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Theorem lvolbase 30312
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 3625 . . . 4  |-  ( X  e.  V  ->  -.  V  =  (/) )
2 lvolbase.v . . . . 5  |-  V  =  ( LVols `  K )
32eqeq1i 2442 . . . 4  |-  ( V  =  (/)  <->  ( LVols `  K
)  =  (/) )
41, 3sylnib 296 . . 3  |-  ( X  e.  V  ->  -.  ( LVols `  K )  =  (/) )
5 fvprc 5714 . . 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 2435 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
9 eqid 2435 . . . 4  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
107, 8, 9, 2islvol 30307 . . 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 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948   (/)c0 3620   class class class wbr 4204   ` cfv 5446   Basecbs 13461    <o ccvr 29997   LPlanesclpl 30226   LVolsclvol 30227
This theorem is referenced by:  islvol2  30314  lvolnle3at  30316  lvolneatN  30322  lvolnelln  30323  lvolnelpln  30324  lplncvrlvol2  30349  lvolcmp  30351  2lplnja  30353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-lvols 30234
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