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Theorem lbssp 20555
Description: The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
lbsss.v 𝑉 = (Baseβ€˜π‘Š)
lbsss.j 𝐽 = (LBasisβ€˜π‘Š)
lbssp.n 𝑁 = (LSpanβ€˜π‘Š)
Assertion
Ref Expression
lbssp (𝐡 ∈ 𝐽 β†’ (π‘β€˜π΅) = 𝑉)

Proof of Theorem lbssp
Dummy variables 𝑦 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6880 . . . . 5 (𝐡 ∈ (LBasisβ€˜π‘Š) β†’ π‘Š ∈ dom LBasis)
2 lbsss.j . . . . 5 𝐽 = (LBasisβ€˜π‘Š)
31, 2eleq2s 2852 . . . 4 (𝐡 ∈ 𝐽 β†’ π‘Š ∈ dom LBasis)
4 lbsss.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
5 eqid 2733 . . . . 5 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
6 eqid 2733 . . . . 5 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
7 eqid 2733 . . . . 5 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
8 lbssp.n . . . . 5 𝑁 = (LSpanβ€˜π‘Š)
9 eqid 2733 . . . . 5 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
104, 5, 6, 7, 2, 8, 9islbs 20552 . . . 4 (π‘Š ∈ dom LBasis β†’ (𝐡 ∈ 𝐽 ↔ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (𝑦( ·𝑠 β€˜π‘Š)π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))))
113, 10syl 17 . . 3 (𝐡 ∈ 𝐽 β†’ (𝐡 ∈ 𝐽 ↔ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (𝑦( ·𝑠 β€˜π‘Š)π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))))
1211ibi 267 . 2 (𝐡 ∈ 𝐽 β†’ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (𝑦( ·𝑠 β€˜π‘Š)π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯}))))
1312simp2d 1144 1 (𝐡 ∈ 𝐽 β†’ (π‘β€˜π΅) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   βˆ– cdif 3908   βŠ† wss 3911  {csn 4587  dom cdm 5634  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  Scalarcsca 17141   ·𝑠 cvsca 17142  0gc0g 17326  LSpanclspn 20447  LBasisclbs 20550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-lbs 20551
This theorem is referenced by:  islbs2  20631  islbs3  20632  frlmup3  21222  frlmup4  21223  lmimlbs  21258  lbslcic  21263  lbslsp  32212  lvecdim0i  32358  dimkerim  32379  lindsdom  36118  matunitlindflem2  36121  aacllem  47334
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