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Theorem lvolbase 33528
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 3740 . . . 4  |-  ( X  e.  V  ->  -.  V  =  (/) )
2 lvolbase.v . . . . 5  |-  V  =  ( LVols `  K )
32eqeq1i 2458 . . . 4  |-  ( V  =  (/)  <->  ( LVols `  K
)  =  (/) )
41, 3sylnib 304 . . 3  |-  ( X  e.  V  ->  -.  ( LVols `  K )  =  (/) )
5 fvprc 5783 . . 3  |-  ( -.  K  e.  _V  ->  (
LVols `  K )  =  (/) )
64, 5nsyl2 127 . 2  |-  ( X  e.  V  ->  K  e.  _V )
7 lvolbase.b . . . 4  |-  B  =  ( Base `  K
)
8 eqid 2451 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
9 eqid 2451 . . . 4  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
107, 8, 9, 2islvol 33523 . . 3  |-  ( K  e.  _V  ->  ( X  e.  V  <->  ( X  e.  B  /\  E. x  e.  ( LPlanes `  K )
x (  <o  `  K
) X ) ) )
1110simprbda 623 . 2  |-  ( ( K  e.  _V  /\  X  e.  V )  ->  X  e.  B )
126, 11mpancom 669 1  |-  ( X  e.  V  ->  X  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   E.wrex 2796   _Vcvv 3068   (/)c0 3735   class class class wbr 4390   ` cfv 5516   Basecbs 14276    <o ccvr 33213   LPlanesclpl 33442   LVolsclvol 33443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-iota 5479  df-fun 5518  df-fv 5524  df-lvols 33450
This theorem is referenced by:  islvol2  33530  lvolnle3at  33532  lvolneatN  33538  lvolnelln  33539  lvolnelpln  33540  lplncvrlvol2  33565  lvolcmp  33567  2lplnja  33569
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