Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2llnma1 Structured version   Unicode version

Theorem 2llnma1 33739
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 988 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  K  e.  HL )
2 simp21 1021 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  P  e.  A )
3 eqid 2451 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
4 2llnm.a . . . 4  |-  A  =  ( Atoms `  K )
53, 4atbase 33242 . . 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 1022 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  Q  e.  A )
8 simp23 1023 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  R  e.  A )
9 simp3 990 . . 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 33320 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
121, 2, 7, 11syl3anc 1219 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
1312breq2d 4404 . . 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 33738 . 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 1232 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 965    = wceq 1370    e. wcel 1758   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   Basecbs 14278   lecple 14349   joincjn 15218   meetcmee 15219   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-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304
This theorem is referenced by:  2llnma3r  33740  2llnma2  33741  cdleme17c  34240
  Copyright terms: Public domain W3C validator