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Theorem lplnn0N 29662
Description: A lattice plane is non-zero. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lplnn0.z  |-  .0.  =  ( 0. `  K )
lplnn0.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
lplnn0N  |-  ( ( K  e.  HL  /\  X  e.  P )  ->  X  =/=  .0.  )

Proof of Theorem lplnn0N
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 eqid 2388 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
21atex 29521 . . . 4  |-  ( K  e.  HL  ->  ( Atoms `  K )  =/=  (/) )
3 n0 3581 . . . 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.  P )  ->  E. p  p  e.  ( Atoms `  K )
)
6 eqid 2388 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
7 lplnn0.p . . . . 5  |-  P  =  ( LPlanes `  K )
86, 1, 7lplnnleat 29657 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  p  e.  ( Atoms `  K ) )  ->  -.  X ( le `  K ) p )
983expa 1153 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P )  /\  p  e.  (
Atoms `  K ) )  ->  -.  X ( le `  K ) p )
10 hlop 29478 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
1110ad2antrr 707 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  P )  /\  p  e.  (
Atoms `  K ) )  ->  K  e.  OP )
12 eqid 2388 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1312, 1atbase 29405 . . . . . . 7  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  ( Base `  K )
)
1413adantl 453 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  P )  /\  p  e.  (
Atoms `  K ) )  ->  p  e.  (
Base `  K )
)
15 lplnn0.z . . . . . . 7  |-  .0.  =  ( 0. `  K )
1612, 6, 15op0le 29302 . . . . . 6  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  ->  .0.  ( le `  K
) p )
1711, 14, 16syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  P )  /\  p  e.  (
Atoms `  K ) )  ->  .0.  ( le `  K ) p )
18 breq1 4157 . . . . 5  |-  ( X  =  .0.  ->  ( X ( le `  K ) p  <->  .0.  ( le `  K ) p ) )
1917, 18syl5ibrcom 214 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P )  /\  p  e.  (
Atoms `  K ) )  ->  ( X  =  .0.  ->  X ( le `  K ) p ) )
2019necon3bd 2588 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P )  /\  p  e.  (
Atoms `  K ) )  ->  ( -.  X
( le `  K
) p  ->  X  =/=  .0.  ) )
219, 20mpd 15 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P )  /\  p  e.  (
Atoms `  K ) )  ->  X  =/=  .0.  )
225, 21exlimddv 1645 1  |-  ( ( K  e.  HL  /\  X  e.  P )  ->  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 2551   (/)c0 3572   class class class wbr 4154   ` cfv 5395   Basecbs 13397   lecple 13464   0.cp0 14394   OPcops 29288   Atomscatm 29379   HLchlt 29466   LPlanesclpl 29607
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-llines 29613  df-lplanes 29614
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