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Theorem 2llnma1 34458
Description: Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012.)
Hypotheses
Ref Expression
2llnm.l  |-  .<_  =  ( le `  K )
2llnm.j  |-  .\/  =  ( join `  K )
2llnm.m  |-  ./\  =  ( meet `  K )
2llnm.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2llnma1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  (
( Q  .\/  P
)  ./\  ( Q  .\/  R ) )  =  Q )

Proof of Theorem 2llnma1
StepHypRef Expression
1 simp1 991 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  K  e.  HL )
2 simp21 1024 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  P  e.  A )
3 eqid 2460 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
4 2llnm.a . . . 4  |-  A  =  ( Atoms `  K )
53, 4atbase 33961 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
62, 5syl 16 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  P  e.  ( Base `  K
) )
7 simp22 1025 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  Q  e.  A )
8 simp23 1026 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  R  e.  A )
9 simp3 993 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  -.  R  .<_  ( P  .\/  Q ) )
10 2llnm.j . . . . . 6  |-  .\/  =  ( join `  K )
1110, 4hlatjcom 34039 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
121, 2, 7, 11syl3anc 1223 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
1312breq2d 4452 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( R  .<_  ( P  .\/  Q )  <->  R  .<_  ( Q 
.\/  P ) ) )
149, 13mtbid 300 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  -.  R  .<_  ( Q  .\/  P ) )
15 2llnm.l . . 3  |-  .<_  =  ( le `  K )
16 2llnm.m . . 3  |-  ./\  =  ( meet `  K )
173, 15, 10, 16, 42llnma1b 34457 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  ( Base `  K )  /\  Q  e.  A  /\  R  e.  A )  /\  -.  R  .<_  ( Q 
.\/  P ) )  ->  ( ( Q 
.\/  P )  ./\  ( Q  .\/  R ) )  =  Q )
181, 6, 7, 8, 14, 17syl131anc 1236 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  (
( Q  .\/  P
)  ./\  ( Q  .\/  R ) )  =  Q )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Atomscatm 33935   HLchlt 34022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023
This theorem is referenced by:  2llnma3r  34459  2llnma2  34460  cdleme17c  34959
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