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Theorem lbslinds 26974
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 2389 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 lbslinds.j . . . 4  |-  J  =  (LBasis `  W )
3 eqid 2389 . . . 4  |-  ( LSpan `  W )  =  (
LSpan `  W )
41, 2, 3islbs4 26973 . . 3  |-  ( a  e.  J  <->  ( a  e.  (LIndS `  W )  /\  ( ( LSpan `  W
) `  a )  =  ( Base `  W
) ) )
54simplbi 447 . 2  |-  ( a  e.  J  ->  a  e.  (LIndS `  W )
)
65ssriv 3297 1  |-  J  C_  (LIndS `  W )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717    C_ wss 3265   ` cfv 5396   Basecbs 13398   LSpanclspn 15976  LBasisclbs 16075  LIndSclinds 26946
This theorem is referenced by:  islinds4  26976  lmimlbs  26977  lbslcic  26982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-lbs 16076  df-lindf 26947  df-linds 26948
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