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Theorem lbsss 20680
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 6925 . . . . 5 (𝐡 ∈ (LBasisβ€˜π‘Š) β†’ π‘Š ∈ dom LBasis)
2 lbsss.j . . . . 5 𝐽 = (LBasisβ€˜π‘Š)
31, 2eleq2s 2851 . . . 4 (𝐡 ∈ 𝐽 β†’ π‘Š ∈ dom LBasis)
4 lbsss.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
5 eqid 2732 . . . . 5 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
6 eqid 2732 . . . . 5 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
7 eqid 2732 . . . . 5 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
8 eqid 2732 . . . . 5 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
9 eqid 2732 . . . . 5 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
104, 5, 6, 7, 2, 8, 9islbs 20679 . . . 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 266 . 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 1142 1 (𝐡 ∈ 𝐽 β†’ 𝐡 βŠ† 𝑉)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βˆ– cdif 3944   βŠ† wss 3947  {csn 4627  dom cdm 5675  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  LSpanclspn 20574  LBasisclbs 20677
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-lbs 20678
This theorem is referenced by:  lbsel  20681  lbspss  20685  islbs2  20759  islbs3  20760  lmimlbs  21382  lbslsp  32481  lmimdim  32677  lvecdim0  32679  lssdimle  32680  lbsdiflsp0  32699  dimkerim  32700  fedgmullem1  32702  fedgmullem2  32703  fedgmul  32704  extdg1id  32730
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