Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2llnma3r Unicode version

Theorem 2llnma3r 30270
Description: Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 30-Apr-2013.)
Hypotheses
Ref Expression
2llnm.l  |-  .<_  =  ( le `  K )
2llnm.j  |-  .\/  =  ( join `  K )
2llnm.m  |-  ./\  =  ( meet `  K )
2llnm.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2llnma3r  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( ( P 
.\/  R )  ./\  ( Q  .\/  R ) )  =  R )

Proof of Theorem 2llnma3r
StepHypRef Expression
1 simp1 957 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  K  e.  HL )
2 simp21 990 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  P  e.  A
)
3 simp23 992 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  R  e.  A
)
4 2llnm.j . . . . 5  |-  .\/  =  ( join `  K )
5 2llnm.a . . . . 5  |-  A  =  ( Atoms `  K )
64, 5hlatjcom 29850 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  =  ( R 
.\/  P ) )
71, 2, 3, 6syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( P  .\/  R )  =  ( R 
.\/  P ) )
8 simp22 991 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  Q  e.  A
)
94, 5hlatjcom 29850 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  =  ( R 
.\/  Q ) )
101, 8, 3, 9syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( Q  .\/  R )  =  ( R 
.\/  Q ) )
117, 10oveq12d 6058 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( ( P 
.\/  R )  ./\  ( Q  .\/  R ) )  =  ( ( R  .\/  P ) 
./\  ( R  .\/  Q ) ) )
12 simpr 448 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  Q  =  R )
1312oveq2d 6056 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  ( R  .\/  Q )  =  ( R  .\/  R ) )
14 simpl1 960 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  K  e.  HL )
15 simpl23 1037 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  R  e.  A )
164, 5hlatjidm 29851 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A )  ->  ( R  .\/  R
)  =  R )
1714, 15, 16syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  ( R  .\/  R )  =  R )
1813, 17eqtrd 2436 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  ( R  .\/  Q )  =  R )
1918oveq2d 6056 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  ( ( R  .\/  P )  ./\  ( R  .\/  Q ) )  =  ( ( R  .\/  P ) 
./\  R ) )
20 2llnm.l . . . . . . . 8  |-  .<_  =  ( le `  K )
2120, 4, 5hlatlej1 29857 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  R  .<_  ( R  .\/  P ) )
221, 3, 2, 21syl3anc 1184 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  R  .<_  ( R 
.\/  P ) )
23 hllat 29846 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
24233ad2ant1 978 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  K  e.  Lat )
25 eqid 2404 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2625, 5atbase 29772 . . . . . . . 8  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
273, 26syl 16 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  R  e.  (
Base `  K )
)
2825, 4, 5hlatjcl 29849 . . . . . . . 8  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  ( R  .\/  P
)  e.  ( Base `  K ) )
291, 3, 2, 28syl3anc 1184 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( R  .\/  P )  e.  ( Base `  K ) )
30 2llnm.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
3125, 20, 30latleeqm2 14464 . . . . . . 7  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  ( R  .\/  P )  e.  ( Base `  K
) )  ->  ( R  .<_  ( R  .\/  P )  <->  ( ( R 
.\/  P )  ./\  R )  =  R ) )
3224, 27, 29, 31syl3anc 1184 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( R  .<_  ( R  .\/  P )  <-> 
( ( R  .\/  P )  ./\  R )  =  R ) )
3322, 32mpbid 202 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( ( R 
.\/  P )  ./\  R )  =  R )
3433adantr 452 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  ( ( R  .\/  P )  ./\  R )  =  R )
3519, 34eqtrd 2436 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  ( ( R  .\/  P )  ./\  ( R  .\/  Q ) )  =  R )
