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Theorem 3noncolr1N 33402
Description: Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
3noncol.l  |-  .<_  =  ( le `  K )
3noncol.j  |-  .\/  =  ( join `  K )
3noncol.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
3noncolr1N  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  -> 
( R  =/=  P  /\  -.  Q  .<_  ( R 
.\/  P ) ) )

Proof of Theorem 3noncolr1N
StepHypRef Expression
1 simp1 988 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  K  e.  HL )
2 simp22 1022 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  Q  e.  A )
3 simp23 1023 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  R  e.  A )
4 simp21 1021 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  P  e.  A )
5 3noncol.l . . 3  |-  .<_  =  ( le `  K )
6 3noncol.j . . 3  |-  .\/  =  ( join `  K )
7 3noncol.a . . 3  |-  A  =  ( Atoms `  K )
85, 6, 73noncolr2 33401 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  -> 
( Q  =/=  R  /\  -.  P  .<_  ( Q 
.\/  R ) ) )
95, 6, 73noncolr2 33401 . 2  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  ( Q  =/=  R  /\  -.  P  .<_  ( Q  .\/  R
) ) )  -> 
( R  =/=  P  /\  -.  Q  .<_  ( R 
.\/  P ) ) )
101, 2, 3, 4, 8, 9syl131anc 1232 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  -> 
( R  =/=  P  /\  -.  Q  .<_  ( R 
.\/  P ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   lecple 14349   joincjn 15218   Atomscatm 33216   HLchlt 33303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-lat 15320  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304
This theorem is referenced by:  lplnribN  33503
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