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Theorem lplnn0N 30281
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 2435 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
21atex 30140 . . . 4  |-  ( K  e.  HL  ->  ( Atoms `  K )  =/=  (/) )
3 n0 3629 . . . 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 2435 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
7 lplnn0.p . . . . 5  |-  P  =  ( LPlanes `  K )
86, 1, 7lplnnleat 30276 . . . 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 30097 . . . . . . 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 2435 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1312, 1atbase 30024 . . . . . . 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 29921 . . . . . 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 4207 . . . . 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 2635 . . 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 1648 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 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   (/)c0 3620   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   0.cp0 14458   OPcops 29907   Atomscatm 29998   HLchlt 30085   LPlanesclpl 30226
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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  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-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233
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