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Theorem lbsss 15846
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 5570 . . . . 5  |-  ( B  e.  (LBasis `  W
)  ->  W  e.  dom LBasis )
2 lbsss.j . . . . 5  |-  J  =  (LBasis `  W )
31, 2eleq2s 2388 . . . 4  |-  ( B  e.  J  ->  W  e.  dom LBasis )
4 lbsss.v . . . . 5  |-  V  =  ( Base `  W
)
5 eqid 2296 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
6 eqid 2296 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
7 eqid 2296 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
8 eqid 2296 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
9 eqid 2296 . . . . 5  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
104, 5, 6, 7, 2, 8, 9islbs 15845 . . . 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 15 . . 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 232 . 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 967 1  |-  ( B  e.  J  ->  B  C_  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    \ cdif 3162    C_ wss 3165   {csn 3653   dom cdm 4705   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   LSpanclspn 15744  LBasisclbs 15843
This theorem is referenced by:  lbsel  15847  lbspss  15851  islbs2  15923  islbs3  15924  lmimlbs  27409
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
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-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-lbs 15844
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