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

Theorem 2llnneN 33392
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 988 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  K  e.  HL )
2 simp21 1021 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  P  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 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  P  e.  A )
5 simp23 1023 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  R  e.  A )
6 simp22 1022 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  Q  e.  A )
74, 5, 63jca 1168 . . . . . . 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 33379 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A  /\  Q  e.  A
)  /\  P  =/=  Q )  ->  ( P  .<_  ( R  .\/  Q
)  ->  R  .<_  ( P  .\/  Q ) ) )
127, 11syld3an2 1266 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  ( P  .<_  ( R  .\/  Q
)  ->  R  .<_  ( P  .\/  Q ) ) )
1312con3d 133 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  ->  ( -.  R  .<_  ( P  .\/  Q )  ->  -.  P  .<_  ( R  .\/  Q
) ) )
14133exp 1187 . . . 4  |-  ( K  e.  HL  ->  (
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  ->  ( P  =/=  Q  ->  ( -.  R  .<_  ( P  .\/  Q )  ->  -.  P  .<_  ( R  .\/  Q
) ) ) ) )
1514imp4a 589 . . 3  |-  ( K  e.  HL  ->  (
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  ->  ( ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  -.  P  .<_  ( R  .\/  Q ) ) ) )
16153imp 1182 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  -.  P  .<_  ( R 
.\/  Q ) )
178, 9, 102llnne2N 33391 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A
)  /\  -.  P  .<_  ( R  .\/  Q
) )  ->  ( R  .\/  P )  =/=  ( R  .\/  Q
) )
181, 2, 3, 16, 17syl121anc 1224 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   lecple 14365   joincjn 15234   Atomscatm 33247   HLchlt 33334
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-lat 15336  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator