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Theorem lplnribN 30348
Description: Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
islpln2a.l  |-  .<_  =  ( le `  K )
islpln2a.j  |-  .\/  =  ( join `  K )
islpln2a.a  |-  A  =  ( Atoms `  K )
islpln2a.p  |-  P  =  ( LPlanes `  K )
islpln2a.y  |-  Y  =  ( ( Q  .\/  R )  .\/  S )
Assertion
Ref Expression
lplnribN  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  ->  -.  R  .<_  ( Q  .\/  S ) )

Proof of Theorem lplnribN
StepHypRef Expression
1 islpln2a.l . . . . . 6  |-  .<_  =  ( le `  K )
2 islpln2a.j . . . . . 6  |-  .\/  =  ( join `  K )
3 islpln2a.a . . . . . 6  |-  A  =  ( Atoms `  K )
41, 2, 33noncolr1N 30247 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  -> 
( S  =/=  Q  /\  -.  R  .<_  ( S 
.\/  Q ) ) )
54simprd 450 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  -.  R  .<_  ( S 
.\/  Q ) )
653expia 1155 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( Q  =/=  R  /\  -.  S  .<_  ( Q 
.\/  R ) )  ->  -.  R  .<_  ( S  .\/  Q ) ) )
7 islpln2a.p . . . 4  |-  P  =  ( LPlanes `  K )
8 islpln2a.y . . . 4  |-  Y  =  ( ( Q  .\/  R )  .\/  S )
91, 2, 3, 7, 8islpln2ah 30346 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( Y  e.  P  <->  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) ) )
102, 3hlatjcom 30165 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  ( Q  .\/  S
)  =  ( S 
.\/  Q ) )
11103adant3r2 1163 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( Q  .\/  S )  =  ( S  .\/  Q
) )
1211breq2d 4224 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( R  .<_  ( Q  .\/  S )  <->  R  .<_  ( S 
.\/  Q ) ) )
1312notbid 286 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( -.  R  .<_  ( Q 
.\/  S )  <->  -.  R  .<_  ( S  .\/  Q
) ) )
146, 9, 133imtr4d 260 . 2  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( Y  e.  P  ->  -.  R  .<_  ( Q  .\/  S ) ) )
15143impia 1150 1  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  ->  -.  R  .<_  ( Q  .\/  S ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> 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   HLchlt 30148   LPlanesclpl 30289
This theorem is referenced by:  lplnri3N  30352
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-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296
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