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Theorem 2llnm4 35691
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 35482 . . 3  |-  ( K  e.  HL  ->  K  e.  AtLat )
213ad2ant1 1015 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  K  e.  AtLat )
3 hllat 35485 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
433ad2ant1 1015 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  K  e.  Lat )
5 simp22 1028 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  X  e.  N )
6 eqid 2454 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
7 2llnm4.n . . . . 5  |-  N  =  ( LLines `  K )
86, 7llnbase 35630 . . . 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 1029 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  Y  e.  N )
116, 7llnbase 35630 . . . 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 15881 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
154, 9, 12, 14syl3anc 1226 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
16 simp21 1027 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  P  e.  A )
17 simp3 996 . . 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 35411 . . . . 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 15907 . . . 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 1228 . . 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 210 . 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 35432 . 2  |-  ( ( ( K  e.  AtLat  /\  ( X  ./\  Y
)  e.  ( Base `  K )  /\  P  e.  A )  /\  P  .<_  ( X  ./\  Y
) )  ->  ( X  ./\  Y )  =/= 
.0.  )
272, 15, 16, 24, 26syl31anc 1229 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  ( X  ./\  Y )  =/= 
.0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   meetcmee 15773   0.cp0 15866   Latclat 15874   Atomscatm 35385   AtLatcal 35386   HLchlt 35472   LLinesclln 35612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-lat 15875  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619
This theorem is referenced by:  2llnmeqat  35692  dalem2  35782
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