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Theorem cvlsupr7 34020
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 33998 . . . . . 6  |-  ( K  e.  CvLat  ->  K  e.  Lat )
213ad2ant1 1012 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  K  e.  Lat )
3 simp21 1024 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  P  e.  A )
4 eqid 2460 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
5 cvlsupr5.a . . . . . . 7  |-  A  =  ( Atoms `  K )
64, 5atbase 33961 . . . . . 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 1026 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  e.  A )
94, 5atbase 33961 . . . . . 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 2460 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
12 cvlsupr5.j . . . . . 6  |-  .\/  =  ( join `  K )
134, 11, 12latlej1 15536 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  P
( le `  K
) ( P  .\/  R ) )
142, 7, 10, 13syl3anc 1223 . . . 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 1020 . . . 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 4464 . . 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 1025 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  Q  e.  A )
184, 5atbase 33961 . . . . 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 15535 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  =  ( R  .\/  Q
) )
212, 19, 10, 20syl3anc 1223 . . 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 4464 . 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 991 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  K  e.  CvLat
)
24 simp3l 1019 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  P  =/=  Q )
2511, 12, 5cvlatexchb2 34007 . . 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 1236 . 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   Latclat 15521   Atomscatm 33935   CvLatclc 33937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994
This theorem is referenced by:  cvlsupr8  34021  4atexlemswapqr  34734  4atexlemcnd  34743  cdleme21c  34998
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