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Metamath Proof Explorer

Theorem lbsss

Description: A basis is a set of vectors. (Contributed by Mario Carneiro, 24-Jun-2014)

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

Proof

Step Hyp Ref Expression
1 lbsss.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 lbsss.j ⊢ 𝐽 = ( LBasis ‘ 𝑊 )
3 elfvdm ⊢ ( 𝐵 ∈ ( LBasis ‘ 𝑊 ) → 𝑊 ∈ dom LBasis )
4 3 2 eleq2s ⊢ ( 𝐵 ∈ 𝐽 → 𝑊 ∈ dom LBasis )
5 eqid ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
6 eqid ⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 )
7 eqid ⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
8 eqid ⊢ ( LSpan ‘ 𝑊 ) = ( LSpan ‘ 𝑊 )
9 eqid ⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) )
10 1 5 6 7 2 8 9 islbs ⊢ ( 𝑊 ∈ dom LBasis → ( 𝐵 ∈ 𝐽 ↔ ( 𝐵 ⊆ 𝑉 ∧ ( ( LSpan ‘ 𝑊 ) ‘ 𝐵 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑦 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐵 ∖ { 𝑥 } ) ) ) ) )
11 4 10 syl ⊢ ( 𝐵 ∈ 𝐽 → ( 𝐵 ∈ 𝐽 ↔ ( 𝐵 ⊆ 𝑉 ∧ ( ( LSpan ‘ 𝑊 ) ‘ 𝐵 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑦 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐵 ∖ { 𝑥 } ) ) ) ) )
12 11 ibi ⊢ ( 𝐵 ∈ 𝐽 → ( 𝐵 ⊆ 𝑉 ∧ ( ( LSpan ‘ 𝑊 ) ‘ 𝐵 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑦 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐵 ∖ { 𝑥 } ) ) ) )
13 12 simp1d ⊢ ( 𝐵 ∈ 𝐽 → 𝐵 ⊆ 𝑉 )