Theorem lbsind 19854

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 4721	. 2 ⊢ (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ))
2		elfvdm 6704	. . . . . . . 8 ⊢ (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis)
3		lbsss.j	. . . . . . . 8 ⊢ 𝐽 = (LBasis‘𝑊)
4	2, 3	eleq2s 2933	. . . . . . 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 19850	. . . . . . 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 17	. . . . . 6 ⊢ (𝐵 ∈ 𝐽 → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
13	12	ibi 269	. . . . 5 ⊢ (𝐵 ∈ 𝐽 → (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
14	13	simp3d 1140	. . . 4 ⊢ (𝐵 ∈ 𝐽 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))
15		oveq2 7166	. . . . . . 7 ⊢ (𝑥 = 𝐸 → (𝑦 · 𝑥) = (𝑦 · 𝐸))
16		sneq 4579	. . . . . . . . 9 ⊢ (𝑥 = 𝐸 → {𝑥} = {𝐸})
17	16	difeq2d 4101	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝐸}))
18	17	fveq2d 6676	. . . . . . 7 ⊢ (𝑥 = 𝐸 → (𝑁‘(𝐵 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝐸})))
19	15, 18	eleq12d 2909	. . . . . 6 ⊢ (𝑥 = 𝐸 → ((𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})) ↔ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
20	19	notbid 320	. . . . 5 ⊢ (𝑥 = 𝐸 → (¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
21		oveq1 7165	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 · 𝐸) = (𝐴 · 𝐸))
22	21	eleq1d 2899	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})) ↔ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
23	22	notbid 320	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})) ↔ ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
24	20, 23	rspc2v 3635	. . . 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 31	. . 3 ⊢ (𝐵 ∈ 𝐽 → ((𝐸 ∈ 𝐵 ∧ 𝐴 ∈ (𝐾 ∖ { 0 })) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
26	25	impl 458	. 2 ⊢ (((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) ∧ 𝐴 ∈ (𝐾 ∖ { 0 })) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))
27	1, 26	sylan2br 596	1 ⊢ (((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140   ∖ cdif 3935   ⊆ wss 3938  {csn 4569  dom cdm 5557  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  Scalarcsca 16570   ·_𝑠 cvsca 16571  0_gc0g 16715  LSpanclspn 19745  LBasisclbs 19848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-lbs 19849

This theorem is referenced by:  lbsind2  19855