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Theorem islbs4 20090
Description: A basis is an independent spanning set. This could have been used as alternative definition of a basis: LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ (((LSpan‘𝑤) 𝑏) = (Base‘𝑤) ∧ 𝑏 ∈ (LIndS‘𝑤))}). (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
islbs4.b 𝐵 = (Base‘𝑊)
islbs4.j 𝐽 = (LBasis‘𝑊)
islbs4.k 𝐾 = (LSpan‘𝑊)
Assertion
Ref Expression
islbs4 (𝑋𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵))

Proof of Theorem islbs4
Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6178 . . 3 (𝑋 ∈ (LBasis‘𝑊) → 𝑊 ∈ V)
2 islbs4.j . . 3 𝐽 = (LBasis‘𝑊)
31, 2eleq2s 2716 . 2 (𝑋𝐽𝑊 ∈ V)
4 elfvex 6178 . . 3 (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ V)
54adantr 481 . 2 ((𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵) → 𝑊 ∈ V)
6 islbs4.b . . . 4 𝐵 = (Base‘𝑊)
7 eqid 2621 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
8 eqid 2621 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
9 eqid 2621 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
10 islbs4.k . . . 4 𝐾 = (LSpan‘𝑊)
11 eqid 2621 . . . 4 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
126, 7, 8, 9, 2, 10, 11islbs 18995 . . 3 (𝑊 ∈ V → (𝑋𝐽 ↔ (𝑋𝐵 ∧ (𝐾𝑋) = 𝐵 ∧ ∀𝑥𝑋𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥})))))
136, 8, 10, 7, 9, 11islinds2 20071 . . . . 5 (𝑊 ∈ V → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋𝐵 ∧ ∀𝑥𝑋𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥})))))
1413anbi1d 740 . . . 4 (𝑊 ∈ V → ((𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵) ↔ ((𝑋𝐵 ∧ ∀𝑥𝑋𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))) ∧ (𝐾𝑋) = 𝐵)))
15 3anan32 1048 . . . 4 ((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵 ∧ ∀𝑥𝑋𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))) ↔ ((𝑋𝐵 ∧ ∀𝑥𝑋𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))) ∧ (𝐾𝑋) = 𝐵))
1614, 15syl6rbbr 279 . . 3 (𝑊 ∈ V → ((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵 ∧ ∀𝑥𝑋𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))) ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵)))
1712, 16bitrd 268 . 2 (𝑊 ∈ V → (𝑋𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵)))
183, 5, 17pm5.21nii 368 1 (𝑋𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cdif 3552  wss 3555  {csn 4148  cfv 5847  (class class class)co 6604  Basecbs 15781  Scalarcsca 15865   ·𝑠 cvsca 15866  0gc0g 16021  LSpanclspn 18890  LBasisclbs 18993  LIndSclinds 20063
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-lbs 18994  df-lindf 20064  df-linds 20065
This theorem is referenced by:  lbslinds  20091  islinds3  20092  lmimlbs  20094  lindsenlbs  33036
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