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Theorem cvlsupr3 32879
Description: Two equivalent ways of expressing that  R is a superposition of  P and  Q, which can replace the superposition part of ishlat1 32887,  ( x  =/=  y  ->  E. z  e.  A ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x  .\/  y ) )  ), with the simpler  E. z  e.  A ( x  .\/  z )  =  ( y  .\/  z ) as shown in ishlat3N 32889. (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
cvlsupr2.a  |-  A  =  ( Atoms `  K )
cvlsupr2.l  |-  .<_  =  ( le `  K )
cvlsupr2.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
cvlsupr3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
)  ->  ( ( P  .\/  R )  =  ( Q  .\/  R
)  <->  ( P  =/= 
Q  ->  ( R  =/=  P  /\  R  =/= 
Q  /\  R  .<_  ( P  .\/  Q ) ) ) ) )

Proof of Theorem cvlsupr3
StepHypRef Expression
1 df-ne 2616 . . . 4  |-  ( P  =/=  Q  <->  -.  P  =  Q )
21imbi1i 326 . . 3  |-  ( ( P  =/=  Q  -> 
( P  .\/  R
)  =  ( Q 
.\/  R ) )  <-> 
( -.  P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
3 oveq1 6312 . . . 4  |-  ( P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R
) )
43biantrur 508 . . 3  |-  ( ( -.  P  =  Q  ->  ( P  .\/  R )  =  ( Q 
.\/  R ) )  <-> 
( ( P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( -.  P  =  Q  -> 
( P  .\/  R
)  =  ( Q 
.\/  R ) ) ) )
5 pm4.83 937 . . 3  |-  ( ( ( P  =  Q  ->  ( P  .\/  R )  =  ( Q 
.\/  R ) )  /\  ( -.  P  =  Q  ->  ( P 
.\/  R )  =  ( Q  .\/  R
) ) )  <->  ( P  .\/  R )  =  ( Q  .\/  R ) )
62, 4, 53bitrri 275 . 2  |-  ( ( P  .\/  R )  =  ( Q  .\/  R )  <->  ( P  =/= 
Q  ->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
7 cvlsupr2.a . . . . 5  |-  A  =  ( Atoms `  K )
8 cvlsupr2.l . . . . 5  |-  .<_  =  ( le `  K )
9 cvlsupr2.j . . . . 5  |-  .\/  =  ( join `  K )
107, 8, 9cvlsupr2 32878 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  R )  =  ( Q  .\/  R
)  <->  ( R  =/= 
P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) ) )
11103expia 1207 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
)  ->  ( P  =/=  Q  ->  ( ( P  .\/  R )  =  ( Q  .\/  R
)  <->  ( R  =/= 
P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) ) ) )
1211pm5.74d 250 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
)  ->  ( ( P  =/=  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R
) )  <->  ( P  =/=  Q  ->  ( R  =/=  P  /\  R  =/= 
Q  /\  R  .<_  ( P  .\/  Q ) ) ) ) )
136, 12syl5bb 260 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
)  ->  ( ( P  .\/  R )  =  ( Q  .\/  R
)  <->  ( P  =/= 
Q  ->  ( R  =/=  P  /\  R  =/= 
Q  /\  R  .<_  ( P  .\/  Q ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   class class class wbr 4423   ` cfv 5601  (class class class)co 6305   lecple 15196   joincjn 16188   Atomscatm 32798   CvLatclc 32800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16172  df-poset 16190  df-plt 16203  df-lub 16219  df-glb 16220  df-join 16221  df-meet 16222  df-p0 16284  df-lat 16291  df-covers 32801  df-ats 32802  df-atl 32833  df-cvlat 32857
This theorem is referenced by:  ishlat3N  32889  hlsupr2  32921
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