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Theorem lbssp 19282
Description: The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
lbsss.v 𝑉 = (Base‘𝑊)
lbsss.j 𝐽 = (LBasis‘𝑊)
lbssp.n 𝑁 = (LSpan‘𝑊)
Assertion
Ref Expression
lbssp (𝐵𝐽 → (𝑁𝐵) = 𝑉)

Proof of Theorem lbssp
Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6436 . . . . 5 (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis)
2 lbsss.j . . . . 5 𝐽 = (LBasis‘𝑊)
31, 2eleq2s 2903 . . . 4 (𝐵𝐽𝑊 ∈ dom LBasis)
4 lbsss.v . . . . 5 𝑉 = (Base‘𝑊)
5 eqid 2806 . . . . 5 (Scalar‘𝑊) = (Scalar‘𝑊)
6 eqid 2806 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
7 eqid 2806 . . . . 5 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
8 lbssp.n . . . . 5 𝑁 = (LSpan‘𝑊)
9 eqid 2806 . . . . 5 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
104, 5, 6, 7, 2, 8, 9islbs 19279 . . . 4 (𝑊 ∈ dom LBasis → (𝐵𝐽 ↔ (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑦( ·𝑠𝑊)𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
113, 10syl 17 . . 3 (𝐵𝐽 → (𝐵𝐽 ↔ (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑦( ·𝑠𝑊)𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
1211ibi 258 . 2 (𝐵𝐽 → (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑦( ·𝑠𝑊)𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
1312simp2d 1166 1 (𝐵𝐽 → (𝑁𝐵) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  w3a 1100   = wceq 1637  wcel 2156  wral 3096  cdif 3766  wss 3769  {csn 4370  dom cdm 5311  cfv 6097  (class class class)co 6870  Basecbs 16064  Scalarcsca 16152   ·𝑠 cvsca 16153  0gc0g 16301  LSpanclspn 19174  LBasisclbs 19277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6060  df-fun 6099  df-fv 6105  df-ov 6873  df-lbs 19278
This theorem is referenced by:  islbs2  19359  islbs3  19360  frlmup3  20345  frlmup4  20346  lmimlbs  20381  lbslcic  20386  lindsdom  33711  matunitlindflem2  33714  aacllem  43112
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