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Theorem islbs4 19990
 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 6131 . . 3 (𝑋 ∈ (LBasis‘𝑊) → 𝑊 ∈ V)
2 islbs4.j . . 3 𝐽 = (LBasis‘𝑊)
31, 2eleq2s 2706 . 2 (𝑋𝐽𝑊 ∈ V)
4 elfvex 6131 . . 3 (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ V)
54adantr 480 . 2 ((𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵) → 𝑊 ∈ V)
6 islbs4.b . . . 4 𝐵 = (Base‘𝑊)
7 eqid 2610 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
8 eqid 2610 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
9 eqid 2610 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
10 islbs4.k . . . 4 𝐾 = (LSpan‘𝑊)
11 eqid 2610 . . . 4 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
126, 7, 8, 9, 2, 10, 11islbs 18897 . . 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 19971 . . . . 5 (𝑊 ∈ V → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋𝐵 ∧ ∀𝑥𝑋𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥})))))
1413anbi1d 737 . . . 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 1043 . . . 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 278 . . 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 267 . 2 (𝑊 ∈ V → (𝑋𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵)))
183, 5, 17pm5.21nii 367 1 (𝑋𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923  LSpanclspn 18792  LBasisclbs 18895  LIndSclinds 19963 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-lbs 18896  df-lindf 19964  df-linds 19965 This theorem is referenced by:  lbslinds  19991  islinds3  19992  lmimlbs  19994  lindsenlbs  32574
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