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Theorem cvlsupr3 32994
Description: Two equivalent ways of expressing that  R is a superposition of  P and  Q, which can replace the superposition part of ishlat1 33002,  ( 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 33004. (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 2613 . . . 4  |-  ( P  =/=  Q  <->  -.  P  =  Q )
21imbi1i 325 . . 3  |-  ( ( P  =/=  Q  -> 
( P  .\/  R
)  =  ( Q 
.\/  R ) )  <-> 
( -.  P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
3 oveq1 6103 . . . 4  |-  ( P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R
) )
43biantrur 506 . . 3  |-  ( ( -.  P  =  Q  ->  ( P  .\/  R )  =  ( Q 
.\/  R ) )  <-> 
( ( P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( -.  P  =  Q  -> 
( P  .\/  R
)  =  ( Q 
.\/  R ) ) ) )
5 pm4.83 920 . . 3  |-  ( ( ( P  =  Q  ->  ( P  .\/  R )  =  ( Q 
.\/  R ) )  /\  ( -.  P  =  Q  ->  ( P 
.\/  R )  =  ( Q  .\/  R
) ) )  <->  ( P  .\/  R )  =  ( Q  .\/  R ) )
62, 4, 53bitrri 272 . 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 32993 . . . 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 1189 . . 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 247 . 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 257 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   lecple 14250   joincjn 15119   Atomscatm 32913   CvLatclc 32915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-lat 15221  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972
This theorem is referenced by:  ishlat3N  33004  hlsupr2  33036
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