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Theorem cvlsupr3 34497
Description: Two equivalent ways of expressing that  R is a superposition of  P and  Q, which can replace the superposition part of ishlat1 34505,  ( 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 34507. (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 2664 . . . 4  |-  ( P  =/=  Q  <->  -.  P  =  Q )
21imbi1i 325 . . 3  |-  ( ( P  =/=  Q  -> 
( P  .\/  R
)  =  ( Q 
.\/  R ) )  <-> 
( -.  P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
3 oveq1 6302 . . . 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 pm4.83 927 . . 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 34496 . . . 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 1198 . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   lecple 14579   joincjn 15448   Atomscatm 34416   CvLatclc 34418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-covers 34419  df-ats 34420  df-atl 34451  df-cvlat 34475
This theorem is referenced by:  ishlat3N  34507  hlsupr2  34539
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