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Theorem llnn0 29705
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 2283 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
21atex 29595 . . . 4  |-  ( K  e.  HL  ->  ( Atoms `  K )  =/=  (/) )
3 n0 3464 . . . 4  |-  ( (
Atoms `  K )  =/=  (/) 
<->  E. p  p  e.  ( Atoms `  K )
)
42, 3sylib 188 . . 3  |-  ( K  e.  HL  ->  E. p  p  e.  ( Atoms `  K ) )
54adantr 451 . 2  |-  ( ( K  e.  HL  /\  X  e.  N )  ->  E. p  p  e.  ( Atoms `  K )
)
6 eqid 2283 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
7 llnn0.n . . . . . . 7  |-  N  =  ( LLines `  K )
86, 1, 7llnnleat 29702 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  N  /\  p  e.  ( Atoms `  K ) )  ->  -.  X ( le `  K ) p )
983expa 1151 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  -.  X ( le `  K ) p )
10 hlop 29552 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OP )
1110ad2antrr 706 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  K  e.  OP )
12 eqid 2283 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
1312, 1atbase 29479 . . . . . . . . 9  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  ( Base `  K )
)
1413adantl 452 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  p  e.  (
Base `  K )
)
15 llnn0.z . . . . . . . . 9  |-  .0.  =  ( 0. `  K )
1612, 6, 15op0le 29376 . . . . . . . 8  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  ->  .0.  ( le `  K
) p )
1711, 14, 16syl2anc 642 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  .0.  ( le `  K ) p )
18 breq1 4026 . . . . . . 7  |-  ( X  =  .0.  ->  ( X ( le `  K ) p  <->  .0.  ( le `  K ) p ) )
1917, 18syl5ibrcom 213 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  ( X  =  .0.  ->  X ( le `  K ) p ) )
2019necon3bd 2483 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  ( -.  X
( le `  K
) p  ->  X  =/=  .0.  ) )
219, 20mpd 14 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  X  =/=  .0.  )
2221ex 423 . . 3  |-  ( ( K  e.  HL  /\  X  e.  N )  ->  ( p  e.  (
Atoms `  K )  ->  X  =/=  .0.  ) )
2322exlimdv 1664 . 2  |-  ( ( K  e.  HL  /\  X  e.  N )  ->  ( E. p  p  e.  ( Atoms `  K
)  ->  X  =/=  .0.  ) )
245, 23mpd 14 1  |-  ( ( K  e.  HL  /\  X  e.  N )  ->  X  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   0.cp0 14143   OPcops 29362   Atomscatm 29453   HLchlt 29540   LLinesclln 29680
This theorem is referenced by:  2llnm3N  29758  cdleme22b  30530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687
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