Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lvolbase Unicode version

Theorem lvolbase 29767
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 3460 . . . 4  |-  ( X  e.  V  ->  -.  V  =  (/) )
2 lvolbase.v . . . . 5  |-  V  =  ( LVols `  K )
32eqeq1i 2290 . . . 4  |-  ( V  =  (/)  <->  ( LVols `  K
)  =  (/) )
41, 3sylnib 295 . . 3  |-  ( X  e.  V  ->  -.  ( LVols `  K )  =  (/) )
5 fvprc 5519 . . 3  |-  ( -.  K  e.  _V  ->  (
LVols `  K )  =  (/) )
64, 5nsyl2 119 . 2  |-  ( X  e.  V  ->  K  e.  _V )
7 lvolbase.b . . . 4  |-  B  =  ( Base `  K
)
8 eqid 2283 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
9 eqid 2283 . . . 4  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
107, 8, 9, 2islvol 29762 . . 3  |-  ( K  e.  _V  ->  ( X  e.  V  <->  ( X  e.  B  /\  E. x  e.  ( LPlanes `  K )
x (  <o  `  K
) X ) ) )
1110simprbda 606 . 2  |-  ( ( K  e.  _V  /\  X  e.  V )  ->  X  e.  B )
126, 11mpancom 650 1  |-  ( X  e.  V  ->  X  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   (/)c0 3455   class class class wbr 4023   ` cfv 5255   Basecbs 13148    <o ccvr 29452   LPlanesclpl 29681   LVolsclvol 29682
This theorem is referenced by:  islvol2  29769  lvolnle3at  29771  lvolneatN  29777  lvolnelln  29778  lvolnelpln  29779  lplncvrlvol2  29804  lvolcmp  29806  2lplnja  29808
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-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-lvols 29689
  Copyright terms: Public domain W3C validator