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Theorem islbs4 20388
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 6362 . . 3 (𝑋 ∈ (LBasis‘𝑊) → 𝑊 ∈ V)
2 islbs4.j . . 3 𝐽 = (LBasis‘𝑊)
31, 2eleq2s 2868 . 2 (𝑋𝐽𝑊 ∈ V)
4 elfvex 6362 . . 3 (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ V)
54adantr 466 . 2 ((𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵) → 𝑊 ∈ V)
6 islbs4.b . . . 4 𝐵 = (Base‘𝑊)
7 eqid 2771 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
8 eqid 2771 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
9 eqid 2771 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
10 islbs4.k . . . 4 𝐾 = (LSpan‘𝑊)
11 eqid 2771 . . . 4 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
126, 7, 8, 9, 2, 10, 11islbs 19289 . . 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 20369 . . . . 5 (𝑊 ∈ V → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋𝐵 ∧ ∀𝑥𝑋𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥})))))
1413anbi1d 615 . . . 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 1082 . . . 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 367 1 (𝑋𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cdif 3720  wss 3723  {csn 4316  cfv 6031  (class class class)co 6793  Basecbs 16064  Scalarcsca 16152   ·𝑠 cvsca 16153  0gc0g 16308  LSpanclspn 19184  LBasisclbs 19287  LIndSclinds 20361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-lbs 19288  df-lindf 20362  df-linds 20363
This theorem is referenced by:  lbslinds  20389  islinds3  20390  lmimlbs  20392  lindsenlbs  33737
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