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Theorem lbsind 20923
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 4790 . 2 (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴𝐾𝐴0 ))
2 elfvdm 6928 . . . . . . . 8 (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis)
3 lbsss.j . . . . . . . 8 𝐽 = (LBasis‘𝑊)
42, 3eleq2s 2850 . . . . . . 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 20919 . . . . . . 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 267 . . . . 5 (𝐵𝐽 → (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
1413simp3d 1143 . . . 4 (𝐵𝐽 → ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))
15 oveq2 7420 . . . . . . 7 (𝑥 = 𝐸 → (𝑦 · 𝑥) = (𝑦 · 𝐸))
16 sneq 4638 . . . . . . . . 9 (𝑥 = 𝐸 → {𝑥} = {𝐸})
1716difeq2d 4122 . . . . . . . 8 (𝑥 = 𝐸 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝐸}))
1817fveq2d 6895 . . . . . . 7 (𝑥 = 𝐸 → (𝑁‘(𝐵 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝐸})))
1915, 18eleq12d 2826 . . . . . 6 (𝑥 = 𝐸 → ((𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})) ↔ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
2019notbid 318 . . . . 5 (𝑥 = 𝐸 → (¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
21 oveq1 7419 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 · 𝐸) = (𝐴 · 𝐸))
2221eleq1d 2817 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})) ↔ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
2322notbid 318 . . . . 5 (𝑦 = 𝐴 → (¬ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})) ↔ ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
2420, 23rspc2v 3622 . . . 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 455 . 2 (((𝐵𝐽𝐸𝐵) ∧ 𝐴 ∈ (𝐾 ∖ { 0 })) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))
271, 26sylan2br 594 1 (((𝐵𝐽𝐸𝐵) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939  wral 3060  cdif 3945  wss 3948  {csn 4628  dom cdm 5676  cfv 6543  (class class class)co 7412  Basecbs 17151  Scalarcsca 17207   ·𝑠 cvsca 17208  0gc0g 17392  LSpanclspn 20814  LBasisclbs 20917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-lbs 20918
This theorem is referenced by:  lbsind2  20924
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