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Theorem lbssp 20689
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 6928 . . . . 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 lbssp.n . . . . 5 𝑁 = (LSpanβ€˜π‘Š)
9 eqid 2732 . . . . 5 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
104, 5, 6, 7, 2, 8, 9islbs 20686 . . . 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 266 . 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 1143 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 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βˆ– cdif 3945   βŠ† wss 3948  {csn 4628  dom cdm 5676  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  Scalarcsca 17199   ·𝑠 cvsca 17200  0gc0g 17384  LSpanclspn 20581  LBasisclbs 20684
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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
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 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-lbs 20685
This theorem is referenced by:  islbs2  20766  islbs3  20767  frlmup3  21354  frlmup4  21355  lmimlbs  21390  lbslcic  21395  lbslsp  32488  lvecdim0i  32685  dimkerim  32707  lindsdom  36477  matunitlindflem2  36480  aacllem  47838
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