Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lvoln0N Structured version   Unicode version

Theorem lvoln0N 33075
Description: A lattice volume is non-zero. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lvoln0.z  |-  .0.  =  ( 0. `  K )
lvoln0.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
lvoln0N  |-  ( ( K  e.  HL  /\  X  e.  V )  ->  X  =/=  .0.  )

Proof of Theorem lvoln0N
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
21atex 32890 . . . 4  |-  ( K  e.  HL  ->  ( Atoms `  K )  =/=  (/) )
3 n0 3641 . . . 4  |-  ( (
Atoms `  K )  =/=  (/) 
<->  E. p  p  e.  ( Atoms `  K )
)
42, 3sylib 196 . . 3  |-  ( K  e.  HL  ->  E. p  p  e.  ( Atoms `  K ) )
54adantr 465 . 2  |-  ( ( K  e.  HL  /\  X  e.  V )  ->  E. p  p  e.  ( Atoms `  K )
)
6 eqid 2438 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
7 lvoln0.v . . . . 5  |-  V  =  ( LVols `  K )
86, 1, 7lvolnleat 33067 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  V  /\  p  e.  ( Atoms `  K ) )  ->  -.  X ( le `  K ) p )
983expa 1187 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  V )  /\  p  e.  (
Atoms `  K ) )  ->  -.  X ( le `  K ) p )
10 hlop 32847 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
1110ad2antrr 725 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  V )  /\  p  e.  (
Atoms `  K ) )  ->  K  e.  OP )
12 eqid 2438 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1312, 1atbase 32774 . . . . . . 7  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  ( Base `  K )
)
1413adantl 466 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  V )  /\  p  e.  (
Atoms `  K ) )  ->  p  e.  (
Base `  K )
)
15 lvoln0.z . . . . . . 7  |-  .0.  =  ( 0. `  K )
1612, 6, 15op0le 32671 . . . . . 6  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  ->  .0.  ( le `  K
) p )
1711, 14, 16syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  V )  /\  p  e.  (
Atoms `  K ) )  ->  .0.  ( le `  K ) p )
18 breq1 4290 . . . . 5  |-  ( X  =  .0.  ->  ( X ( le `  K ) p  <->  .0.  ( le `  K ) p ) )
1917, 18syl5ibrcom 222 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  V )  /\  p  e.  (
Atoms `  K ) )  ->  ( X  =  .0.  ->  X ( le `  K ) p ) )
2019necon3bd 2640 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  V )  /\  p  e.  (
Atoms `  K ) )  ->  ( -.  X
( le `  K
) p  ->  X  =/=  .0.  ) )
219, 20mpd 15 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  V )  /\  p  e.  (
Atoms `  K ) )  ->  X  =/=  .0.  )
225, 21exlimddv 1692 1  |-  ( ( K  e.  HL  /\  X  e.  V )  ->  X  =/=  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2601   (/)c0 3632   class class class wbr 4287   ` cfv 5413   Basecbs 14166   lecple 14237   0.cp0 15199   OPcops 32657   Atomscatm 32748   HLchlt 32835   LVolsclvol 32977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-llines 32982  df-lplanes 32983  df-lvols 32984
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator