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Theorem cvlatcvr2 34014
Description: An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
cvlatcvr1.j  |-  .\/  =  ( join `  K )
cvlatcvr1.c  |-  C  =  (  <o  `  K )
cvlatcvr1.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvlatcvr2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  P C
( Q  .\/  P
) ) )

Proof of Theorem cvlatcvr2
StepHypRef Expression
1 cvlatcvr1.j . . 3  |-  .\/  =  ( join `  K )
2 cvlatcvr1.c . . 3  |-  C  =  (  <o  `  K )
3 cvlatcvr1.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 3cvlatcvr1 34013 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  P C
( P  .\/  Q
) ) )
5 simp13 1023 . . . . 5  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  CvLat )
6 cvllat 33998 . . . . 5  |-  ( K  e.  CvLat  ->  K  e.  Lat )
75, 6syl 16 . . . 4  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  Lat )
8 eqid 2460 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
98, 3atbase 33961 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1093ad2ant2 1013 . . . 4  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  P  e.  ( Base `  K
) )
118, 3atbase 33961 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
12113ad2ant3 1014 . . . 4  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  Q  e.  ( Base `  K
) )
138, 1latjcom 15535 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
147, 10, 12, 13syl3anc 1223 . . 3  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
1514breq2d 4452 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  ( P C ( P  .\/  Q )  <->  P C ( Q 
.\/  P ) ) )
164, 15bitrd 253 1  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  P C
( Q  .\/  P
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   joincjn 15420   Latclat 15521   CLatccla 15583   OMLcoml 33847    <o ccvr 33934   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-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994
This theorem is referenced by:  atcvr2  34089
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