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Theorem lvoln0N 29707
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 2389 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
21atex 29522 . . . 4  |-  ( K  e.  HL  ->  ( Atoms `  K )  =/=  (/) )
3 n0 3582 . . . 4  |-  ( (
Atoms `  K )  =/=  (/) 
<->  E. p  p  e.  ( Atoms `  K )
)
42, 3sylib 189 . . 3  |-  ( K  e.  HL  ->  E. p  p  e.  ( Atoms `  K ) )
54adantr 452 . 2  |-  ( ( K  e.  HL  /\  X  e.  V )  ->  E. p  p  e.  ( Atoms `  K )
)
6 eqid 2389 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
7 lvoln0.v . . . . 5  |-  V  =  ( LVols `  K )
86, 1, 7lvolnleat 29699 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  V  /\  p  e.  ( Atoms `  K ) )  ->  -.  X ( le `  K ) p )
983expa 1153 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  V )  /\  p  e.  (
Atoms `  K ) )  ->  -.  X ( le `  K ) p )
10 hlop 29479 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
1110ad2antrr 707 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  V )  /\  p  e.  (
Atoms `  K ) )  ->  K  e.  OP )
12 eqid 2389 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1312, 1atbase 29406 . . . . . . 7  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  ( Base `  K )
)
1413adantl 453 . . . . . 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 29303 . . . . . 6  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  ->  .0.  ( le `  K
) p )
1711, 14, 16syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  V )  /\  p  e.  (
Atoms `  K ) )  ->  .0.  ( le `  K ) p )
18 breq1 4158 . . . . 5  |-  ( X  =  .0.  ->  ( X ( le `  K ) p  <->  .0.  ( le `  K ) p ) )
1917, 18syl5ibrcom 214 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  V )  /\  p  e.  (
Atoms `  K ) )  ->  ( X  =  .0.  ->  X ( le `  K ) p ) )
2019necon3bd 2589 . . 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 1645 1  |-  ( ( K  e.  HL  /\  X  e.  V )  ->  X  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2552   (/)c0 3573   class class class wbr 4155   ` cfv 5396   Basecbs 13398   lecple 13465   0.cp0 14395   OPcops 29289   Atomscatm 29380   HLchlt 29467   LVolsclvol 29609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-llines 29614  df-lplanes 29615  df-lvols 29616
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