6		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
7		eqid	⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 )
8		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
9		eqid	⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) )
10	1 6 7 8 2 3 9	islbs	⊢ ( 𝑊 ∈ dom LBasis → ( 𝐵 ∈ 𝐽 ↔ ( 𝐵 ⊆ 𝑉 ∧ ( 𝑁 ‘ 𝐵 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑦 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( 𝑁 ‘ ( 𝐵 ∖ { 𝑥 } ) ) ) ) )
11	5 10	syl	⊢ ( 𝐵 ∈ 𝐽 → ( 𝐵 ∈ 𝐽 ↔ ( 𝐵 ⊆ 𝑉 ∧ ( 𝑁 ‘ 𝐵 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑦 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( 𝑁 ‘ ( 𝐵 ∖ { 𝑥 } ) ) ) ) )
12	11	ibi	⊢ ( 𝐵 ∈ 𝐽 → ( 𝐵 ⊆ 𝑉 ∧ ( 𝑁 ‘ 𝐵 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑦 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( 𝑁 ‘ ( 𝐵 ∖ { 𝑥 } ) ) ) )
13	12	simp2d	⊢ ( 𝐵 ∈ 𝐽 → ( 𝑁 ‘ 𝐵 ) = 𝑉 )