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Theorem cvlatexch3 33322
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 988 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  K  e.  CvLat )
2 simp21 1021 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  P  e.  A )
3 simp23 1023 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  R  e.  A )
4 simp22 1022 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  Q  e.  A )
5 simp3l 1016 . . . . 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 33318 . . . . 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 1232 . . . 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 991 . . . . 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 33310 . . . . 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 1066 . . . . 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 2454 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1716, 8atbase 33273 . . . . 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 1067 . . . . 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 33273 . . . . 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 15349 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
2314, 18, 21, 22syl3anc 1219 . . 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 33319 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( Q  .\/  R
)  <->  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )
25243adant3l 1215 . . . 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 2505 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   Latclat 15335   Atomscatm 33247   CvLatclc 33249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-lat 15336  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306
This theorem is referenced by:  cdleme21ct  34312
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