Theorem lindsind 22030

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 781	. 2 ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐸 ∈ 𝐹)
2		eldifsn 4748	. . 3 ⊢ (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ))
3	2	bilanri 512	. 2 ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐴 ∈ (𝐾 ∖ { 0 }))
4		elfvdm 6912	. . . . . 6 ⊢ (𝐹 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS)
5		eqid 2760	. . . . . . 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 22026	. . . . . 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 501	. . 3 ⊢ (𝐹 ∈ (LIndS‘𝑊) → ∀𝑒 ∈ 𝐹 ∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))
15	14	ad2antrr 739	. 2 ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ∀𝑒 ∈ 𝐹 ∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))
16		oveq2 7421	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑎 · 𝑒) = (𝑎 · 𝐸))
17		sneq 4594	. . . . . . 7 ⊢ (𝑒 = 𝐸 → {𝑒} = {𝐸})
18	17	difeq2d 4074	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝐹 ∖ {𝑒}) = (𝐹 ∖ {𝐸}))
19	18	fveq2d 6882	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑁‘(𝐹 ∖ {𝑒})) = (𝑁‘(𝐹 ∖ {𝐸})))
20	16, 19	eleq12d 2854	. . . 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 7420	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 · 𝐸) = (𝐴 · 𝐸))
23	22	eleq1d 2845	. . . 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 3588	. 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 851	1 ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076   ∖ cdif 3896   ⊆ wss 3899  {csn 4584  dom cdm 5655  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  Scalarcsca 17345   ·_𝑠 cvsca 17346  0_gc0g 17524  LSpanclspn 21155  LIndSclinds 22018

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-lindf 22019  df-linds 22020

This theorem is used by: (None)