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Theorem cvlatexch3 34803
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 997 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  K  e.  CvLat )
2 simp21 1030 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  P  e.  A )
3 simp23 1032 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  R  e.  A )
4 simp22 1031 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  Q  e.  A )
5 simp3l 1025 . . . . 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 34799 . . . . 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 1242 . . . 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 1000 . . . . 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 34791 . . . . 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 1075 . . . . 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 2443 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1716, 8atbase 34754 . . . . 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 1076 . . . . 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 34754 . . . . 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 15667 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
2314, 18, 21, 22syl3anc 1229 . . 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 34800 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( Q  .\/  R
)  <->  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )
25243adant3l 1225 . . . 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 2494 . 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14613   lecple 14685   joincjn 15551   Latclat 15653   Atomscatm 34728   CvLatclc 34730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-preset 15535  df-poset 15553  df-plt 15566  df-lub 15582  df-glb 15583  df-join 15584  df-meet 15585  df-p0 15647  df-lat 15654  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787
This theorem is referenced by:  cdleme21ct  35795
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