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Theorem 2llnm3N 33516
Description: Two lattice lines in a lattice plane always meet. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2llnm3.l  |-  .<_  =  ( le `  K )
2llnm3.m  |-  ./\  =  ( meet `  K )
2llnm3.z  |-  .0.  =  ( 0. `  K )
2llnm3.n  |-  N  =  ( LLines `  K )
2llnm3.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
2llnm3N  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  ->  ( X  ./\  Y )  =/= 
.0.  )

Proof of Theorem 2llnm3N
StepHypRef Expression
1 oveq1 6194 . . 3  |-  ( X  =  Y  ->  ( X  ./\  Y )  =  ( Y  ./\  Y
) )
21neeq1d 2723 . 2  |-  ( X  =  Y  ->  (
( X  ./\  Y
)  =/=  .0.  <->  ( Y  ./\ 
Y )  =/=  .0.  ) )
3 simpl1 991 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  /\  X  =/=  Y )  ->  K  e.  HL )
4 hlatl 33308 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
53, 4syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  /\  X  =/=  Y )  ->  K  e.  AtLat )
6 simpl2 992 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  /\  X  =/=  Y )  ->  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P )
)
7 simpl3l 1043 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  /\  X  =/=  Y )  ->  X  .<_  W )
8 simpl3r 1044 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  /\  X  =/=  Y )  ->  Y  .<_  W )
9 simpr 461 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  /\  X  =/=  Y )  ->  X  =/=  Y )
10 2llnm3.l . . . . 5  |-  .<_  =  ( le `  K )
11 2llnm3.m . . . . 5  |-  ./\  =  ( meet `  K )
12 eqid 2451 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
13 2llnm3.n . . . . 5  |-  N  =  ( LLines `  K )
14 2llnm3.p . . . . 5  |-  P  =  ( LPlanes `  K )
1510, 11, 12, 13, 142llnm2N 33515 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X  ./\  Y )  e.  ( Atoms `  K )
)
163, 6, 7, 8, 9, 15syl113anc 1231 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  /\  X  =/=  Y )  ->  ( X  ./\  Y )  e.  ( Atoms `  K )
)
17 2llnm3.z . . . 4  |-  .0.  =  ( 0. `  K )
1817, 12atn0 33256 . . 3  |-  ( ( K  e.  AtLat  /\  ( X  ./\  Y )  e.  ( Atoms `  K )
)  ->  ( X  ./\ 
Y )  =/=  .0.  )
195, 16, 18syl2anc 661 . 2  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  /\  X  =/=  Y )  ->  ( X  ./\  Y )  =/= 
.0.  )
20 hllat 33311 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
21203ad2ant1 1009 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  ->  K  e.  Lat )
22 simp22 1022 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  ->  Y  e.  N )
23 eqid 2451 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2423, 13llnbase 33456 . . . . 5  |-  ( Y  e.  N  ->  Y  e.  ( Base `  K
) )
2522, 24syl 16 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  ->  Y  e.  ( Base `  K
) )
2623, 11latmidm 15355 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K ) )  -> 
( Y  ./\  Y
)  =  Y )
2721, 25, 26syl2anc 661 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  ->  ( Y  ./\  Y )  =  Y )
28 simp1 988 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  ->  K  e.  HL )
2917, 13llnn0 33463 . . . 4  |-  ( ( K  e.  HL  /\  Y  e.  N )  ->  Y  =/=  .0.  )
3028, 22, 29syl2anc 661 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  ->  Y  =/=  .0.  )
3127, 30eqnetrd 2739 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  ->  ( Y  ./\  Y )  =/= 
.0.  )
322, 19, 31pm2.61ne 2761 1  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W ) )  ->  ( X  ./\  Y )  =/= 
.0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2642   class class class wbr 4387   ` cfv 5513  (class class class)co 6187   Basecbs 14273   lecple 14344   meetcmee 15214   0.cp0 15306   Latclat 15314   Atomscatm 33211   AtLatcal 33212   HLchlt 33298   LLinesclln 33438   LPlanesclpl 33439
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 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-poset 15215  df-plt 15227  df-lub 15243  df-glb 15244  df-join 15245  df-meet 15246  df-p0 15308  df-p1 15309  df-lat 15315  df-clat 15377  df-oposet 33124  df-ol 33126  df-oml 33127  df-covers 33214  df-ats 33215  df-atl 33246  df-cvlat 33270  df-hlat 33299  df-llines 33445  df-lplanes 33446
This theorem is referenced by: (None)
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