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Theorem cvlatexch3 34010
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 991 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  K  e.  CvLat )
2 simp21 1024 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  P  e.  A )
3 simp23 1026 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  R  e.  A )
4 simp22 1025 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  Q  e.  A )
5 simp3l 1019 . . . . 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 34006 . . . . 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 1236 . . . 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 484 . . 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 994 . . . . 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 33998 . . . . 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 1069 . . . . 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 2460 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1716, 8atbase 33961 . . . . 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 1070 . . . . 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 33961 . . . . 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 15535 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
2314, 18, 21, 22syl3anc 1223 . . 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 34007 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( Q  .\/  R
)  <->  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )
25243adant3l 1219 . . . 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 484 . . 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 2511 . 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 434 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   Latclat 15521   Atomscatm 33935   CvLatclc 33937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994
This theorem is referenced by:  cdleme21ct  35000
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