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Theorem 2llnma3r 30647
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 958 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  K  e.  HL )
2 simp21 991 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  P  e.  A
)
3 simp23 993 . . . 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 30227 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  =  ( R 
.\/  P ) )
71, 2, 3, 6syl3anc 1185 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( P  .\/  R )  =  ( R 
.\/  P ) )
8 simp22 992 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  Q  e.  A
)
94, 5hlatjcom 30227 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  =  ( R 
.\/  Q ) )
101, 8, 3, 9syl3anc 1185 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( Q  .\/  R )  =  ( R 
.\/  Q ) )
117, 10oveq12d 6101 . 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 449 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  Q  =  R )
1312oveq2d 6099 . . . . . 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 961 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  K  e.  HL )
15 simpl23 1038 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  R  e.  A )
164, 5hlatjidm 30228 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A )  ->  ( R  .\/  R
)  =  R )
1714, 15, 16syl2anc 644 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  ( R  .\/  R )  =  R )
1813, 17eqtrd 2470 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  ( R  .\/  Q )  =  R )
1918oveq2d 6099 . . . 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 30234 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  R  .<_  ( R  .\/  P ) )
221, 3, 2, 21syl3anc 1185 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  R  .<_  ( R 
.\/  P ) )
23 hllat 30223 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
24233ad2ant1 979 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  K  e.  Lat )
25 eqid 2438 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2625, 5atbase 30149 . . . . . . . 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 30226 . . . . . . . 8  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  ( R  .\/  P
)  e.  ( Base `  K ) )
291, 3, 2, 28syl3anc 1185 . . . . . . 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 14511 . . . . . . 7  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  ( R  .\/  P )  e.  ( Base `  K
) )  ->  ( R  .<_  ( R  .\/  P )  <->  ( ( R 
.\/  P )  ./\  R )  =  R ) )
3224, 27, 29, 31syl3anc 1185 . . . . . 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 203 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( ( R 
.\/  P )  ./\  R )  =  R )
3433adantr 453 . . . 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 2470 . . 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 961 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  K  e.  HL )
37 simpl21 1036 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  P  e.  A )
38 simpl23 1038 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  R  e.  A )
39 simpl22 1037 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =/=  R
)  ->  Q  e.  A )
40 simpl3 963 . . . . 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 30235 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  R  .<_  ( P  .\/  R ) )
421, 2, 3, 41syl3anc 1185 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  R  .<_  ( P 
.\/  R ) )
4325, 5atbase 30149 . . . . . . . . . . . 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 30226 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
461, 2, 3, 45syl3anc 1185 . . . . . . . . . . 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 14493 . . . . . . . . . . 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 1187 . . . . . . . . . 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 200 . . . . . . . . 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 657 . . . . . . . 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 453 . . . . . . 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 449 . . . . . . . . . 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 30336 . . . . . . . . . 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 1203 . . . . . . . . 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 200 . . . . . . . 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 2440 . . . . . . . 8  |-  ( ( Q  .\/  R )  =  ( P  .\/  R )  <->  ( P  .\/  R )  =  ( Q 
.\/  R ) )
5755, 56syl6ib 219 . . . . . . 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 43 . . . . . 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 2639 . . . . 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 30646 . . . 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 1198 . . 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 2684 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( ( R 
.\/  P )  ./\  ( R  .\/  Q ) )  =  R )
6411, 63eqtrd 2470 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Latclat 14476   Atomscatm 30123   HLchlt 30210
This theorem is referenced by:  cdlemg9a  31491  cdlemg12a  31502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211
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