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Theorem cvlatexch3 35476
Description: Atom exchange property. (Contributed by NM, 29-Nov-2012.)
Hypotheses
Ref Expression
cvlatexch.l  |-  .<_  =  ( le `  K )
cvlatexch.j  |-  .\/  =  ( join `  K )
cvlatexch.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvlatexch3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  ( P  .<_  ( Q  .\/  R )  ->  ( P  .\/  Q )  =  ( P  .\/  R ) ) )

Proof of Theorem cvlatexch3
StepHypRef Expression
1 simp1 994 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  K  e.  CvLat )
2 simp21 1027 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  P  e.  A )
3 simp23 1029 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  R  e.  A )
4 simp22 1028 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  Q  e.  A )
5 simp3l 1022 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  P  =/=  Q )
6 cvlatexch.l . . . . . 6  |-  .<_  =  ( le `  K )
7 cvlatexch.j . . . . . 6  |-  .\/  =  ( join `  K )
8 cvlatexch.a . . . . . 6  |-  A  =  ( Atoms `  K )
96, 7, 8cvlatexchb1 35472 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  R  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .<_  ( Q  .\/  R
)  <->  ( Q  .\/  P )  =  ( Q 
.\/  R ) ) )
101, 2, 3, 4, 5, 9syl131anc 1239 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  ( P  .<_  ( Q  .\/  R )  <->  ( Q  .\/  P )  =  ( Q 
.\/  R ) ) )
1110biimpa 482 . . 3  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  P  =/= 
R ) )  /\  P  .<_  ( Q  .\/  R ) )  ->  ( Q  .\/  P )  =  ( Q  .\/  R
) )
12 simpl1 997 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  P  =/= 
R ) )  /\  P  .<_  ( Q  .\/  R ) )  ->  K  e.  CvLat )
13 cvllat 35464 . . . . 5  |-  ( K  e.  CvLat  ->  K  e.  Lat )
1412, 13syl 16 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  P  =/= 
R ) )  /\  P  .<_  ( Q  .\/  R ) )  ->  K  e.  Lat )
15 simpl21 1072 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  P  =/= 
R ) )  /\  P  .<_  ( Q  .\/  R ) )  ->  P  e.  A )
16 eqid 2382 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1716, 8atbase 35427 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1815, 17syl 16 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  P  =/= 
R ) )  /\  P  .<_  ( Q  .\/  R ) )  ->  P  e.  ( Base `  K
) )
19 simpl22 1073 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  P  =/= 
R ) )  /\  P  .<_  ( Q  .\/  R ) )  ->  Q  e.  A )
2016, 8atbase 35427 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2119, 20syl 16 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  P  =/= 
R ) )  /\  P  .<_  ( Q  .\/  R ) )  ->  Q  e.  ( Base `  K
) )
2216, 7latjcom 15806 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
2314, 18, 21, 22syl3anc 1226 . . 3  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  P  =/= 
R ) )  /\  P  .<_  ( Q  .\/  R ) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
246, 7, 8cvlatexchb2 35473 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( Q  .\/  R
)  <->  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )
25243adant3l 1222 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  ( P  .<_  ( Q  .\/  R )  <->  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )
2625biimpa 482 . . 3  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  P  =/= 
R ) )  /\  P  .<_  ( Q  .\/  R ) )  ->  ( P  .\/  R )  =  ( Q  .\/  R
) )
2711, 23, 263eqtr4d 2433 . 2  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  P  =/= 
R ) )  /\  P  .<_  ( Q  .\/  R ) )  ->  ( P  .\/  Q )  =  ( P  .\/  R
) )
2827ex 432 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  ( P  .<_  ( Q  .\/  R )  ->  ( P  .\/  Q )  =  ( P  .\/  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   Basecbs 14634   lecple 14709   joincjn 15690   Latclat 15792   Atomscatm 35401   CvLatclc 35403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-preset 15674  df-poset 15692  df-plt 15705  df-lub 15721  df-glb 15722  df-join 15723  df-meet 15724  df-p0 15786  df-lat 15793  df-covers 35404  df-ats 35405  df-atl 35436  df-cvlat 35460
This theorem is referenced by:  cdleme21ct  36468
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