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Theorem 2llnm4 30541
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 30332 . . 3  |-  ( K  e.  HL  ->  K  e.  AtLat )
213ad2ant1 979 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  K  e.  AtLat )
3 hllat 30335 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
433ad2ant1 979 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  K  e.  Lat )
5 simp22 992 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  X  e.  N )
6 eqid 2443 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
7 2llnm4.n . . . . 5  |-  N  =  ( LLines `  K )
86, 7llnbase 30480 . . . 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 993 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  Y  e.  N )
116, 7llnbase 30480 . . . 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 14518 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
154, 9, 12, 14syl3anc 1185 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
16 simp21 991 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  P  e.  A )
17 simp3 960 . . 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 30261 . . . . 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 14545 . . . 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 1187 . . 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 203 . 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 30282 . 2  |-  ( ( ( K  e.  AtLat  /\  ( X  ./\  Y
)  e.  ( Base `  K )  /\  P  e.  A )  /\  P  .<_  ( X  ./\  Y
) )  ->  ( X  ./\  Y )  =/= 
.0.  )
272, 15, 16, 24, 26syl31anc 1188 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 178    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728    =/= wne 2606   class class class wbr 4243   ` cfv 5489  (class class class)co 6117   Basecbs 13507   lecple 13574   meetcmee 14440   0.cp0 14504   Latclat 14512   Atomscatm 30235   AtLatcal 30236   HLchlt 30322   LLinesclln 30462
This theorem is referenced by:  2llnmeqat  30542  dalem2  30632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-undef 6579  df-riota 6585  df-poset 14441  df-plt 14453  df-glb 14470  df-meet 14472  df-lat 14513  df-covers 30238  df-ats 30239  df-atl 30270  df-cvlat 30294  df-hlat 30323  df-llines 30469
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