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Theorem cvlatexchb2 32317
Description: A version of cvlexchb2 32313 for atoms. (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
cvlatexch.l  |-  .<_  =  ( le `  K )
cvlatexch.j  |-  .\/  =  ( join `  K )
cvlatexch.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvlatexchb2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( Q  .\/  R
)  <->  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )

Proof of Theorem cvlatexchb2
StepHypRef Expression
1 cvlatexch.l . . 3  |-  .<_  =  ( le `  K )
2 cvlatexch.j . . 3  |-  .\/  =  ( join `  K )
3 cvlatexch.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 3cvlatexchb1 32316 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( R  .\/  Q
)  <->  ( R  .\/  P )  =  ( R 
.\/  Q ) ) )
5 cvllat 32308 . . . . 5  |-  ( K  e.  CvLat  ->  K  e.  Lat )
653ad2ant1 1016 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  K  e.  Lat )
7 simp22 1029 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  Q  e.  A )
8 eqid 2400 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
98, 3atbase 32271 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
107, 9syl 17 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  Q  e.  ( Base `  K )
)
11 simp23 1030 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  R  e.  A )
128, 3atbase 32271 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1311, 12syl 17 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  R  e.  ( Base `  K )
)
148, 2latjcom 15903 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  =  ( R  .\/  Q
) )
156, 10, 13, 14syl3anc 1228 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( Q  .\/  R )  =  ( R  .\/  Q ) )
1615breq2d 4404 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( Q  .\/  R
)  <->  P  .<_  ( R 
.\/  Q ) ) )
17 simp21 1028 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  P  e.  A )
188, 3atbase 32271 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1917, 18syl 17 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  P  e.  ( Base `  K )
)
208, 2latjcom 15903 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( P  .\/  R )  =  ( R  .\/  P
) )
216, 19, 13, 20syl3anc 1228 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .\/  R )  =  ( R  .\/  P ) )
2221, 15eqeq12d 2422 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( ( P  .\/  R )  =  ( Q  .\/  R
)  <->  ( R  .\/  P )  =  ( R 
.\/  Q ) ) )
234, 16, 223bitr4d 285 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( Q  .\/  R
)  <->  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787   Latclat 15889   Atomscatm 32245   CvLatclc 32247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-lat 15890  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304
This theorem is referenced by:  cvlatexch3  32320  cvlsupr2  32325  cvlsupr7  32330  hlatexchb2  32375
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