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Theorem cvlatexch3 29528
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 955 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  K  e.  CvLat )
2 simp21 988 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  P  e.  A )
3 simp23 990 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  R  e.  A )
4 simp22 989 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  P  =/=  R
) )  ->  Q  e.  A )
5 simp3l 983 . . . . 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 29524 . . . . 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 1195 . . . 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 470 . . 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 958 . . . . 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 29516 . . . . 5  |-  ( K  e.  CvLat  ->  K  e.  Lat )
1412, 13syl 15 . . . 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 1033 . . . . 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 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1716, 8atbase 29479 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1815, 17syl 15 . . . 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 1034 . . . . 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 29479 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2119, 20syl 15 . . . 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 14165 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
2314, 18, 21, 22syl3anc 1182 . . 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 29525 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( Q  .\/  R
)  <->  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )
25243adant3l 1178 . . . 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 470 . . 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 2325 . 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 423 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 29453   CvLatclc 29455
This theorem is referenced by:  cdleme21ct  30518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-join 14110  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512
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