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Theorem llnn0 34187
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 2460 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
21atex 34077 . . . 4  |-  ( K  e.  HL  ->  ( Atoms `  K )  =/=  (/) )
3 n0 3787 . . . 4  |-  ( (
Atoms `  K )  =/=  (/) 
<->  E. p  p  e.  ( Atoms `  K )
)
42, 3sylib 196 . . 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 2460 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
7 llnn0.n . . . . 5  |-  N  =  ( LLines `  K )
86, 1, 7llnnleat 34184 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  p  e.  ( Atoms `  K ) )  ->  -.  X ( le `  K ) p )
983expa 1191 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  -.  X ( le `  K ) p )
10 hlop 34034 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
1110ad2antrr 725 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  K  e.  OP )
12 eqid 2460 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1312, 1atbase 33961 . . . . . . 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 33858 . . . . . 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 4443 . . . . 5  |-  ( X  =  .0.  ->  ( X ( le `  K ) p  <->  .0.  ( le `  K ) p ) )
1917, 18syl5ibrcom 222 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N )  /\  p  e.  (
Atoms `  K ) )  ->  ( X  =  .0.  ->  X ( le `  K ) p ) )
2019necon3bd 2672 . . 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 1697 1  |-  ( ( K  e.  HL  /\  X  e.  N )  ->  X  =/=  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2655   (/)c0 3778   class class class wbr 4440   ` cfv 5579   Basecbs 14479   lecple 14551   0.cp0 15513   OPcops 33844   Atomscatm 33935   HLchlt 34022   LLinesclln 34162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169
This theorem is referenced by:  2llnm3N  34240  cdleme22b  35012
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