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	biimpri	⊢ ( ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ ( 𝐾 ∖ { 0 } ) )
9	8	adantl	⊢ ( ( ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝐸 ∈ 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → 𝐴 ∈ ( 𝐾 ∖ { 0 } ) )
10		elfvdm	⊢ ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) → 𝑊 ∈ dom LIndS )
11		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
12	11 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 } ) ) ) ) )
13	10 12	syl	⊢ ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) → ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ↔ ( 𝐹 ⊆ ( Base ‘ 𝑊 ) ∧ ∀ 𝑒 ∈ 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) ) ) )
14	13	ibi	⊢ ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) → ( 𝐹 ⊆ ( Base ‘ 𝑊 ) ∧ ∀ 𝑒 ∈ 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) ) )
15	14	simprd	⊢ ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) → ∀ 𝑒 ∈ 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) )
16	15	ad2antrr	⊢ ( ( ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝐸 ∈ 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → ∀ 𝑒 ∈ 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) )
17		oveq2	⊢ ( 𝑒 = 𝐸 → ( 𝑎 · 𝑒 ) = ( 𝑎 · 𝐸 ) )
18		sneq	⊢ ( 𝑒 = 𝐸 → { 𝑒 } = { 𝐸 } )
19	18	difeq2d	⊢ ( 𝑒 = 𝐸 → ( 𝐹 ∖ { 𝑒 } ) = ( 𝐹 ∖ { 𝐸 } ) )
20	19	fveq2d	⊢ ( 𝑒 = 𝐸 → ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) = ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) )
21	17 20	eleq12d	⊢ ( 𝑒 = 𝐸 → ( ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) ↔ ( 𝑎 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ) )
22	21	notbid	⊢ ( 𝑒 = 𝐸 → ( ¬ ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) ↔ ¬ ( 𝑎 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ) )
23		oveq1	⊢ ( 𝑎 = 𝐴 → ( 𝑎 · 𝐸 ) = ( 𝐴 · 𝐸 ) )
24	23	eleq1d	⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ↔ ( 𝐴 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ) )
25	24	notbid	⊢ ( 𝑎 = 𝐴 → ( ¬ ( 𝑎 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ↔ ¬ ( 𝐴 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) ) )
26	22 25	rspc2va	⊢ ( ( ( 𝐸 ∈ 𝐹 ∧ 𝐴 ∈ ( 𝐾 ∖ { 0 } ) ) ∧ ∀ 𝑒 ∈ 𝐹 ∀ 𝑎 ∈ ( 𝐾 ∖ { 0 } ) ¬ ( 𝑎 · 𝑒 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝑒 } ) ) ) → ¬ ( 𝐴 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) )
27	6 9 16 26	syl21anc	⊢ ( ( ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝐸 ∈ 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → ¬ ( 𝐴 · 𝐸 ) ∈ ( 𝑁 ‘ ( 𝐹 ∖ { 𝐸 } ) ) )