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Theorem 2llnma3r 30599
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 955 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  K  e.  HL )
2 simp21 988 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  P  e.  A
)
3 simp23 990 . . . 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 30179 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  =  ( R 
.\/  P ) )
71, 2, 3, 6syl3anc 1182 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( P  .\/  R )  =  ( R 
.\/  P ) )
8 simp22 989 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  Q  e.  A
)
94, 5hlatjcom 30179 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  =  ( R 
.\/  Q ) )
101, 8, 3, 9syl3anc 1182 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( Q  .\/  R )  =  ( R 
.\/  Q ) )
117, 10oveq12d 5892 . 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 447 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  Q  =  R )
1312oveq2d 5890 . . . . . 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 958 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  K  e.  HL )
15 simpl23 1035 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  R  e.  A )
164, 5hlatjidm 30180 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A )  ->  ( R  .\/  R
)  =  R )
1714, 15, 16syl2anc 642 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  ( R  .\/  R )  =  R )
1813, 17eqtrd 2328 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  ( R  .\/  Q )  =  R )
1918oveq2d 5890 . . . 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 30186 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  R  .<_  ( R  .\/  P ) )
221, 3, 2, 21syl3anc 1182 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  R  .<_  ( R 
.\/  P ) )
23 hllat 30175 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
24233ad2ant1 976 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  K  e.  Lat )
25 eqid 2296 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2625, 5atbase 30101 . . . . . . . 8  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
273, 26syl 15 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  R  e.  (
Base `  K )
)
2825, 4, 5hlatjcl 30178 . . . . . . . 8  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  ( R  .\/  P
)  e.  ( Base `  K ) )
291, 3, 2, 28syl3anc 1182 . . . . . . 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 14202 . . . . . . 7  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  ( R  .\/  P )  e.  ( Base `  K
) )  ->  ( R  .<_  ( R  .\/  P )  <->  ( ( R 
.\/  P )  ./\  R )  =  R ) )
3224, 27, 29, 31syl3anc 1182 . . . . . 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 201 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( ( R 
.\/  P )  ./\  R )  =  R )
3433adantr 451 . . . 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 2328 . . 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 958 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  K  e.  HL )
37 simpl21 1033 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  P  e.  A )
38 simpl23 1035 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  R  e.  A )
39 simpl22 1034 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  Q  e.  A )
40 simpl3 960 . . . . 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 30187 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  R  .<_  ( P  .\/  R ) )
421, 2, 3, 41syl3anc 1182 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  R  .<_  ( P 
.\/  R ) )
4325, 5atbase 30101 . . . . . . . . . . . 12  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
448, 43syl 15 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  Q  e.  (
Base `  K )
)
4525, 4, 5hlatjcl 30178 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
461, 2, 3, 45syl3anc 1182 . . . . . . . . . . 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 14184 . . . . . . . . . . 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 1184 . . . . . . . . . 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 198 . . . . . . . . 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 655 . . . . . . . 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 451 . . . . . . 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 447 . . . . . . . . . 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 30288 . . . . . . . . . 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 1200 . . . . . . . . 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 198 . . . . . . . 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 2298 . . . . . . . 8  |-  ( ( Q  .\/  R )  =  ( P  .\/  R )  <->  ( P  .\/  R )  =  ( Q 
.\/  R ) )
5755, 56syl6ib 217 . . . . . . 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 40 . . . . . 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 2495 . . . . 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 14 . . . 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 30598 . . . 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 1195 . . 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 2537 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( ( R 
.\/  P )  ./\  ( R  .\/  Q ) )  =  R )
6411, 63eqtrd 2328 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  cdlemg9a  31443  cdlemg12a  31454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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