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Theorem 2llnma3r 29977
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 29557 . . . 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 29557 . . . 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 5876 . 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 5874 . . . . . 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 29558 . . . . . . 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 2315 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  /\  Q  =  R )  ->  ( R  .\/  Q )  =  R )
1918oveq2d 5874 . . . 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 29564 . . . . . . 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 29553 . . . . . . . 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 2283 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2625, 5atbase 29479 . . . . . . . 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 29556 . . . . . . . 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 14186 . . . . . . 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 2315 . . 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 29565 . . . . . . . . . 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 29479 . . . . . . . . . . . 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 29556 . . . . . . . . . . . 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 14168 . . . . . . . . . . 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 29666 . . . . . . . . . 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 2285 . . . . . . . 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 2482 . . . . 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 29976 . . . 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 2524 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  ( ( R 
.\/  P )  ./\  ( R  .\/  Q ) )  =  R )
6411, 63eqtrd 2315 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 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   HLchlt 29540
This theorem is referenced by:  cdlemg9a  30821  cdlemg12a  30832
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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