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	biimpri	⊢ ( ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ ( 𝐾 ∖ { 0 } ) )
9	8	adantl	⊢ ( ( ( 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → 𝐴 ∈ ( 𝐾 ∖ { 0 } ) )
10		simpll	⊢ ( ( ( 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → 𝐹 LIndF 𝑊 )
11	3 5	elbasfv	⊢ ( 𝐴 ∈ 𝐾 → 𝑊 ∈ V )
12	11	ad2antrl	⊢ ( ( ( 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → 𝑊 ∈ V )
13		rellindf	⊢ Rel LIndF
14	13	brrelex1i	⊢ ( 𝐹 LIndF 𝑊 → 𝐹 ∈ V )
15	14	ad2antrr	⊢ ( ( ( 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → 𝐹 ∈ V )
16		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
17	16 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 } ) ) ) ) ) )
18	12 15 17	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 } ) ) ) ) ) )
19	10 18	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 } ) ) ) ) )
20	19	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 } ) ) ) )
21		fveq2	⊢ ( 𝑒 = 𝐸 → ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝐸 ) )
22	21	oveq2d	⊢ ( 𝑒 = 𝐸 → ( 𝑎 · ( 𝐹 ‘ 𝑒 ) ) = ( 𝑎 · ( 𝐹 ‘ 𝐸 ) ) )
23		sneq	⊢ ( 𝑒 = 𝐸 → { 𝑒 } = { 𝐸 } )
24	23	difeq2d	⊢ ( 𝑒 = 𝐸 → ( dom 𝐹 ∖ { 𝑒 } ) = ( dom 𝐹 ∖ { 𝐸 } ) )
25	24	imaeq2d	⊢ ( 𝑒 = 𝐸 → ( 𝐹 “ ( dom 𝐹 ∖ { 𝑒 } ) ) = ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) )
26	25	fveq2d	⊢ ( 𝑒 = 𝐸 → ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑒 } ) ) ) = ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) )
27	22 26	eleq12d	⊢ ( 𝑒 = 𝐸 → ( ( 𝑎 · ( 𝐹 ‘ 𝑒 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑒 } ) ) ) ↔ ( 𝑎 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) ) )
28	27	notbid	⊢ ( 𝑒 = 𝐸 → ( ¬ ( 𝑎 · ( 𝐹 ‘ 𝑒 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝑒 } ) ) ) ↔ ¬ ( 𝑎 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) ) )
29		oveq1	⊢ ( 𝑎 = 𝐴 → ( 𝑎 · ( 𝐹 ‘ 𝐸 ) ) = ( 𝐴 · ( 𝐹 ‘ 𝐸 ) ) )
30	29	eleq1d	⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) ↔ ( 𝐴 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) ) )
31	30	notbid	⊢ ( 𝑎 = 𝐴 → ( ¬ ( 𝑎 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) ↔ ¬ ( 𝐴 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) ) )
32	28 31	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 } ) ) ) )
33	6 9 20 32	syl21anc	⊢ ( ( ( 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹 ) ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) ) → ¬ ( 𝐴 · ( 𝐹 ‘ 𝐸 ) ) ∈ ( 𝑁 ‘ ( 𝐹 “ ( dom 𝐹 ∖ { 𝐸 } ) ) ) )