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Theorem cvlatexch1 33287
Description: Atom exchange property. (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
cvlatexch1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( R  .\/  Q
)  ->  Q  .<_  ( R  .\/  P ) ) )

Proof of Theorem cvlatexch1
StepHypRef Expression
1 cvlatexch.l . . 3  |-  .<_  =  ( le `  K )
2 cvlatexch.j . . 3  |-  .\/  =  ( join `  K )
3 cvlatexch.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 3cvlatexchb1 33285 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( R  .\/  Q
)  <->  ( R  .\/  P )  =  ( R 
.\/  Q ) ) )
5 cvllat 33277 . . . . 5  |-  ( K  e.  CvLat  ->  K  e.  Lat )
653ad2ant1 1009 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  K  e.  Lat )
7 simp23 1023 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  R  e.  A )
8 eqid 2451 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
98, 3atbase 33240 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
107, 9syl 16 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  R  e.  ( Base `  K )
)
11 simp22 1022 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  Q  e.  A )
128, 3atbase 33240 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1311, 12syl 16 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  Q  e.  ( Base `  K )
)
148, 1, 2latlej2 15333 . . . 4  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  Q  .<_  ( R  .\/  Q
) )
156, 10, 13, 14syl3anc 1219 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  Q  .<_  ( R  .\/  Q ) )
16 breq2 4394 . . 3  |-  ( ( R  .\/  P )  =  ( R  .\/  Q )  ->  ( Q  .<_  ( R  .\/  P
)  <->  Q  .<_  ( R 
.\/  Q ) ) )
1715, 16syl5ibrcom 222 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( ( R  .\/  P )  =  ( R  .\/  Q
)  ->  Q  .<_  ( R  .\/  P ) ) )
184, 17sylbid 215 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( R  .\/  Q
)  ->  Q  .<_  ( R  .\/  P ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   Basecbs 14276   lecple 14347   joincjn 15216   Latclat 15317   Atomscatm 33214   CvLatclc 33216
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 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-poset 15218  df-plt 15230  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-lat 15318  df-covers 33217  df-ats 33218  df-atl 33249  df-cvlat 33273
This theorem is referenced by:  cvlatexch2  33288  cvlsupr2  33294  hlatexch1  33345  4atex  34026  cdleme20zN  34251  cdleme20y  34252  cdleme19a  34253  cdleme21b  34276  cdleme21c  34277  cdleme22g  34298  cdlemf1  34511
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