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Theorem islbs4 20530
 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 6683 . . 3 (𝑋 ∈ (LBasis‘𝑊) → 𝑊 ∈ V)
2 islbs4.j . . 3 𝐽 = (LBasis‘𝑊)
31, 2eleq2s 2908 . 2 (𝑋𝐽𝑊 ∈ V)
4 elfvex 6683 . . 3 (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ V)
54adantr 484 . 2 ((𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵) → 𝑊 ∈ V)
6 islbs4.b . . . 4 𝐵 = (Base‘𝑊)
7 eqid 2798 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
8 eqid 2798 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
9 eqid 2798 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
10 islbs4.k . . . 4 𝐾 = (LSpan‘𝑊)
11 eqid 2798 . . . 4 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
126, 7, 8, 9, 2, 10, 11islbs 19849 . . 3 (𝑊 ∈ V → (𝑋𝐽 ↔ (𝑋𝐵 ∧ (𝐾𝑋) = 𝐵 ∧ ∀𝑥𝑋𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥})))))
13 3anan32 1094 . . . 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))
146, 8, 10, 7, 9, 11islinds2 20511 . . . . 5 (𝑊 ∈ V → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋𝐵 ∧ ∀𝑥𝑋𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥})))))
1514anbi1d 632 . . . 4 (𝑊 ∈ V → ((𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵) ↔ ((𝑋𝐵 ∧ ∀𝑥𝑋𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))) ∧ (𝐾𝑋) = 𝐵)))
1613, 15bitr4id 293 . . 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 282 . 2 (𝑊 ∈ V → (𝑋𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵)))
183, 5, 17pm5.21nii 383 1 (𝑋𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  {csn 4525  ‘cfv 6327  (class class class)co 7140  Basecbs 16482  Scalarcsca 16567   ·𝑠 cvsca 16568  0gc0g 16712  LSpanclspn 19744  LBasisclbs 19847  LIndSclinds 20503 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7143  df-lbs 19848  df-lindf 20504  df-linds 20505 This theorem is referenced by:  lbslinds  20531  islinds3  20532  lmimlbs  20534  lindflbs  31013  rgmoddim  31132  dimkerim  31147  fedgmullem1  31149  fedgmul  31151  ccfldextdgrr  31181  lindsenlbs  35092
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