Theorem lbsind 21182

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 4754	. 2 ⊢ (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ))
2		elfvdm 6917	. . . . . . . 8 ⊢ (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis)
3		lbsss.j	. . . . . . . 8 ⊢ 𝐽 = (LBasis‘𝑊)
4	2, 3	eleq2s 2881	. . . . . . 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 21178	. . . . . . 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 7420	. . . . . . 7 ⊢ (𝑥 = 𝐸 → (𝑦 · 𝑥) = (𝑦 · 𝐸))
16		sneq 4600	. . . . . . . . 9 ⊢ (𝑥 = 𝐸 → {𝑥} = {𝐸})
17	16	difeq2d 4082	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝐸}))
18	17	fveq2d 6887	. . . . . . 7 ⊢ (𝑥 = 𝐸 → (𝑁‘(𝐵 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝐸})))
19	15, 18	eleq12d 2857	. . . . . 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 7419	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 · 𝐸) = (𝐴 · 𝐸))
22	21	eleq1d 2848	. . . . . 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 3593	. . . 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 460	. 2 ⊢ (((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) ∧ 𝐴 ∈ (𝐾 ∖ { 0 })) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))
27	1, 26	sylan2br 606	1 ⊢ (((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079   ∖ cdif 3903   ⊆ wss 3906  {csn 4590  dom cdm 5663  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17493  LSpanclspn 21073  LBasisclbs 21176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-lbs 21177

This theorem is referenced by:  lbsind2  21183