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Theorem 2llnneN 29574
Description: Condition implying that two intersecting lines are different. (Contributed by NM, 29-May-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2lnne.l  |-  .<_  =  ( le `  K )
2lnne.j  |-  .\/  =  ( join `  K )
2lnne.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2llnneN  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  -> 
( R  .\/  P
)  =/=  ( R 
.\/  Q ) )

Proof of Theorem 2llnneN
StepHypRef Expression
1 simp1 957 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  K  e.  HL )
2 simp21 990 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  P  e.  A )
3 simp23 992 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  R  e.  A )
4 simp21 990 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  P  e.  A )
5 simp23 992 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  R  e.  A )
6 simp22 991 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  Q  e.  A )
74, 5, 63jca 1134 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  ( P  e.  A  /\  R  e.  A  /\  Q  e.  A ) )
8 2lnne.l . . . . . . . 8  |-  .<_  =  ( le `  K )
9 2lnne.j . . . . . . . 8  |-  .\/  =  ( join `  K )
10 2lnne.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
118, 9, 10hlatexch2 29561 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A  /\  Q  e.  A
)  /\  P  =/=  Q )  ->  ( P  .<_  ( R  .\/  Q
)  ->  R  .<_  ( P  .\/  Q ) ) )
127, 11syld3an2 1231 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  ( P  .<_  ( R  .\/  Q
)  ->  R  .<_  ( P  .\/  Q ) ) )
1312con3d 127 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  ( -.  R  .<_  ( P  .\/  Q )  ->  -.  P  .<_  ( R  .\/  Q
) ) )
14133exp 1152 . . . 4  |-  ( K  e.  HL  ->  (
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  ->  ( P  =/=  Q  ->  ( -.  R  .<_  ( P  .\/  Q )  ->  -.  P  .<_  ( R  .\/  Q
) ) ) ) )
1514imp4a 573 . . 3  |-  ( K  e.  HL  ->  (
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  ->  ( ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  -.  P  .<_  ( R  .\/  Q ) ) ) )
16153imp 1147 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  -.  P  .<_  ( R 
.\/  Q ) )
178, 9, 102llnne2N 29573 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( R  .\/  Q
) )  ->  ( R  .\/  P )  =/=  ( R  .\/  Q
) )
181, 2, 3, 16, 17syl121anc 1189 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  -> 
( R  .\/  P
)  =/=  ( R 
.\/  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   lecple 13456   joincjn 14321   Atomscatm 29429   HLchlt 29516
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-poset 14323  df-plt 14335  df-lub 14351  df-join 14353  df-lat 14395  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517
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