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Theorem islbs4 21716
Description: A basis is an independent spanning set. This could have been used as alternative definition of a basis: LBasis = (𝑀 ∈ V ↦ {𝑏 ∈ 𝒫 (Baseβ€˜π‘€) ∣ (((LSpanβ€˜π‘€) β€˜π‘) = (Baseβ€˜π‘€) ∧ 𝑏 ∈ (LIndSβ€˜π‘€))}). (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
islbs4.b 𝐡 = (Baseβ€˜π‘Š)
islbs4.j 𝐽 = (LBasisβ€˜π‘Š)
islbs4.k 𝐾 = (LSpanβ€˜π‘Š)
Assertion
Ref Expression
islbs4 (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndSβ€˜π‘Š) ∧ (πΎβ€˜π‘‹) = 𝐡))

Proof of Theorem islbs4
Dummy variables π‘˜ π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6920 . . 3 (𝑋 ∈ (LBasisβ€˜π‘Š) β†’ π‘Š ∈ V)
2 islbs4.j . . 3 𝐽 = (LBasisβ€˜π‘Š)
31, 2eleq2s 2843 . 2 (𝑋 ∈ 𝐽 β†’ π‘Š ∈ V)
4 elfvex 6920 . . 3 (𝑋 ∈ (LIndSβ€˜π‘Š) β†’ π‘Š ∈ V)
54adantr 480 . 2 ((𝑋 ∈ (LIndSβ€˜π‘Š) ∧ (πΎβ€˜π‘‹) = 𝐡) β†’ π‘Š ∈ V)
6 islbs4.b . . . 4 𝐡 = (Baseβ€˜π‘Š)
7 eqid 2724 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
8 eqid 2724 . . . 4 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
9 eqid 2724 . . . 4 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
10 islbs4.k . . . 4 𝐾 = (LSpanβ€˜π‘Š)
11 eqid 2724 . . . 4 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
126, 7, 8, 9, 2, 10, 11islbs 20920 . . 3 (π‘Š ∈ V β†’ (𝑋 ∈ 𝐽 ↔ (𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡 ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) ∈ (πΎβ€˜(𝑋 βˆ– {π‘₯})))))
13 3anan32 1094 . . . 4 ((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡 ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) ∈ (πΎβ€˜(𝑋 βˆ– {π‘₯}))) ↔ ((𝑋 βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) ∈ (πΎβ€˜(𝑋 βˆ– {π‘₯}))) ∧ (πΎβ€˜π‘‹) = 𝐡))
146, 8, 10, 7, 9, 11islinds2 21697 . . . . 5 (π‘Š ∈ V β†’ (𝑋 ∈ (LIndSβ€˜π‘Š) ↔ (𝑋 βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) ∈ (πΎβ€˜(𝑋 βˆ– {π‘₯})))))
1514anbi1d 629 . . . 4 (π‘Š ∈ V β†’ ((𝑋 ∈ (LIndSβ€˜π‘Š) ∧ (πΎβ€˜π‘‹) = 𝐡) ↔ ((𝑋 βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) ∈ (πΎβ€˜(𝑋 βˆ– {π‘₯}))) ∧ (πΎβ€˜π‘‹) = 𝐡)))
1613, 15bitr4id 290 . . 3 (π‘Š ∈ V β†’ ((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡 ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) ∈ (πΎβ€˜(𝑋 βˆ– {π‘₯}))) ↔ (𝑋 ∈ (LIndSβ€˜π‘Š) ∧ (πΎβ€˜π‘‹) = 𝐡)))
1712, 16bitrd 279 . 2 (π‘Š ∈ V β†’ (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndSβ€˜π‘Š) ∧ (πΎβ€˜π‘‹) = 𝐡)))
183, 5, 17pm5.21nii 378 1 (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndSβ€˜π‘Š) ∧ (πΎβ€˜π‘‹) = 𝐡))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  Vcvv 3466   βˆ– cdif 3938   βŠ† wss 3941  {csn 4621  β€˜cfv 6534  (class class class)co 7402  Basecbs 17149  Scalarcsca 17205   ·𝑠 cvsca 17206  0gc0g 17390  LSpanclspn 20814  LBasisclbs 20918  LIndSclinds 21689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-lbs 20919  df-lindf 21690  df-linds 21691
This theorem is referenced by:  lbslinds  21717  islinds3  21718  lmimlbs  21720  lindflbs  32991  rlmdim  33202  rgmoddimOLD  33203  dimkerim  33220  fedgmullem1  33222  fedgmul  33224  ccfldextdgrr  33255  lindsenlbs  36987
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