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Theorem 2llnm4 30098
Description: Two lattice lines that majorize the same atom always meet. (Contributed by NM, 20-Jul-2012.)
Hypotheses
Ref Expression
2llnm4.l  |-  .<_  =  ( le `  K )
2llnm4.m  |-  ./\  =  ( meet `  K )
2llnm4.z  |-  .0.  =  ( 0. `  K )
2llnm4.a  |-  A  =  ( Atoms `  K )
2llnm4.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
2llnm4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  ( X  ./\  Y )  =/= 
.0.  )

Proof of Theorem 2llnm4
StepHypRef Expression
1 hlatl 29889 . . 3  |-  ( K  e.  HL  ->  K  e.  AtLat )
213ad2ant1 978 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  K  e.  AtLat )
3 hllat 29892 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
433ad2ant1 978 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  K  e.  Lat )
5 simp22 991 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  X  e.  N )
6 eqid 2430 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
7 2llnm4.n . . . . 5  |-  N  =  ( LLines `  K )
86, 7llnbase 30037 . . . 4  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
95, 8syl 16 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  X  e.  ( Base `  K
) )
10 simp23 992 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  Y  e.  N )
116, 7llnbase 30037 . . . 4  |-  ( Y  e.  N  ->  Y  e.  ( Base `  K
) )
1210, 11syl 16 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  Y  e.  ( Base `  K
) )
13 2llnm4.m . . . 4  |-  ./\  =  ( meet `  K )
146, 13latmcl 14463 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
154, 9, 12, 14syl3anc 1184 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
16 simp21 990 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  P  e.  A )
17 simp3 959 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  ( P  .<_  X  /\  P  .<_  Y ) )
18 2llnm4.a . . . . . 6  |-  A  =  ( Atoms `  K )
196, 18atbase 29818 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2016, 19syl 16 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  P  e.  ( Base `  K
) )
21 2llnm4.l . . . . 5  |-  .<_  =  ( le `  K )
226, 21, 13latlem12 14490 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  X  /\  P  .<_  Y )  <-> 
P  .<_  ( X  ./\  Y ) ) )
234, 20, 9, 12, 22syl13anc 1186 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  (
( P  .<_  X  /\  P  .<_  Y )  <->  P  .<_  ( X  ./\  Y )
) )
2417, 23mpbid 202 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  P  .<_  ( X  ./\  Y
) )
25 2llnm4.z . . 3  |-  .0.  =  ( 0. `  K )
266, 21, 25, 18atlen0 29839 . 2  |-  ( ( ( K  e.  AtLat  /\  ( X  ./\  Y
)  e.  ( Base `  K )  /\  P  e.  A )  /\  P  .<_  ( X  ./\  Y
) )  ->  ( X  ./\  Y )  =/= 
.0.  )
272, 15, 16, 24, 26syl31anc 1187 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  ( X  ./\  Y )  =/= 
.0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2593   class class class wbr 4199   ` cfv 5440  (class class class)co 6067   Basecbs 13452   lecple 13519   meetcmee 14385   0.cp0 14449   Latclat 14457   Atomscatm 29792   AtLatcal 29793   HLchlt 29879   LLinesclln 30019
This theorem is referenced by:  2llnmeqat  30099  dalem2  30189
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 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
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 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-undef 6529  df-riota 6535  df-poset 14386  df-plt 14398  df-glb 14415  df-meet 14417  df-lat 14458  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880  df-llines 30026
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