Metamath Proof Explorer

Theorem lindfind

Description: A linearly independent family 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 lindfind ( ( ( 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → ¬ ( 𝐴 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 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 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → 𝐸 ∈ dom 𝐹 )
7 eldifsn ⊢ ( 𝐴 ∈ ( 𝐾 ∖ { 0 } ) ↔ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) )
8 7 bilanri ⊢ ( ( ( 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → 𝐴 ∈ ( 𝐾 ∖ { 0 } ) )
9 simpll ⊢ ( ( ( 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → 𝐹 LIndF 𝑊 )
10 3 5 elbasfv ⊢ ( 𝐴 ∈ 𝐾 → 𝑊 ∈ V )
11 10 ad2antrl ⊢ ( ( ( 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → 𝑊 ∈ V )
12 rellindf ⊢ Rel LIndF
13 12 brrelex1i ⊢ ( 𝐹 LIndF 𝑊 → 𝐹 ∈ V )
14 13 ad2antrr ⊢ ( ( ( 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → 𝐹 ∈ V )
15 eqid ⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
16 15 1 2 3 5 4 islindf ⊢ ( ( 𝑊 ∈ V ∧ 𝐹 ∈ V ) → ( 𝐹 LIndF 𝑊 ↔ ( 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑊 ) ∧ ∀ 𝑒 ∈ dom 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · ( 𝐹 ‘ 𝑒 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑒 } ) ) ) ) ) )
17 11 14 16 syl2anc ⊢ ( ( ( 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → ( 𝐹 LIndF 𝑊 ↔ ( 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑊 ) ∧ ∀ 𝑒 ∈ dom 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · ( 𝐹 ‘ 𝑒 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑒 } ) ) ) ) ) )
18 9 17 mpbid ⊢ ( ( ( 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → ( 𝐹 : dom 𝐹 ⟶ ( Base ‘ 𝑊 ) ∧ ∀ 𝑒 ∈ dom 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · ( 𝐹 ‘ 𝑒 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑒 } ) ) ) ) )
19 18 simprd ⊢ ( ( ( 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → ∀ 𝑒 ∈ dom 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · ( 𝐹 ‘ 𝑒 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑒 } ) ) ) )
20 fveq2 ⊢ ( 𝑒 = 𝐸 → ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝐸 ) )
21 20 oveq2d ⊢ ( 𝑒 = 𝐸 → ( 𝑎 · ( 𝐹 ‘ 𝑒 ) ) = ( 𝑎 · ( 𝐹 ‘ 𝐸 ) ) )
22 sneq ⊢ ( 𝑒 = 𝐸 → { 𝑒 } = { 𝐸 } )
23 22 difeq2d ⊢ ( 𝑒 = 𝐸 → ( dom 𝐹 ∖ { 𝑒 } ) = ( dom 𝐹 ∖ { 𝐸 } ) )
24 23 imaeq2d ⊢ ( 𝑒 = 𝐸 → ( 𝐹 “ ( dom 𝐹 ∖ { 𝑒 } ) ) = ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) )
25 24 fveq2d ⊢ ( 𝑒 = 𝐸 → ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑒 } ) ) ) = ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) )
26 21 25 eleq12d ⊢ ( 𝑒 = 𝐸 → ( ( 𝑎 · ( 𝐹 ‘ 𝑒 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑒 } ) ) ) ↔ ( 𝑎 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) ) )
27 26 notbid ⊢ ( 𝑒 = 𝐸 → ( ¬ ( 𝑎 · ( 𝐹 ‘ 𝑒 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑒 } ) ) ) ↔ ¬ ( 𝑎 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) ) )
28 oveq1 ⊢ ( 𝑎 = 𝐴 → ( 𝑎 · ( 𝐹 ‘ 𝐸 ) ) = ( 𝐴 · ( 𝐹 ‘ 𝐸 ) ) )
29 28 eleq1d ⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) ↔ ( 𝐴 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) ) )
30 29 notbid ⊢ ( 𝑎 = 𝐴 → ( ¬ ( 𝑎 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) ↔ ¬ ( 𝐴 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) ) )
31 27 30 rspc2va ⊢ ( ( ( 𝐸 ∈ dom 𝐹 ∧ 𝐴 ∈ ( 𝐾 ∖ { 0 } ) ) ∧ ∀ 𝑒 ∈ dom 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · ( 𝐹 ‘ 𝑒 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑒 } ) ) ) ) → ¬ ( 𝐴 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) )
32 6 8 19 31 syl21anc ⊢ ( ( ( 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → ¬ ( 𝐴 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) )