Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2llnneN Structured version   Unicode version

Theorem 2llnneN 30304
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 958 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  K  e.  HL )
2 simp21 991 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  P  e.  A )
3 simp23 993 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  R  e.  A )
4 simp21 991 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  P  e.  A )
5 simp23 993 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  R  e.  A )
6 simp22 992 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  Q  e.  A )
74, 5, 63jca 1135 . . . . . . 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 30291 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A  /\  Q  e.  A
)  /\  P  =/=  Q )  ->  ( P  .<_  ( R  .\/  Q
)  ->  R  .<_  ( P  .\/  Q ) ) )
127, 11syld3an2 1232 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  ( P  .<_  ( R  .\/  Q
)  ->  R  .<_  ( P  .\/  Q ) ) )
1312con3d 128 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  ( -.  R  .<_  ( P  .\/  Q )  ->  -.  P  .<_  ( R  .\/  Q
) ) )
14133exp 1153 . . . 4  |-  ( K  e.  HL  ->  (
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  ->  ( P  =/=  Q  ->  ( -.  R  .<_  ( P  .\/  Q )  ->  -.  P  .<_  ( R  .\/  Q
) ) ) ) )
1514imp4a 574 . . 3  |-  ( K  e.  HL  ->  (
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  ->  ( ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  -.  P  .<_  ( R  .\/  Q ) ) ) )
16153imp 1148 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  -.  P  .<_  ( R 
.\/  Q ) )
178, 9, 102llnne2N 30303 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( R  .\/  Q
) )  ->  ( R  .\/  P )  =/=  ( R  .\/  Q
) )
181, 2, 3, 16, 17syl121anc 1190 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 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   lecple 13567   joincjn 14432   Atomscatm 30159   HLchlt 30246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-poset 14434  df-plt 14446  df-lub 14462  df-join 14464  df-lat 14506  df-covers 30162  df-ats 30163  df-atl 30194  df-cvlat 30218  df-hlat 30247
  Copyright terms: Public domain W3C validator