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Theorem lbsss 16077
Description: A basis is a set of vectors. (Contributed by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
lbsss.v  |-  V  =  ( Base `  W
)
lbsss.j  |-  J  =  (LBasis `  W )
Assertion
Ref Expression
lbsss  |-  ( B  e.  J  ->  B  C_  V )

Proof of Theorem lbsss
Dummy variables  y  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 5698 . . . . 5  |-  ( B  e.  (LBasis `  W
)  ->  W  e.  dom LBasis )
2 lbsss.j . . . . 5  |-  J  =  (LBasis `  W )
31, 2eleq2s 2480 . . . 4  |-  ( B  e.  J  ->  W  e.  dom LBasis )
4 lbsss.v . . . . 5  |-  V  =  ( Base `  W
)
5 eqid 2388 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
6 eqid 2388 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
7 eqid 2388 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
8 eqid 2388 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
9 eqid 2388 . . . . 5  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
104, 5, 6, 7, 2, 8, 9islbs 16076 . . . 4  |-  ( W  e.  dom LBasis  ->  ( B  e.  J  <->  ( B  C_  V  /\  ( (
LSpan `  W ) `  B )  =  V  /\  A. x  e.  B  A. y  e.  ( ( Base `  (Scalar `  W ) )  \  { ( 0g `  (Scalar `  W ) ) } )  -.  (
y ( .s `  W ) x )  e.  ( ( LSpan `  W ) `  ( B  \  { x }
) ) ) ) )
113, 10syl 16 . . 3  |-  ( B  e.  J  ->  ( B  e.  J  <->  ( B  C_  V  /\  ( (
LSpan `  W ) `  B )  =  V  /\  A. x  e.  B  A. y  e.  ( ( Base `  (Scalar `  W ) )  \  { ( 0g `  (Scalar `  W ) ) } )  -.  (
y ( .s `  W ) x )  e.  ( ( LSpan `  W ) `  ( B  \  { x }
) ) ) ) )
1211ibi 233 . 2  |-  ( B  e.  J  ->  ( B  C_  V  /\  (
( LSpan `  W ) `  B )  =  V  /\  A. x  e.  B  A. y  e.  ( ( Base `  (Scalar `  W ) )  \  { ( 0g `  (Scalar `  W ) ) } )  -.  (
y ( .s `  W ) x )  e.  ( ( LSpan `  W ) `  ( B  \  { x }
) ) ) )
1312simp1d 969 1  |-  ( B  e.  J  ->  B  C_  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650    \ cdif 3261    C_ wss 3264   {csn 3758   dom cdm 4819   ` cfv 5395  (class class class)co 6021   Basecbs 13397  Scalarcsca 13460   .scvsca 13461   0gc0g 13651   LSpanclspn 15975  LBasisclbs 16074
This theorem is referenced by:  lbsel  16078  lbspss  16082  islbs2  16154  islbs3  16155  lmimlbs  26976
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403  df-ov 6024  df-lbs 16075
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