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Theorem lbslinds 21728
Description: A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypothesis
Ref Expression
lbslinds.j 𝐽 = (LBasisβ€˜π‘Š)
Assertion
Ref Expression
lbslinds 𝐽 βŠ† (LIndSβ€˜π‘Š)

Proof of Theorem lbslinds
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 eqid 2726 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 lbslinds.j . . . 4 𝐽 = (LBasisβ€˜π‘Š)
3 eqid 2726 . . . 4 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
41, 2, 3islbs4 21727 . . 3 (π‘Ž ∈ 𝐽 ↔ (π‘Ž ∈ (LIndSβ€˜π‘Š) ∧ ((LSpanβ€˜π‘Š)β€˜π‘Ž) = (Baseβ€˜π‘Š)))
54simplbi 497 . 2 (π‘Ž ∈ 𝐽 β†’ π‘Ž ∈ (LIndSβ€˜π‘Š))
65ssriv 3981 1 𝐽 βŠ† (LIndSβ€˜π‘Š)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  β€˜cfv 6537  Basecbs 17153  LSpanclspn 20818  LBasisclbs 20922  LIndSclinds 21700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-lbs 20923  df-lindf 21701  df-linds 21702
This theorem is referenced by:  islinds4  21730  lmimlbs  21731  lbslcic  21736  lvecdim0  33209  lssdimle  33210  lbsdiflsp0  33229  dimkerim  33230  fedgmullem2  33233  fedgmul  33234  extdg1id  33260
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