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Theorem lbsss 16141
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 5749 . . . . 5  |-  ( B  e.  (LBasis `  W
)  ->  W  e.  dom LBasis )
2 lbsss.j . . . . 5  |-  J  =  (LBasis `  W )
31, 2eleq2s 2527 . . . 4  |-  ( B  e.  J  ->  W  e.  dom LBasis )
4 lbsss.v . . . . 5  |-  V  =  ( Base `  W
)
5 eqid 2435 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
6 eqid 2435 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
7 eqid 2435 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
8 eqid 2435 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
9 eqid 2435 . . . . 5  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
104, 5, 6, 7, 2, 8, 9islbs 16140 . . . 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 1652    e. wcel 1725   A.wral 2697    \ cdif 3309    C_ wss 3312   {csn 3806   dom cdm 4870   ` cfv 5446  (class class class)co 6073   Basecbs 13461  Scalarcsca 13524   .scvsca 13525   0gc0g 13715   LSpanclspn 16039  LBasisclbs 16138
This theorem is referenced by:  lbsel  16142  lbspss  16146  islbs2  16218  islbs3  16219  lmimlbs  27274
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-lbs 16139
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