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Theorem 2llnma1 30028
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 955 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  K  e.  HL )
2 simp21 988 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  P  e.  A )
3 eqid 2358 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
4 2llnm.a . . . 4  |-  A  =  ( Atoms `  K )
53, 4atbase 29531 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
62, 5syl 15 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  P  e.  ( Base `  K
) )
7 simp22 989 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  Q  e.  A )
8 simp23 990 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  R  e.  A )
9 simp3 957 . . 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 29609 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
121, 2, 7, 11syl3anc 1182 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
1312breq2d 4114 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( R  .<_  ( P  .\/  Q )  <->  R  .<_  ( Q 
.\/  P ) ) )
149, 13mtbid 291 . 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 30027 . 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 1195 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 934    = wceq 1642    e. wcel 1710   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   Basecbs 13239   lecple 13306   joincjn 14171   meetcmee 14172   Atomscatm 29505   HLchlt 29592
This theorem is referenced by:  2llnma3r  30029  2llnma2  30030  cdleme17c  30529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-undef 6382  df-riota 6388  df-poset 14173  df-plt 14185  df-lub 14201  df-glb 14202  df-join 14203  df-meet 14204  df-p0 14238  df-lat 14245  df-clat 14307  df-oposet 29418  df-ol 29420  df-oml 29421  df-covers 29508  df-ats 29509  df-atl 29540  df-cvlat 29564  df-hlat 29593
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