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Theorem lvolbase 29693
Description: A 3-dim lattice volume is a lattice element. (Contributed by NM, 1-Jul-2012.)
Hypotheses
Ref Expression
lvolbase.b  |-  B  =  ( Base `  K
)
lvolbase.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
lvolbase  |-  ( X  e.  V  ->  X  e.  B )

Proof of Theorem lvolbase
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 n0i 3577 . . . 4  |-  ( X  e.  V  ->  -.  V  =  (/) )
2 lvolbase.v . . . . 5  |-  V  =  ( LVols `  K )
32eqeq1i 2395 . . . 4  |-  ( V  =  (/)  <->  ( LVols `  K
)  =  (/) )
41, 3sylnib 296 . . 3  |-  ( X  e.  V  ->  -.  ( LVols `  K )  =  (/) )
5 fvprc 5663 . . 3  |-  ( -.  K  e.  _V  ->  (
LVols `  K )  =  (/) )
64, 5nsyl2 121 . 2  |-  ( X  e.  V  ->  K  e.  _V )
7 lvolbase.b . . . 4  |-  B  =  ( Base `  K
)
8 eqid 2388 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
9 eqid 2388 . . . 4  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
107, 8, 9, 2islvol 29688 . . 3  |-  ( K  e.  _V  ->  ( X  e.  V  <->  ( X  e.  B  /\  E. x  e.  ( LPlanes `  K )
x (  <o  `  K
) X ) ) )
1110simprbda 607 . 2  |-  ( ( K  e.  _V  /\  X  e.  V )  ->  X  e.  B )
126, 11mpancom 651 1  |-  ( X  e.  V  ->  X  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   E.wrex 2651   _Vcvv 2900   (/)c0 3572   class class class wbr 4154   ` cfv 5395   Basecbs 13397    <o ccvr 29378   LPlanesclpl 29607   LVolsclvol 29608
This theorem is referenced by:  islvol2  29695  lvolnle3at  29697  lvolneatN  29703  lvolnelln  29704  lvolnelpln  29705  lplncvrlvol2  29730  lvolcmp  29732  2lplnja  29734
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-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-lvols 29615
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