Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvlsupr5 Unicode version

Theorem cvlsupr5 29829
Description: Consequence of superposition condition  ( P  .\/  R
)  =  ( Q 
.\/  R ). (Contributed by NM, 9-Nov-2012.)
Hypotheses
Ref Expression
cvlsupr5.a  |-  A  =  ( Atoms `  K )
cvlsupr5.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
cvlsupr5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  =/=  P )

Proof of Theorem cvlsupr5
StepHypRef Expression
1 cvlsupr5.a . . . . . 6  |-  A  =  ( Atoms `  K )
2 eqid 2404 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
3 cvlsupr5.j . . . . . 6  |-  .\/  =  ( join `  K )
41, 2, 3cvlsupr2 29826 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  R )  =  ( Q  .\/  R
)  <->  ( R  =/= 
P  /\  R  =/=  Q  /\  R ( le
`  K ) ( P  .\/  Q ) ) ) )
5 simp1 957 . . . . 5  |-  ( ( R  =/=  P  /\  R  =/=  Q  /\  R
( le `  K
) ( P  .\/  Q ) )  ->  R  =/=  P )
64, 5syl6bi 220 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  R )  =  ( Q  .\/  R
)  ->  R  =/=  P ) )
763exp 1152 . . 3  |-  ( K  e.  CvLat  ->  ( ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  ->  ( P  =/=  Q  ->  ( ( P  .\/  R )  =  ( Q 
.\/  R )  ->  R  =/=  P ) ) ) )
87imp4a 573 . 2  |-  ( K  e.  CvLat  ->  ( ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  ->  ( ( P  =/= 
Q  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  ->  R  =/=  P ) ) )
983imp 1147 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  =/=  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   Atomscatm 29746   CvLatclc 29748
This theorem is referenced by:  4atexlemswapqr  30545  4atexlemntlpq  30550  cdleme21b  30808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-join 14388  df-lat 14430  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805
  Copyright terms: Public domain W3C validator