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Theorem lbssp 15832
Description: The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
lbsss.v  |-  V  =  ( Base `  W
)
lbsss.j  |-  J  =  (LBasis `  W )
lbssp.n  |-  N  =  ( LSpan `  W )
Assertion
Ref Expression
lbssp  |-  ( B  e.  J  ->  ( N `  B )  =  V )

Proof of Theorem lbssp
Dummy variables  y  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 5554 . . . . 5  |-  ( B  e.  (LBasis `  W
)  ->  W  e.  dom LBasis )
2 lbsss.j . . . . 5  |-  J  =  (LBasis `  W )
31, 2eleq2s 2375 . . . 4  |-  ( B  e.  J  ->  W  e.  dom LBasis )
4 lbsss.v . . . . 5  |-  V  =  ( Base `  W
)
5 eqid 2283 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
6 eqid 2283 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
7 eqid 2283 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
8 lbssp.n . . . . 5  |-  N  =  ( LSpan `  W )
9 eqid 2283 . . . . 5  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
104, 5, 6, 7, 2, 8, 9islbs 15829 . . . 4  |-  ( W  e.  dom LBasis  ->  ( B  e.  J  <->  ( B  C_  V  /\  ( N `
 B )  =  V  /\  A. x  e.  B  A. y  e.  ( ( Base `  (Scalar `  W ) )  \  { ( 0g `  (Scalar `  W ) ) } )  -.  (
y ( .s `  W ) x )  e.  ( N `  ( B  \  { x } ) ) ) ) )
113, 10syl 15 . . 3  |-  ( B  e.  J  ->  ( B  e.  J  <->  ( B  C_  V  /\  ( N `
 B )  =  V  /\  A. x  e.  B  A. y  e.  ( ( Base `  (Scalar `  W ) )  \  { ( 0g `  (Scalar `  W ) ) } )  -.  (
y ( .s `  W ) x )  e.  ( N `  ( B  \  { x } ) ) ) ) )
1211ibi 232 . 2  |-  ( B  e.  J  ->  ( B  C_  V  /\  ( N `  B )  =  V  /\  A. x  e.  B  A. y  e.  ( ( Base `  (Scalar `  W ) )  \  { ( 0g `  (Scalar `  W ) ) } )  -.  (
y ( .s `  W ) x )  e.  ( N `  ( B  \  { x } ) ) ) )
1312simp2d 968 1  |-  ( B  e.  J  ->  ( N `  B )  =  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   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:  islbs2  15907  islbs3  15908  frlmup3  27252  frlmup4  27253  lmimlbs  27306  lbslcic  27311
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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