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Theorem islbs4 21766
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 6935 . . 3 (𝑋 ∈ (LBasisβ€˜π‘Š) β†’ π‘Š ∈ V)
2 islbs4.j . . 3 𝐽 = (LBasisβ€˜π‘Š)
31, 2eleq2s 2847 . 2 (𝑋 ∈ 𝐽 β†’ π‘Š ∈ V)
4 elfvex 6935 . . 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 2728 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
8 eqid 2728 . . . 4 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
9 eqid 2728 . . . 4 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
10 islbs4.k . . . 4 𝐾 = (LSpanβ€˜π‘Š)
11 eqid 2728 . . . 4 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
126, 7, 8, 9, 2, 10, 11islbs 20961 . . 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 1095 . . . 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 21747 . . . . 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 630 . . . 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 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  Vcvv 3471   βˆ– cdif 3944   βŠ† wss 3947  {csn 4629  β€˜cfv 6548  (class class class)co 7420  Basecbs 17180  Scalarcsca 17236   ·𝑠 cvsca 17237  0gc0g 17421  LSpanclspn 20855  LBasisclbs 20959  LIndSclinds 21739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-lbs 20960  df-lindf 21740  df-linds 21741
This theorem is referenced by:  lbslinds  21767  islinds3  21768  lmimlbs  21770  lindflbs  33107  rlmdim  33307  rgmoddimOLD  33308  dimkerim  33325  fedgmullem1  33327  fedgmul  33329  ccfldextdgrr  33360  lindsenlbs  37088
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