Theorem lindsind 21932

Description: A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)

Hypotheses

Ref	Expression
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

Ref	Expression
lindsind	⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))

Proof of Theorem lindsind

Dummy variables 𝑎 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 780	. 2 ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐸 ∈ 𝐹)
2		eldifsn 4755	. . 3 ⊢ (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ))
3	2	bilanri 511	. 2 ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐴 ∈ (𝐾 ∖ { 0 }))
4		elfvdm 6913	. . . . . 6 ⊢ (𝐹 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS)
5		eqid 2769	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
6		lindfind.s	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊)
7		lindfind.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑊)
8		lindfind.l	. . . . . . 7 ⊢ 𝐿 = (Scalar‘𝑊)
9		lindfind.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐿)
10		lindfind.z	. . . . . . 7 ⊢ 0 = (0g‘𝐿)
11	5, 6, 7, 8, 9, 10	islinds2 21928	. . . . . 6 ⊢ (𝑊 ∈ dom LIndS → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ (Base‘𝑊) ∧ ∀𝑒 ∈ 𝐹 ∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))))
12	4, 11	syl 18	. . . . 5 ⊢ (𝐹 ∈ (LIndS‘𝑊) → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ (Base‘𝑊) ∧ ∀𝑒 ∈ 𝐹 ∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))))
13	12	ibi 270	. . . 4 ⊢ (𝐹 ∈ (LIndS‘𝑊) → (𝐹 ⊆ (Base‘𝑊) ∧ ∀𝑒 ∈ 𝐹 ∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒}))))
14	13	simprd 500	. . 3 ⊢ (𝐹 ∈ (LIndS‘𝑊) → ∀𝑒 ∈ 𝐹 ∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))
15	14	ad2antrr 738	. 2 ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ∀𝑒 ∈ 𝐹 ∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))
16		oveq2 7416	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑎 · 𝑒) = (𝑎 · 𝐸))
17		sneq 4601	. . . . . . 7 ⊢ (𝑒 = 𝐸 → {𝑒} = {𝐸})
18	17	difeq2d 4089	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝐹 ∖ {𝑒}) = (𝐹 ∖ {𝐸}))
19	18	fveq2d 6883	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑁‘(𝐹 ∖ {𝑒})) = (𝑁‘(𝐹 ∖ {𝐸})))
20	16, 19	eleq12d 2863	. . . 4 ⊢ (𝑒 = 𝐸 → ((𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})) ↔ (𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
21	20	notbid 321	. . 3 ⊢ (𝑒 = 𝐸 → (¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})) ↔ ¬ (𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
22		oveq1 7415	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 · 𝐸) = (𝐴 · 𝐸))
23	22	eleq1d 2854	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})) ↔ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
24	23	notbid 321	. . 3 ⊢ (𝑎 = 𝐴 → (¬ (𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})) ↔ ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
25	21, 24	rspc2va 3602	. 2 ⊢ (((𝐸 ∈ 𝐹 ∧ 𝐴 ∈ (𝐾 ∖ { 0 })) ∧ ∀𝑒 ∈ 𝐹 ∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒}))) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))
26	1, 3, 15, 25	syl21anc 850	1 ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085   ∖ cdif 3910   ⊆ wss 3913  {csn 4591  dom cdm 5659  ‘cfv 6534  (class class class)co 7408  Basecbs 17265  Scalarcsca 17309   ·_𝑠 cvsca 17310  0_gc0g 17488  LSpanclspn 21066  LIndSclinds 21920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-lindf 21921  df-linds 21922

This theorem is referenced by: (None)