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Theorem lbsind 16152
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 3927 . 2  |-  ( A  e.  ( K  \  {  .0.  } )  <->  ( A  e.  K  /\  A  =/= 
.0.  ) )
2 elfvdm 5757 . . . . . . . 8  |-  ( B  e.  (LBasis `  W
)  ->  W  e.  dom LBasis )
3 lbsss.j . . . . . . . 8  |-  J  =  (LBasis `  W )
42, 3eleq2s 2528 . . . . . . 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 16148 . . . . . . 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 16 . . . . . 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 233 . . . . 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 971 . . . 4  |-  ( B  e.  J  ->  A. x  e.  B  A. y  e.  ( K  \  {  .0.  } )  -.  (
y  .x.  x )  e.  ( N `  ( B  \  { x }
) ) )
15 oveq2 6089 . . . . . . 7  |-  ( x  =  E  ->  (
y  .x.  x )  =  ( y  .x.  E ) )
16 sneq 3825 . . . . . . . . 9  |-  ( x  =  E  ->  { x }  =  { E } )
1716difeq2d 3465 . . . . . . . 8  |-  ( x  =  E  ->  ( B  \  { x }
)  =  ( B 
\  { E }
) )
1817fveq2d 5732 . . . . . . 7  |-  ( x  =  E  ->  ( N `  ( B  \  { x } ) )  =  ( N `
 ( B  \  { E } ) ) )
1915, 18eleq12d 2504 . . . . . 6  |-  ( x  =  E  ->  (
( y  .x.  x
)  e.  ( N `
 ( B  \  { x } ) )  <->  ( y  .x.  E )  e.  ( N `  ( B 
\  { E }
) ) ) )
2019notbid 286 . . . . 5  |-  ( x  =  E  ->  ( -.  ( y  .x.  x
)  e.  ( N `
 ( B  \  { x } ) )  <->  -.  ( y  .x.  E )  e.  ( N `  ( B 
\  { E }
) ) ) )
21 oveq1 6088 . . . . . . 7  |-  ( y  =  A  ->  (
y  .x.  E )  =  ( A  .x.  E ) )
2221eleq1d 2502 . . . . . 6  |-  ( y  =  A  ->  (
( y  .x.  E
)  e.  ( N `
 ( B  \  { E } ) )  <-> 
( A  .x.  E
)  e.  ( N `
 ( B  \  { E } ) ) ) )
2322notbid 286 . . . . 5  |-  ( y  =  A  ->  ( -.  ( y  .x.  E
)  e.  ( N `
 ( B  \  { E } ) )  <->  -.  ( A  .x.  E
)  e.  ( N `
 ( B  \  { E } ) ) ) )
2420, 23rspc2v 3058 . . . 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 28 . . 3  |-  ( B  e.  J  ->  (
( E  e.  B  /\  A  e.  ( K  \  {  .0.  }
) )  ->  -.  ( A  .x.  E )  e.  ( N `  ( B  \  { E } ) ) ) )
2625impl 604 . 2  |-  ( ( ( B  e.  J  /\  E  e.  B
)  /\  A  e.  ( K  \  {  .0.  } ) )  ->  -.  ( A  .x.  E )  e.  ( N `  ( B  \  { E } ) ) )
271, 26sylan2br 463 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705    \ cdif 3317    C_ wss 3320   {csn 3814   dom cdm 4878   ` cfv 5454  (class class class)co 6081   Basecbs 13469  Scalarcsca 13532   .scvsca 13533   0gc0g 13723   LSpanclspn 16047  LBasisclbs 16146
This theorem is referenced by:  lbsind2  16153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-lbs 16147
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