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Theorem llnn0 32546
Description: A lattice line is non-zero. (Contributed by NM, 15-Jul-2012.)
Hypotheses
Ref Expression
llnn0.z  |-  .0.  =  ( 0. `  K )
llnn0.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
llnn0  |-  ( ( K  e.  HL  /\  X  e.  N )  ->  X  =/=  .0.  )

Proof of Theorem llnn0
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
21atex 32436 . . . 4  |-  ( K  e.  HL  ->  ( Atoms `  K )  =/=  (/) )
3 n0 3750 . . . 4  |-  ( (
Atoms `  K )  =/=  (/) 
<->  E. p  p  e.  ( Atoms `  K )
)
42, 3sylib 198 . . 3  |-  ( K  e.  HL  ->  E. p  p  e.  ( Atoms `  K ) )
54adantr 465 . 2  |-  ( ( K  e.  HL  /\  X  e.  N )  ->  E. p  p  e.  ( Atoms `  K )
)
6 eqid 2404 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
7 llnn0.n . . . . 5  |-  N  =  ( LLines `  K )
86, 1, 7llnnleat 32543 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  p  e.  ( Atoms `  K ) )  ->  -.  X ( le `  K ) p )
983expa 1199 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  -.  X ( le `  K ) p )
10 hlop 32393 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
1110ad2antrr 726 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  K  e.  OP )
12 eqid 2404 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1312, 1atbase 32320 . . . . . . 7  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  ( Base `  K )
)
1413adantl 466 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  p  e.  (
Base `  K )
)
15 llnn0.z . . . . . . 7  |-  .0.  =  ( 0. `  K )
1612, 6, 15op0le 32217 . . . . . 6  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  ->  .0.  ( le `  K
) p )
1711, 14, 16syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  .0.  ( le `  K ) p )
18 breq1 4400 . . . . 5  |-  ( X  =  .0.  ->  ( X ( le `  K ) p  <->  .0.  ( le `  K ) p ) )
1917, 18syl5ibrcom 224 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  ( X  =  .0.  ->  X ( le `  K ) p ) )
2019necon3bd 2617 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  ( -.  X
( le `  K
) p  ->  X  =/=  .0.  ) )
219, 20mpd 15 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  X  =/=  .0.  )
225, 21exlimddv 1749 1  |-  ( ( K  e.  HL  /\  X  e.  N )  ->  X  =/=  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1407   E.wex 1635    e. wcel 1844    =/= wne 2600   (/)c0 3740   class class class wbr 4397   ` cfv 5571   Basecbs 14843   lecple 14918   0.cp0 15993   OPcops 32203   Atomscatm 32294   HLchlt 32381   LLinesclln 32521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-preset 15883  df-poset 15901  df-plt 15914  df-lub 15930  df-glb 15931  df-join 15932  df-meet 15933  df-p0 15995  df-p1 15996  df-lat 16002  df-clat 16064  df-oposet 32207  df-ol 32209  df-oml 32210  df-covers 32297  df-ats 32298  df-atl 32329  df-cvlat 32353  df-hlat 32382  df-llines 32528
This theorem is referenced by:  2llnm3N  32599  cdleme22b  33373
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