Theorem lbsind 21127

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 4745	. 2 ⊢ (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ))
2		elfvdm 6897	. . . . . . . 8 ⊢ (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis)
3		lbsss.j	. . . . . . . 8 ⊢ 𝐽 = (LBasis‘𝑊)
4	2, 3	eleq2s 2879	. . . . . . 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 21123	. . . . . . 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 1156	. . . 4 ⊢ (𝐵 ∈ 𝐽 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))
15		oveq2 7400	. . . . . . 7 ⊢ (𝑥 = 𝐸 → (𝑦 · 𝑥) = (𝑦 · 𝐸))
16		sneq 4591	. . . . . . . . 9 ⊢ (𝑥 = 𝐸 → {𝑥} = {𝐸})
17	16	difeq2d 4080	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝐸}))
18	17	fveq2d 6867	. . . . . . 7 ⊢ (𝑥 = 𝐸 → (𝑁‘(𝐵 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝐸})))
19	15, 18	eleq12d 2855	. . . . . 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 7399	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 · 𝐸) = (𝐴 · 𝐸))
22	21	eleq1d 2846	. . . . . 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 3592	. . . 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 459	. 2 ⊢ (((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) ∧ 𝐴 ∈ (𝐾 ∖ { 0 })) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))
27	1, 26	sylan2br 604	1 ⊢ (((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075   ∖ cdif 3901   ⊆ wss 3904  {csn 4581  dom cdm 5645  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  Scalarcsca 17272   ·_𝑠 cvsca 17273  0_gc0g 17451  LSpanclspn 21018  LBasisclbs 21121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-ov 7395  df-lbs 21122

This theorem is referenced by:  lbsind2  21128