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Theorem lbsind 18994
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 · = ( ·𝑠𝑊)
lbsind.k 𝐾 = (Base‘𝐹)
lbsind.z 0 = (0g𝐹)
Assertion
Ref Expression
lbsind (((𝐵𝐽𝐸𝐵) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))

Proof of Theorem lbsind
Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifsn 4292 . 2 (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴𝐾𝐴0 ))
2 elfvdm 6178 . . . . . . . 8 (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis)
3 lbsss.j . . . . . . . 8 𝐽 = (LBasis‘𝑊)
42, 3eleq2s 2722 . . . . . . 7 (𝐵𝐽𝑊 ∈ dom LBasis)
5 lbsss.v . . . . . . . 8 𝑉 = (Base‘𝑊)
6 lbsind.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
7 lbsind.s . . . . . . . 8 · = ( ·𝑠𝑊)
8 lbsind.k . . . . . . . 8 𝐾 = (Base‘𝐹)
9 lbssp.n . . . . . . . 8 𝑁 = (LSpan‘𝑊)
10 lbsind.z . . . . . . . 8 0 = (0g𝐹)
115, 6, 7, 8, 3, 9, 10islbs 18990 . . . . . . 7 (𝑊 ∈ dom LBasis → (𝐵𝐽 ↔ (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
124, 11syl 17 . . . . . 6 (𝐵𝐽 → (𝐵𝐽 ↔ (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
1312ibi 256 . . . . 5 (𝐵𝐽 → (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
1413simp3d 1073 . . . 4 (𝐵𝐽 → ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))
15 oveq2 6613 . . . . . . 7 (𝑥 = 𝐸 → (𝑦 · 𝑥) = (𝑦 · 𝐸))
16 sneq 4163 . . . . . . . . 9 (𝑥 = 𝐸 → {𝑥} = {𝐸})
1716difeq2d 3711 . . . . . . . 8 (𝑥 = 𝐸 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝐸}))
1817fveq2d 6154 . . . . . . 7 (𝑥 = 𝐸 → (𝑁‘(𝐵 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝐸})))
1915, 18eleq12d 2698 . . . . . 6 (𝑥 = 𝐸 → ((𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})) ↔ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
2019notbid 308 . . . . 5 (𝑥 = 𝐸 → (¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
21 oveq1 6612 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 · 𝐸) = (𝐴 · 𝐸))
2221eleq1d 2688 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})) ↔ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
2322notbid 308 . . . . 5 (𝑦 = 𝐴 → (¬ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})) ↔ ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
2420, 23rspc2v 3311 . . . 4 ((𝐸𝐵𝐴 ∈ (𝐾 ∖ { 0 })) → (∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
2514, 24syl5com 31 . . 3 (𝐵𝐽 → ((𝐸𝐵𝐴 ∈ (𝐾 ∖ { 0 })) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
2625impl 649 . 2 (((𝐵𝐽𝐸𝐵) ∧ 𝐴 ∈ (𝐾 ∖ { 0 })) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))
271, 26sylan2br 493 1 (((𝐵𝐽𝐸𝐵) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wral 2912  cdif 3557  wss 3560  {csn 4153  dom cdm 5079  cfv 5850  (class class class)co 6605  Basecbs 15776  Scalarcsca 15860   ·𝑠 cvsca 15861  0gc0g 16016  LSpanclspn 18885  LBasisclbs 18988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-lbs 18989
This theorem is referenced by:  lbsind2  18995
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