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Theorem lbsss 20961
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 6934 . . . . 5 (𝐡 ∈ (LBasisβ€˜π‘Š) β†’ π‘Š ∈ dom LBasis)
2 lbsss.j . . . . 5 𝐽 = (LBasisβ€˜π‘Š)
31, 2eleq2s 2847 . . . 4 (𝐡 ∈ 𝐽 β†’ π‘Š ∈ dom LBasis)
4 lbsss.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
5 eqid 2728 . . . . 5 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
6 eqid 2728 . . . . 5 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
7 eqid 2728 . . . . 5 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
8 eqid 2728 . . . . 5 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
9 eqid 2728 . . . . 5 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
104, 5, 6, 7, 2, 8, 9islbs 20960 . . . 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 1140 1 (𝐡 ∈ 𝐽 β†’ 𝐡 βŠ† 𝑉)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058   βˆ– cdif 3944   βŠ† wss 3947  {csn 4629  dom cdm 5678  β€˜cfv 6548  (class class class)co 7420  Basecbs 17179  Scalarcsca 17235   ·𝑠 cvsca 17236  0gc0g 17420  LSpanclspn 20854  LBasisclbs 20958
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
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-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-lbs 20959
This theorem is referenced by:  lbsel  20962  lbspss  20966  islbs2  21041  islbs3  21042  lmimlbs  21769  lbslsp  33092  lmimdim  33297  lvecdim0  33300  lssdimle  33301  lbsdiflsp0  33320  dimkerim  33321  fedgmullem1  33323  fedgmullem2  33324  fedgmul  33325  extdg1id  33351
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