MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lbssp Structured version   Unicode version

Theorem lbssp 17703
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 5882 . . . . 5  |-  ( B  e.  (LBasis `  W
)  ->  W  e.  dom LBasis )
2 lbsss.j . . . . 5  |-  J  =  (LBasis `  W )
31, 2eleq2s 2551 . . . 4  |-  ( B  e.  J  ->  W  e.  dom LBasis )
4 lbsss.v . . . . 5  |-  V  =  ( Base `  W
)
5 eqid 2443 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
6 eqid 2443 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
7 eqid 2443 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
8 lbssp.n . . . . 5  |-  N  =  ( LSpan `  W )
9 eqid 2443 . . . . 5  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
104, 5, 6, 7, 2, 8, 9islbs 17700 . . . 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 16 . . 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 241 . 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 1010 1  |-  ( B  e.  J  ->  ( N `  B )  =  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793    \ cdif 3458    C_ wss 3461   {csn 4014   dom cdm 4989   ` cfv 5578  (class class class)co 6281   Basecbs 14613  Scalarcsca 14681   .scvsca 14682   0gc0g 14818   LSpanclspn 17595  LBasisclbs 17698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-lbs 17699
This theorem is referenced by:  islbs2  17778  islbs3  17779  frlmup3  18811  frlmup4  18812  lmimlbs  18848  lbslcic  18853
  Copyright terms: Public domain W3C validator