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Theorem lbslinds 27293
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  C_  (LIndS `  W )

Proof of Theorem lbslinds
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 lbslinds.j . . . 4  |-  J  =  (LBasis `  W )
3 eqid 2438 . . . 4  |-  ( LSpan `  W )  =  (
LSpan `  W )
41, 2, 3islbs4 27292 . . 3  |-  ( a  e.  J  <->  ( a  e.  (LIndS `  W )  /\  ( ( LSpan `  W
) `  a )  =  ( Base `  W
) ) )
54simplbi 448 . 2  |-  ( a  e.  J  ->  a  e.  (LIndS `  W )
)
65ssriv 3354 1  |-  J  C_  (LIndS `  W )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726    C_ wss 3322   ` cfv 5457   Basecbs 13474   LSpanclspn 16052  LBasisclbs 16151  LIndSclinds 27265
This theorem is referenced by:  islinds4  27295  lmimlbs  27296  lbslcic  27301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-lbs 16152  df-lindf 27266  df-linds 27267
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