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Theorem atnlej2 30251
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
atnlej2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  P  =/=  R )

Proof of Theorem atnlej2
StepHypRef Expression
1 hllat 30235 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 979 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  K  e.  Lat )
3 simp21 991 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  P  e.  A )
4 eqid 2438 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
5 atnlej.a . . . 4  |-  A  =  ( Atoms `  K )
64, 5atbase 30161 . . 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 992 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  Q  e.  A )
94, 5atbase 30161 . . 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 993 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  R  e.  A )
124, 5atbase 30161 . . 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 960 . 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, 16latnlej1r 14504 . 2  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  /\  -.  P  .<_  ( Q  .\/  R ) )  ->  P  =/=  R )
182, 7, 10, 13, 14, 17syl131anc 1198 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( Q  .\/  R
) )  ->  P  =/=  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   Latclat 14479   Atomscatm 30135   HLchlt 30222
This theorem is referenced by:  lplnri2N  30425  lplnri3N  30426  lplnexllnN  30435  dalem41  30584  paddasslem2  30692  4atexlemc  30940
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 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-lub 14436  df-join 14438  df-lat 14480  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223
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