36 simpl1 960 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  K  e.  HL )
37 simpl21 1035 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  P  e.  A )
38 simpl23 1037 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  R  e.  A )
39 simpl22 1036 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  Q  e.  A )
40 simpl3 962 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  ( P  .\/  R )  =/=  ( Q  .\/  R ) )
4120, 4, 5hlatlej2 29858 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  R  .<_  ( P  .\/  R ) )
421, 2, 3, 41syl3anc 1184 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  R  .<_  ( P 
.\/  R ) )
4325, 5atbase 29772 . . . . . . . . . . . 12  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
448, 43syl 16 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  Q  e.  (
Base `  K )
)
4525, 4, 5hlatjcl 29849 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
461, 2, 3, 45syl3anc 1184 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( P  .\/  R )  e.  ( Base `  K ) )
4725, 20, 4latjle12 14446 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K )  /\  ( P  .\/  R )  e.  ( Base `  K
) ) )  -> 
( ( Q  .<_  ( P  .\/  R )  /\  R  .<_  ( P 
.\/  R ) )  <-> 
( Q  .\/  R
)  .<_  ( P  .\/  R ) ) )
4824, 44, 27, 46, 47syl13anc 1186 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( ( Q 
.<_  ( P  .\/  R
)  /\  R  .<_  ( P  .\/  R ) )  <->  ( Q  .\/  R )  .<_  ( P  .\/  R ) ) )
4948biimpd 199 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( ( Q 
.<_  ( P  .\/  R
)  /\  R  .<_  ( P  .\/  R ) )  ->  ( Q  .\/  R )  .<_  ( P 
.\/  R ) ) )
5042, 49mpan2d 656 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( Q  .<_  ( P  .\/  R )  ->  ( Q  .\/  R )  .<_  ( P  .\/  R ) ) )
5150adantr 452 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  ( Q  .<_  ( P  .\/  R
)  ->  ( Q  .\/  R )  .<_  ( P 
.\/  R ) ) )
52 simpr 448 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  Q  =/=  R )
5320, 4, 5ps-1 29959 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R
)  /\  ( P  e.  A  /\  R  e.  A ) )  -> 
( ( Q  .\/  R )  .<_  ( P  .\/  R )  <->  ( Q  .\/  R )  =  ( P  .\/  R ) ) )
5436, 39, 38, 52, 37, 38, 53syl132anc 1202 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  ( ( Q  .\/  R )  .<_  ( P  .\/  R )  <-> 
( Q  .\/  R
)  =  ( P 
.\/  R ) ) )
5554biimpd 199 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  ( ( Q  .\/  R )  .<_  ( P  .\/  R )  ->  ( Q  .\/  R )  =  ( P 
.\/  R ) ) )
56 eqcom 2406 . . . . . . . 8  |-  ( ( Q  .\/  R )  =  ( P  .\/  R )  <->  ( P  .\/  R )  =  ( Q 
.\/  R ) )
5755, 56syl6ib 218 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  ( ( Q  .\/  R )  .<_  ( P  .\/  R )  ->  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )
5851, 57syld 42 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  ( Q  .<_  ( P  .\/  R
)  ->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
5958necon3ad 2603 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  ( ( P  .\/  R )  =/=  ( Q  .\/  R
)  ->  -.  Q  .<_  ( P  .\/  R
) ) )
6040, 59mpd 15 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  -.  Q  .<_  ( P  .\/  R
) )
6120, 4, 30, 52llnma1 30269 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  R
) )  ->  (
( R  .\/  P
)  ./\  ( R  .\/  Q ) )  =  R )
6236, 37, 38, 39, 60, 61syl131anc 1197 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  ( ( R  .\/  P )  ./\  ( R  .\/  Q ) )  =  R )
6335, 62pm2.61dane 2645 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( ( R 
.\/  P )  ./\  ( R  .\/  Q ) )  =  R )
6411, 63eqtrd 2436 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( ( P 
.\/  R )  ./\  ( Q  .\/  R ) )  =  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429   Atomscatm 29746   HLchlt 29833
This theorem is referenced by:  cdlemg9a  31114  cdlemg12a  31125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834
  Copyright terms: Public domain W3C validator