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Theorem cvlatexch2 34535
Description: Atom exchange property. (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
cvlatexch.l  |-  .<_  =  ( le `  K )
cvlatexch.j  |-  .\/  =  ( join `  K )
cvlatexch.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvlatexch2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( Q  .\/  R
)  ->  Q  .<_  ( P  .\/  R ) ) )

Proof of Theorem cvlatexch2
StepHypRef Expression
1 cvlatexch.l . . 3  |-  .<_  =  ( le `  K )
2 cvlatexch.j . . 3  |-  .\/  =  ( join `  K )
3 cvlatexch.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 3cvlatexch1 34534 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( R  .\/  Q
)  ->  Q  .<_  ( R  .\/  P ) ) )
5 cvllat 34524 . . . . 5  |-  ( K  e.  CvLat  ->  K  e.  Lat )
653ad2ant1 1017 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  K  e.  Lat )
7 simp22 1030 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  Q  e.  A )
8 eqid 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
98, 3atbase 34487 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
107, 9syl 16 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  Q  e.  ( Base `  K )
)
11 simp23 1031 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  R  e.  A )
128, 3atbase 34487 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1311, 12syl 16 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  R  e.  ( Base `  K )
)
148, 2latjcom 15563 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  =  ( R  .\/  Q
) )
156, 10, 13, 14syl3anc 1228 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( Q  .\/  R )  =  ( R  .\/  Q ) )
1615breq2d 4465 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( Q  .\/  R
)  <->  P  .<_  ( R 
.\/  Q ) ) )
17 simp21 1029 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  P  e.  A )
188, 3atbase 34487 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1917, 18syl 16 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  P  e.  ( Base `  K )
)
208, 2latjcom 15563 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( P  .\/  R )  =  ( R  .\/  P
) )
216, 19, 13, 20syl3anc 1228 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .\/  R )  =  ( R  .\/  P ) )
2221breq2d 4465 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( Q  .<_  ( P  .\/  R
)  <->  Q  .<_  ( R 
.\/  P ) ) )
234, 16, 223imtr4d 268 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( Q  .\/  R
)  ->  Q  .<_  ( P  .\/  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   Latclat 15549   Atomscatm 34461   CvLatclc 34463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520
This theorem is referenced by:  hlatexch2  34593  4atexlemnclw  35267  4atexlemex2  35268  cdleme21ct  35526  cdleme22f  35543  cdleme22f2  35544  cdlemf1  35758
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