Theorem lbsind 21238

Description: A basis is linearly independent; that is, every element has a span which trivially intersects the span of the remainder of the basis. (Contributed by Mario Carneiro, 12-Jan-2015.)

Hypotheses

Ref	Expression
lbsss.v	⊢ 𝑉 = (Base‘𝑊)
lbsss.j	⊢ 𝐽 = (LBasis‘𝑊)
lbssp.n	⊢ 𝑁 = (LSpan‘𝑊)
lbsind.f	⊢ 𝐹 = (Scalar‘𝑊)
lbsind.s	⊢ · = ( ·𝑠 ‘𝑊)
lbsind.k	⊢ 𝐾 = (Base‘𝐹)
lbsind.z	⊢ 0 = (0g‘𝐹)

Assertion

Ref	Expression
lbsind	⊢ (((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))

Proof of Theorem lbsind

Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifsn 4758	. 2 ⊢ (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ))
2		elfvdm 6922	. . . . . . . 8 ⊢ (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis)
3		lbsss.j	. . . . . . . 8 ⊢ 𝐽 = (LBasis‘𝑊)
4	2, 3	eleq2s 2884	. . . . . . 7 ⊢ (𝐵 ∈ 𝐽 → 𝑊 ∈ dom LBasis)
5		lbsss.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
6		lbsind.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
7		lbsind.s	. . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝑊)
8		lbsind.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹)
9		lbssp.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊)
10		lbsind.z	. . . . . . . 8 ⊢ 0 = (0g‘𝐹)
11	5, 6, 7, 8, 3, 9, 10	islbs 21234	. . . . . . 7 ⊢ (𝑊 ∈ dom LBasis → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
12	4, 11	syl 18	. . . . . 6 ⊢ (𝐵 ∈ 𝐽 → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
13	12	ibi 270	. . . . 5 ⊢ (𝐵 ∈ 𝐽 → (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
14	13	simp3d 1162	. . . 4 ⊢ (𝐵 ∈ 𝐽 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))
15		oveq2 7431	. . . . . . 7 ⊢ (𝑥 = 𝐸 → (𝑦 · 𝑥) = (𝑦 · 𝐸))
16		sneq 4604	. . . . . . . . 9 ⊢ (𝑥 = 𝐸 → {𝑥} = {𝐸})
17	16	difeq2d 4084	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝐸}))
18	17	fveq2d 6892	. . . . . . 7 ⊢ (𝑥 = 𝐸 → (𝑁‘(𝐵 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝐸})))
19	15, 18	eleq12d 2860	. . . . . 6 ⊢ (𝑥 = 𝐸 → ((𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})) ↔ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
20	19	notbid 321	. . . . 5 ⊢ (𝑥 = 𝐸 → (¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
21		oveq1 7430	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 · 𝐸) = (𝐴 · 𝐸))
22	21	eleq1d 2851	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})) ↔ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
23	22	notbid 321	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})) ↔ ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
24	20, 23	rspc2v 3595	. . . 4 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝐴 ∈ (𝐾 ∖ { 0 })) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
25	14, 24	syl5com 32	. . 3 ⊢ (𝐵 ∈ 𝐽 → ((𝐸 ∈ 𝐵 ∧ 𝐴 ∈ (𝐾 ∖ { 0 })) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
26	25	impl 461	. 2 ⊢ (((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) ∧ 𝐴 ∈ (𝐾 ∖ { 0 })) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))
27	1, 26	sylan2br 607	1 ⊢ (((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082   ∖ cdif 3905   ⊆ wss 3908  {csn 4594  dom cdm 5666  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  Scalarcsca 17338   ·_𝑠 cvsca 17339  0_gc0g 17517  LSpanclspn 21129  LBasisclbs 21232

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-lbs 21233

This theorem is used by:  lbsind2  21239