Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atnlej1 Unicode version

Theorem atnlej1 29861
Description: If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)
Hypotheses
Ref Expression
atnlej.l  |-  .<_  =  ( le `  K )
atnlej.j  |-  .\/  =  ( join `  K )
atnlej.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atnlej1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  P  =/=  Q )

Proof of Theorem atnlej1
StepHypRef Expression
1 hllat 29846 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 978 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  K  e.  Lat )
3 simp21 990 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  P  e.  A )
4 eqid 2404 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
5 atnlej.a . . . 4  |-  A  =  ( Atoms `  K )
64, 5atbase 29772 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
73, 6syl 16 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  P  e.  ( Base `  K
) )
8 simp22 991 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  Q  e.  A )
94, 5atbase 29772 . . 3  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
108, 9syl 16 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  Q  e.  ( Base `  K
) )
11 simp23 992 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  R  e.  A )
124, 5atbase 29772 . . 3  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1311, 12syl 16 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  R  e.  ( Base `  K
) )
14 simp3 959 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  -.  P  .<_  ( Q  .\/  R ) )
15 atnlej.l . . 3  |-  .<_  =  ( le `  K )
16 atnlej.j . . 3  |-  .\/  =  ( join `  K )
174, 15, 16latnlej1l 14453 . 2  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  /\  -.  P  .<_  ( Q  .\/  R ) )  ->  P  =/=  Q )
182, 7, 10, 13, 14, 17syl131anc 1197 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  P  =/=  Q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   Latclat 14429   Atomscatm 29746   HLchlt 29833
This theorem is referenced by:  4atlem0be  30077  dalem5  30149  dalem-cly  30153  4atexlemex6  30556  cdleme00a  30691  cdleme21a  30807  cdleme21b  30808  cdleme21c  30809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-lub 14386  df-join 14388  df-lat 14430  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834
  Copyright terms: Public domain W3C validator