Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lbslinds Unicode version

Theorem lbslinds 27406
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 2296 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 lbslinds.j . . . 4  |-  J  =  (LBasis `  W )
3 eqid 2296 . . . 4  |-  ( LSpan `  W )  =  (
LSpan `  W )
41, 2, 3islbs4 27405 . . 3  |-  ( a  e.  J  <->  ( a  e.  (LIndS `  W )  /\  ( ( LSpan `  W
) `  a )  =  ( Base `  W
) ) )
54simplbi 446 . 2  |-  ( a  e.  J  ->  a  e.  (LIndS `  W )
)
65ssriv 3197 1  |-  J  C_  (LIndS `  W )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271   Basecbs 13164   LSpanclspn 15744  LBasisclbs 15843  LIndSclinds 27378
This theorem is referenced by:  islinds4  27408  lmimlbs  27409  lbslcic  27414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-lbs 15844  df-lindf 27379  df-linds 27380
  Copyright terms: Public domain W3C validator