Theorem lindsind 21797

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 769	. 2 ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐸 ∈ 𝐹)
2		eldifsn 4731	. . . 4 ⊢ (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ))
3	2	biimpri 228	. . 3 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ (𝐾 ∖ { 0 }))
4	3	adantl 481	. 2 ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐴 ∈ (𝐾 ∖ { 0 }))
5		elfvdm 6874	. . . . . 6 ⊢ (𝐹 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS)
6		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
7		lindfind.s	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊)
8		lindfind.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑊)
9		lindfind.l	. . . . . . 7 ⊢ 𝐿 = (Scalar‘𝑊)
10		lindfind.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐿)
11		lindfind.z	. . . . . . 7 ⊢ 0 = (0g‘𝐿)
12	6, 7, 8, 9, 10, 11	islinds2 21793	. . . . . 6 ⊢ (𝑊 ∈ dom LIndS → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ (Base‘𝑊) ∧ ∀𝑒 ∈ 𝐹 ∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))))
13	5, 12	syl 17	. . . . 5 ⊢ (𝐹 ∈ (LIndS‘𝑊) → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ (Base‘𝑊) ∧ ∀𝑒 ∈ 𝐹 ∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))))
14	13	ibi 267	. . . 4 ⊢ (𝐹 ∈ (LIndS‘𝑊) → (𝐹 ⊆ (Base‘𝑊) ∧ ∀𝑒 ∈ 𝐹 ∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒}))))
15	14	simprd 495	. . 3 ⊢ (𝐹 ∈ (LIndS‘𝑊) → ∀𝑒 ∈ 𝐹 ∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))
16	15	ad2antrr 727	. 2 ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ∀𝑒 ∈ 𝐹 ∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))
17		oveq2 7375	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑎 · 𝑒) = (𝑎 · 𝐸))
18		sneq 4577	. . . . . . 7 ⊢ (𝑒 = 𝐸 → {𝑒} = {𝐸})
19	18	difeq2d 4066	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝐹 ∖ {𝑒}) = (𝐹 ∖ {𝐸}))
20	19	fveq2d 6844	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑁‘(𝐹 ∖ {𝑒})) = (𝑁‘(𝐹 ∖ {𝐸})))
21	17, 20	eleq12d 2830	. . . 4 ⊢ (𝑒 = 𝐸 → ((𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})) ↔ (𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
22	21	notbid 318	. . 3 ⊢ (𝑒 = 𝐸 → (¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})) ↔ ¬ (𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
23		oveq1 7374	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 · 𝐸) = (𝐴 · 𝐸))
24	23	eleq1d 2821	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})) ↔ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
25	24	notbid 318	. . 3 ⊢ (𝑎 = 𝐴 → (¬ (𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})) ↔ ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
26	22, 25	rspc2va 3576	. 2 ⊢ (((𝐸 ∈ 𝐹 ∧ 𝐴 ∈ (𝐾 ∖ { 0 })) ∧ ∀𝑒 ∈ 𝐹 ∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒}))) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))
27	1, 4, 16, 26	syl21anc 838	1 ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051   ∖ cdif 3886   ⊆ wss 3889  {csn 4567  dom cdm 5631  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Scalarcsca 17223   ·_𝑠 cvsca 17224  0_gc0g 17402  LSpanclspn 20966  LIndSclinds 21785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-lindf 21786  df-linds 21787

This theorem is referenced by: (None)