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Theorem lbsss 20921
Description: A basis is a set of vectors. (Contributed by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
lbsss.v 𝑉 = (Baseβ€˜π‘Š)
lbsss.j 𝐽 = (LBasisβ€˜π‘Š)
Assertion
Ref Expression
lbsss (𝐡 ∈ 𝐽 β†’ 𝐡 βŠ† 𝑉)

Proof of Theorem lbsss
Dummy variables 𝑦 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6919 . . . . 5 (𝐡 ∈ (LBasisβ€˜π‘Š) β†’ π‘Š ∈ dom LBasis)
2 lbsss.j . . . . 5 𝐽 = (LBasisβ€˜π‘Š)
31, 2eleq2s 2843 . . . 4 (𝐡 ∈ 𝐽 β†’ π‘Š ∈ dom LBasis)
4 lbsss.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
5 eqid 2724 . . . . 5 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
6 eqid 2724 . . . . 5 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
7 eqid 2724 . . . . 5 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
8 eqid 2724 . . . . 5 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
9 eqid 2724 . . . . 5 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
104, 5, 6, 7, 2, 8, 9islbs 20920 . . . 4 (π‘Š ∈ dom LBasis β†’ (𝐡 ∈ 𝐽 ↔ (𝐡 βŠ† 𝑉 ∧ ((LSpanβ€˜π‘Š)β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (𝑦( ·𝑠 β€˜π‘Š)π‘₯) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐡 βˆ– {π‘₯})))))
113, 10syl 17 . . 3 (𝐡 ∈ 𝐽 β†’ (𝐡 ∈ 𝐽 ↔ (𝐡 βŠ† 𝑉 ∧ ((LSpanβ€˜π‘Š)β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (𝑦( ·𝑠 β€˜π‘Š)π‘₯) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐡 βˆ– {π‘₯})))))
1211ibi 267 . 2 (𝐡 ∈ 𝐽 β†’ (𝐡 βŠ† 𝑉 ∧ ((LSpanβ€˜π‘Š)β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (𝑦( ·𝑠 β€˜π‘Š)π‘₯) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐡 βˆ– {π‘₯}))))
1312simp1d 1139 1 (𝐡 ∈ 𝐽 β†’ 𝐡 βŠ† 𝑉)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   βˆ– cdif 3938   βŠ† wss 3941  {csn 4621  dom cdm 5667  β€˜cfv 6534  (class class class)co 7402  Basecbs 17149  Scalarcsca 17205   ·𝑠 cvsca 17206  0gc0g 17390  LSpanclspn 20814  LBasisclbs 20918
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
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-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-lbs 20919
This theorem is referenced by:  lbsel  20922  lbspss  20926  islbs2  21001  islbs3  21002  lmimlbs  21720  lbslsp  32988  lmimdim  33195  lvecdim0  33198  lssdimle  33199  lbsdiflsp0  33218  dimkerim  33219  fedgmullem1  33221  fedgmullem2  33222  fedgmul  33223  extdg1id  33249
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