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Theorem cvlsupr7 33302
Description: Consequence of superposition condition  ( P  .\/  R
)  =  ( Q 
.\/  R ). (Contributed by NM, 24-Nov-2012.)
Hypotheses
Ref Expression
cvlsupr5.a  |-  A  =  ( Atoms `  K )
cvlsupr5.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
cvlsupr7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  ( P  .\/  Q )  =  ( R  .\/  Q ) )

Proof of Theorem cvlsupr7
StepHypRef Expression
1 cvllat 33280 . . . . . 6  |-  ( K  e.  CvLat  ->  K  e.  Lat )
213ad2ant1 1009 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  K  e.  Lat )
3 simp21 1021 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  P  e.  A )
4 eqid 2451 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
5 cvlsupr5.a . . . . . . 7  |-  A  =  ( Atoms `  K )
64, 5atbase 33243 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
73, 6syl 16 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  P  e.  ( Base `  K )
)
8 simp23 1023 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  e.  A )
94, 5atbase 33243 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
108, 9syl 16 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  e.  ( Base `  K )
)
11 eqid 2451 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
12 cvlsupr5.j . . . . . 6  |-  .\/  =  ( join `  K )
134, 11, 12latlej1 15341 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  P
( le `  K
) ( P  .\/  R ) )
142, 7, 10, 13syl3anc 1219 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  P ( le `  K ) ( P  .\/  R ) )
15 simp3r 1017 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  ( P  .\/  R )  =  ( Q  .\/  R ) )
1614, 15breqtrd 4417 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  P ( le `  K ) ( Q  .\/  R ) )
17 simp22 1022 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  Q  e.  A )
184, 5atbase 33243 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1917, 18syl 16 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  Q  e.  ( Base `  K )
)
204, 12latjcom 15340 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  =  ( R  .\/  Q
) )
212, 19, 10, 20syl3anc 1219 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  ( Q  .\/  R )  =  ( R  .\/  Q ) )
2216, 21breqtrd 4417 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  P ( le `  K ) ( R  .\/  Q ) )
23 simp1 988 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  K  e.  CvLat
)
24 simp3l 1016 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  P  =/=  Q )
2511, 12, 5cvlatexchb2 33289 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  R  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P
( le `  K
) ( R  .\/  Q )  <->  ( P  .\/  Q )  =  ( R 
.\/  Q ) ) )
2623, 3, 8, 17, 24, 25syl131anc 1232 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  ( P
( le `  K
) ( R  .\/  Q )  <->  ( P  .\/  Q )  =  ( R 
.\/  Q ) ) )
2722, 26mpbid 210 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  ( P  .\/  Q )  =  ( R  .\/  Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   Basecbs 14285   lecple 14356   joincjn 15225   Latclat 15326   Atomscatm 33217   CvLatclc 33219
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 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
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 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-lat 15327  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276
This theorem is referenced by:  cvlsupr8  33303  4atexlemswapqr  34016  4atexlemcnd  34025  cdleme21c  34280
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