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Theorem cvlsupr3 29534
Description: Two equivalent ways of expressing that  R is a superposition of  P and  Q, which can replace the superposition part of ishlat1 29542,  ( 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 29544. (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 2448 . . . 4  |-  ( P  =/=  Q  <->  -.  P  =  Q )
21imbi1i 315 . . 3  |-  ( ( P  =/=  Q  -> 
( P  .\/  R
)  =  ( Q 
.\/  R ) )  <-> 
( -.  P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
3 oveq1 5865 . . . 4  |-  ( P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R
) )
43biantrur 492 . . 3  |-  ( ( -.  P  =  Q  ->  ( P  .\/  R )  =  ( Q 
.\/  R ) )  <-> 
( ( P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( -.  P  =  Q  -> 
( P  .\/  R
)  =  ( Q 
.\/  R ) ) ) )
5 pm4.83 895 . . 3  |-  ( ( ( P  =  Q  ->  ( P  .\/  R )  =  ( Q 
.\/  R ) )  /\  ( -.  P  =  Q  ->  ( P 
.\/  R )  =  ( Q  .\/  R
) ) )  <->  ( P  .\/  R )  =  ( Q  .\/  R ) )
62, 4, 53bitrri 263 . 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 29533 . . . 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 1153 . . 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 238 . 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 248 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453   CvLatclc 29455
This theorem is referenced by:  ishlat3N  29544  hlsupr2  29576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-join 14110  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512
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