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

Theorem lvolbase 30389
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 3473 . . . 4  |-  ( X  e.  V  ->  -.  V  =  (/) )
2 lvolbase.v . . . . 5  |-  V  =  ( LVols `  K )
32eqeq1i 2303 . . . 4  |-  ( V  =  (/)  <->  ( LVols `  K
)  =  (/) )
41, 3sylnib 295 . . 3  |-  ( X  e.  V  ->  -.  ( LVols `  K )  =  (/) )
5 fvprc 5535 . . 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 2296 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
9 eqid 2296 . . . 4  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
107, 8, 9, 2islvol 30384 . . 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 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   (/)c0 3468   class class class wbr 4039   ` cfv 5271   Basecbs 13164    <o ccvr 30074   LPlanesclpl 30303   LVolsclvol 30304
This theorem is referenced by:  islvol2  30391  lvolnle3at  30393  lvolneatN  30399  lvolnelln  30400  lvolnelpln  30401  lplncvrlvol2  30426  lvolcmp  30428  2lplnja  30430
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-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-lvols 30311
  Copyright terms: Public domain W3C validator