Metamath Proof Explorer

Theorem lindsind

Description: A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015)

Ref Expression
Hypotheses lindfind.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
lindfind.n ⊢ 𝑁 = ( LSpan ‘ 𝑊 )
lindfind.l ⊢ 𝐿 = ( Scalar ‘ 𝑊 )
lindfind.z ⊢ 0 = ( 0g ‘ 𝐿 )
lindfind.k ⊢ 𝐾 = ( Base ‘ 𝐿 )
Assertion lindsind ( ( ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝐸 ∈ 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → ¬ ( 𝐴 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) )

Proof

Step Hyp Ref Expression
1 lindfind.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
2 lindfind.n ⊢ 𝑁 = ( LSpan ‘ 𝑊 )
3 lindfind.l ⊢ 𝐿 = ( Scalar ‘ 𝑊 )
4 lindfind.z ⊢ 0 = ( 0g ‘ 𝐿 )
5 lindfind.k ⊢ 𝐾 = ( Base ‘ 𝐿 )
6 simplr ⊢ ( ( ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝐸 ∈ 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → 𝐸 ∈ 𝐹 )
7 eldifsn ⊢ ( 𝐴 ∈ ( 𝐾 ∖ { 0 } ) ↔ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) )
8 7 bilanri ⊢ ( ( ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝐸 ∈ 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → 𝐴 ∈ ( 𝐾 ∖ { 0 } ) )
9 elfvdm ⊢ ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) → 𝑊 ∈ dom LIndS )
10 eqid ⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
11 10 1 2 3 5 4 islinds2 ⊢ ( 𝑊 ∈ dom LIndS → ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ↔ ( 𝐹 ⊆ ( Base ‘ 𝑊 ) ∧ ∀ 𝑒 ∈ 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) ) ) )
12 9 11 syl ⊢ ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) → ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ↔ ( 𝐹 ⊆ ( Base ‘ 𝑊 ) ∧ ∀ 𝑒 ∈ 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) ) ) )
13 12 ibi ⊢ ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) → ( 𝐹 ⊆ ( Base ‘ 𝑊 ) ∧ ∀ 𝑒 ∈ 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) ) )
14 13 simprd ⊢ ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) → ∀ 𝑒 ∈ 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) )
15 14 ad2antrr ⊢ ( ( ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝐸 ∈ 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → ∀ 𝑒 ∈ 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) )
16 oveq2 ⊢ ( 𝑒 = 𝐸 → ( 𝑎 · 𝑒 ) = ( 𝑎 · 𝐸 ) )
17 sneq ⊢ ( 𝑒 = 𝐸 → { 𝑒 } = { 𝐸 } )
18 17 difeq2d ⊢ ( 𝑒 = 𝐸 → ( 𝐹 ∖ { 𝑒 } ) = ( 𝐹 ∖ { 𝐸 } ) )
19 18 fveq2d ⊢ ( 𝑒 = 𝐸 → ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) = ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) )
20 16 19 eleq12d ⊢ ( 𝑒 = 𝐸 → ( ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) ↔ ( 𝑎 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ) )
21 20 notbid ⊢ ( 𝑒 = 𝐸 → ( ¬ ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) ↔ ¬ ( 𝑎 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ) )
22 oveq1 ⊢ ( 𝑎 = 𝐴 → ( 𝑎 · 𝐸 ) = ( 𝐴 · 𝐸 ) )
23 22 eleq1d ⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ↔ ( 𝐴 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ) )
24 23 notbid ⊢ ( 𝑎 = 𝐴 → ( ¬ ( 𝑎 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ↔ ¬ ( 𝐴 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ) )
25 21 24 rspc2va ⊢ ( ( ( 𝐸 ∈ 𝐹 ∧ 𝐴 ∈ ( 𝐾 ∖ { 0 } ) ) ∧ ∀ 𝑒 ∈ 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) ) → ¬ ( 𝐴 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) )
26 6 8 15 25 syl21anc ⊢ ( ( ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝐸 ∈ 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → ¬ ( 𝐴 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) )