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Theorem lbsind 15833
Description: A basis is linearly independent; that is, every element has a span which trivially intersects the span of the remainder of the basis. (Contributed by Mario Carneiro, 12-Jan-2015.)
Hypotheses
Ref Expression
lbsss.v  |-  V  =  ( Base `  W
)
lbsss.j  |-  J  =  (LBasis `  W )
lbssp.n  |-  N  =  ( LSpan `  W )
lbsind.f  |-  F  =  (Scalar `  W )
lbsind.s  |-  .x.  =  ( .s `  W )
lbsind.k  |-  K  =  ( Base `  F
)
lbsind.z  |-  .0.  =  ( 0g `  F )
Assertion
Ref Expression
lbsind  |-  ( ( ( B  e.  J  /\  E  e.  B
)  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  -.  ( A  .x.  E
)  e.  ( N `
 ( B  \  { E } ) ) )

Proof of Theorem lbsind
Dummy variables  y  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifsn 3749 . 2  |-  ( A  e.  ( K  \  {  .0.  } )  <->  ( A  e.  K  /\  A  =/= 
.0.  ) )
2 elfvdm 5554 . . . . . . . 8  |-  ( B  e.  (LBasis `  W
)  ->  W  e.  dom LBasis )
3 lbsss.j . . . . . . . 8  |-  J  =  (LBasis `  W )
42, 3eleq2s 2375 . . . . . . 7  |-  ( B  e.  J  ->  W  e.  dom LBasis )
5 lbsss.v . . . . . . . 8  |-  V  =  ( Base `  W
)
6 lbsind.f . . . . . . . 8  |-  F  =  (Scalar `  W )
7 lbsind.s . . . . . . . 8  |-  .x.  =  ( .s `  W )
8 lbsind.k . . . . . . . 8  |-  K  =  ( Base `  F
)
9 lbssp.n . . . . . . . 8  |-  N  =  ( LSpan `  W )
10 lbsind.z . . . . . . . 8  |-  .0.  =  ( 0g `  F )
115, 6, 7, 8, 3, 9, 10islbs 15829 . . . . . . 7  |-  ( W  e.  dom LBasis  ->  ( B  e.  J  <->  ( B  C_  V  /\  ( N `
 B )  =  V  /\  A. x  e.  B  A. y  e.  ( K  \  {  .0.  } )  -.  (
y  .x.  x )  e.  ( N `  ( B  \  { x }
) ) ) ) )
124, 11syl 15 . . . . . 6  |-  ( B  e.  J  ->  ( B  e.  J  <->  ( B  C_  V  /\  ( N `
 B )  =  V  /\  A. x  e.  B  A. y  e.  ( K  \  {  .0.  } )  -.  (
y  .x.  x )  e.  ( N `  ( B  \  { x }
) ) ) ) )
1312ibi 232 . . . . 5  |-  ( B  e.  J  ->  ( B  C_  V  /\  ( N `  B )  =  V  /\  A. x  e.  B  A. y  e.  ( K  \  {  .0.  } )  -.  (
y  .x.  x )  e.  ( N `  ( B  \  { x }
) ) ) )
1413simp3d 969 . . . 4  |-  ( B  e.  J  ->  A. x  e.  B  A. y  e.  ( K  \  {  .0.  } )  -.  (
y  .x.  x )  e.  ( N `  ( B  \  { x }
) ) )
15 oveq2 5866 . . . . . . 7  |-  ( x  =  E  ->  (
y  .x.  x )  =  ( y  .x.  E ) )
16 sneq 3651 . . . . . . . . 9  |-  ( x  =  E  ->  { x }  =  { E } )
1716difeq2d 3294 . . . . . . . 8  |-  ( x  =  E  ->  ( B  \  { x }
)  =  ( B 
\  { E }
) )
1817fveq2d 5529 . . . . . . 7  |-  ( x  =  E  ->  ( N `  ( B  \  { x } ) )  =  ( N `
 ( B  \  { E } ) ) )
1915, 18eleq12d 2351 . . . . . 6  |-  ( x  =  E  ->  (
( y  .x.  x
)  e.  ( N `
 ( B  \  { x } ) )  <->  ( y  .x.  E )  e.  ( N `  ( B 
\  { E }
) ) ) )
2019notbid 285 . . . . 5  |-  ( x  =  E  ->  ( -.  ( y  .x.  x
)  e.  ( N `
 ( B  \  { x } ) )  <->  -.  ( y  .x.  E )  e.  ( N `  ( B 
\  { E }
) ) ) )
21 oveq1 5865 . . . . . . 7  |-  ( y  =  A  ->  (
y  .x.  E )  =  ( A  .x.  E ) )
2221eleq1d 2349 . . . . . 6  |-  ( y  =  A  ->  (
( y  .x.  E
)  e.  ( N `
 ( B  \  { E } ) )  <-> 
( A  .x.  E
)  e.  ( N `
 ( B  \  { E } ) ) ) )
2322notbid 285 . . . . 5  |-  ( y  =  A  ->  ( -.  ( y  .x.  E
)  e.  ( N `
 ( B  \  { E } ) )  <->  -.  ( A  .x.  E
)  e.  ( N `
 ( B  \  { E } ) ) ) )
2420, 23rspc2v 2890 . . . 4  |-  ( ( E  e.  B  /\  A  e.  ( K  \  {  .0.  } ) )  ->  ( A. x  e.  B  A. y  e.  ( K  \  {  .0.  } )  -.  ( y  .x.  x )  e.  ( N `  ( B 
\  { x }
) )  ->  -.  ( A  .x.  E )  e.  ( N `  ( B  \  { E } ) ) ) )
2514, 24syl5com 26 . . 3  |-  ( B  e.  J  ->  (
( E  e.  B  /\  A  e.  ( K  \  {  .0.  }
) )  ->  -.  ( A  .x.  E )  e.  ( N `  ( B  \  { E } ) ) ) )
2625impl 603 . 2  |-  ( ( ( B  e.  J  /\  E  e.  B
)  /\  A  e.  ( K  \  {  .0.  } ) )  ->  -.  ( A  .x.  E )  e.  ( N `  ( B  \  { E } ) ) )
271, 26sylan2br 462 1  |-  ( ( ( B  e.  J  /\  E  e.  B
)  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  -.  ( A  .x.  E
)  e.  ( N `
 ( B  \  { E } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149    C_ wss 3152   {csn 3640   dom cdm 4689   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   LSpanclspn 15728  LBasisclbs 15827
This theorem is referenced by:  lbsind2  15834
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-lbs 15828
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