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Theorem lbslinds 20389
Description: A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypothesis
Ref Expression
lbslinds.j 𝐽 = (LBasis‘𝑊)
Assertion
Ref Expression
lbslinds 𝐽 ⊆ (LIndS‘𝑊)

Proof of Theorem lbslinds
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eqid 2771 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 lbslinds.j . . . 4 𝐽 = (LBasis‘𝑊)
3 eqid 2771 . . . 4 (LSpan‘𝑊) = (LSpan‘𝑊)
41, 2, 3islbs4 20388 . . 3 (𝑎𝐽 ↔ (𝑎 ∈ (LIndS‘𝑊) ∧ ((LSpan‘𝑊)‘𝑎) = (Base‘𝑊)))
54simplbi 485 . 2 (𝑎𝐽𝑎 ∈ (LIndS‘𝑊))
65ssriv 3756 1 𝐽 ⊆ (LIndS‘𝑊)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  wss 3723  cfv 6030  Basecbs 16064  LSpanclspn 19184  LBasisclbs 19287  LIndSclinds 20361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-lbs 19288  df-lindf 20362  df-linds 20363
This theorem is referenced by:  islinds4  20391  lmimlbs  20392  lbslcic  20397
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