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Theorem lbsind 19128
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 4350 . 2 (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴𝐾𝐴0 ))
2 elfvdm 6258 . . . . . . . 8 (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis)
3 lbsss.j . . . . . . . 8 𝐽 = (LBasis‘𝑊)
42, 3eleq2s 2748 . . . . . . 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 19124 . . . . . . 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 1095 . . . 4 (𝐵𝐽 → ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))
15 oveq2 6698 . . . . . . 7 (𝑥 = 𝐸 → (𝑦 · 𝑥) = (𝑦 · 𝐸))
16 sneq 4220 . . . . . . . . 9 (𝑥 = 𝐸 → {𝑥} = {𝐸})
1716difeq2d 3761 . . . . . . . 8 (𝑥 = 𝐸 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝐸}))
1817fveq2d 6233 . . . . . . 7 (𝑥 = 𝐸 → (𝑁‘(𝐵 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝐸})))
1915, 18eleq12d 2724 . . . . . 6 (𝑥 = 𝐸 → ((𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})) ↔ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
2019notbid 307 . . . . 5 (𝑥 = 𝐸 → (¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
21 oveq1 6697 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 · 𝐸) = (𝐴 · 𝐸))
2221eleq1d 2715 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})) ↔ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
2322notbid 307 . . . . 5 (𝑦 = 𝐴 → (¬ (𝑦 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})) ↔ ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))))
2420, 23rspc2v 3353 . . . 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 492 1 (((𝐵𝐽𝐸𝐵) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  cdif 3604  wss 3607  {csn 4210  dom cdm 5143  cfv 5926  (class class class)co 6690  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147  LSpanclspn 19019  LBasisclbs 19122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-lbs 19123
This theorem is referenced by:  lbsind2  19129
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