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Theorem 2llnm4 33553
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 33344 . . 3  |-  ( K  e.  HL  ->  K  e.  AtLat )
213ad2ant1 1009 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  K  e.  AtLat )
3 hllat 33347 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
433ad2ant1 1009 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  K  e.  Lat )
5 simp22 1022 . . . 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 33492 . . . 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 1023 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  Y  e.  N )
116, 7llnbase 33492 . . . 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 15342 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
154, 9, 12, 14syl3anc 1219 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
16 simp21 1021 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  P  e.  A )
17 simp3 990 . . 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 33273 . . . . 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 15368 . . . 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 1221 . . 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 33294 . 2  |-  ( ( ( K  e.  AtLat  /\  ( X  ./\  Y
)  e.  ( Base `  K )  /\  P  e.  A )  /\  P  .<_  ( X  ./\  Y
) )  ->  ( X  ./\  Y )  =/= 
.0.  )
272, 15, 16, 24, 26syl31anc 1222 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   lecple 14365   meetcmee 15235   0.cp0 15327   Latclat 15335   Atomscatm 33247   AtLatcal 33248   HLchlt 33334   LLinesclln 33474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-lat 15336  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-llines 33481
This theorem is referenced by:  2llnmeqat  33554  dalem2  33644
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