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Theorem cvlsupr6 30145
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
cvlsupr6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  =/=  Q )

Proof of Theorem cvlsupr6
StepHypRef Expression
1 cvlsupr5.a . . . . . 6  |-  A  =  ( Atoms `  K )
2 eqid 2436 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
3 cvlsupr5.j . . . . . 6  |-  .\/  =  ( join `  K )
41, 2, 3cvlsupr2 30141 . . . . 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 simp2 958 . . . . 5  |-  ( ( R  =/=  P  /\  R  =/=  Q  /\  R
( le `  K
) ( P  .\/  Q ) )  ->  R  =/=  Q )
64, 5syl6bi 220 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  R )  =  ( Q  .\/  R
)  ->  R  =/=  Q ) )
763exp 1152 . . 3  |-  ( K  e.  CvLat  ->  ( ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  ->  ( P  =/=  Q  ->  ( ( P  .\/  R )  =  ( Q 
.\/  R )  ->  R  =/=  Q ) ) ) )
87imp4a 573 . 2  |-  ( K  e.  CvLat  ->  ( ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  ->  ( ( P  =/= 
Q  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  ->  R  =/=  Q ) ) )
983imp 1147 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  =/=  Q )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   lecple 13536   joincjn 14401   Atomscatm 30061   CvLatclc 30063
This theorem is referenced by:  4atexlemnclw  30867  4atexlemcnd  30869  cdleme21a  31122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-join 14433  df-lat 14475  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120
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