Metamath Proof Explorer

Theorem 2llnma3r

Description: Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 30-Apr-2013)

Ref Expression
Hypotheses 2llnm.l 
|- .<_ = ( le ` K )
2llnm.j 
|- .\/ = ( join ` K )
2llnm.m 
|- ./\ = ( meet ` K )
2llnm.a 
|- A = ( Atoms ` K )
Assertion 2llnma3r 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> ( ( P .\/ R ) ./\ ( Q .\/ R ) ) = R )

Proof

Step Hyp Ref Expression
1 2llnm.l 
 |-  .<_ = ( le ` K )
2 2llnm.j 
 |-  .\/ = ( join ` K )
3 2llnm.m 
 |-  ./\ = ( meet ` K )
4 2llnm.a 
 |-  A = ( Atoms ` K )
5 simp1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> K e. HL )
6 simp21 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> P e. A )
7 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> R e. A )
8 2 4 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ R e. A ) -> ( P .\/ R ) = ( R .\/ P ) )
9 5 6 7 8 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> ( P .\/ R ) = ( R .\/ P ) )
10 simp22 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> Q e. A )
11 2 4 hlatjcom 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) = ( R .\/ Q ) )
12 5 10 7 11 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> ( Q .\/ R ) = ( R .\/ Q ) )
13 9 12 oveq12d 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> ( ( P .\/ R ) ./\ ( Q .\/ R ) ) = ( ( R .\/ P ) ./\ ( R .\/ Q ) ) )
14 simpr 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q = R ) -> Q = R )
15 14 oveq2d 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q = R ) -> ( R .\/ Q ) = ( R .\/ R ) )
16 simpl1 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q = R ) -> K e. HL )
17 simpl23 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q = R ) -> R e. A )
18 2 4 hlatjidm 
 |-  ( ( K e. HL /\ R e. A ) -> ( R .\/ R ) = R )
19 16 17 18 syl2anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q = R ) -> ( R .\/ R ) = R )
20 15 19 eqtrd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q = R ) -> ( R .\/ Q ) = R )
21 20 oveq2d 
 |-  ( ( ( 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 ) )
22 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ R e. A /\ P e. A ) -> R .<_ ( R .\/ P ) )
23 5 7 6 22 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> R .<_ ( R .\/ P ) )
24 hllat 
 |-  ( K e. HL -> K e. Lat )
25 24 3ad2ant1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> K e. Lat )
26 eqid 
 |-  ( Base ` K ) = ( Base ` K )
27 26 4 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
28 7 27 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> R e. ( Base ` K ) )
29 26 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ P e. A ) -> ( R .\/ P ) e. ( Base ` K ) )
30 5 7 6 29 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> ( R .\/ P ) e. ( Base ` K ) )
31 26 1 3 latleeqm2 
 |-  ( ( K e. Lat /\ R e. ( Base ` K ) /\ ( R .\/ P ) e. ( Base ` K ) ) -> ( R .<_ ( R .\/ P ) <-> ( ( R .\/ P ) ./\ R ) = R ) )
32 25 28 30 31 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> ( R .<_ ( R .\/ P ) <-> ( ( R .\/ P ) ./\ R ) = R ) )
33 23 32 mpbid 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> ( ( R .\/ P ) ./\ R ) = R )
34 33 adantr 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q = R ) -> ( ( R .\/ P ) ./\ R ) = R )
35 21 34 eqtrd 
 |-  ( ( ( 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 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q =/= R ) -> K e. HL )
37 simpl21 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q =/= R ) -> P e. A )
38 simpl23 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q =/= R ) -> R e. A )
39 simpl22 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q =/= R ) -> Q e. A )
40 simpl3 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q =/= R ) -> ( P .\/ R ) =/= ( Q .\/ R ) )
41 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ R e. A ) -> R .<_ ( P .\/ R ) )
42 5 6 7 41 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> R .<_ ( P .\/ R ) )
43 26 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
44 10 43 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> Q e. ( Base ` K ) )
45 26 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ R e. A ) -> ( P .\/ R ) e. ( Base ` K ) )
46 5 6 7 45 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> ( P .\/ R ) e. ( Base ` K ) )
47 26 1 2 latjle12 
 |-  ( ( 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 ) ) )
48 25 44 28 46 47 syl13anc 
 |-  ( ( 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 ) ) )
49 48 biimpd 
 |-  ( ( 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 ) ) )
50 42 49 mpan2d 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> ( Q .<_ ( P .\/ R ) -> ( Q .\/ R ) .<_ ( P .\/ R ) ) )
51 50 adantr 
 |-  ( ( ( 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 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q =/= R ) -> Q =/= R )
53 1 2 4 ps-1 
 |-  ( ( 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 ) ) )
54 36 39 38 52 37 38 53 syl132anc 
 |-  ( ( ( 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 ) ) )
55 54 biimpd 
 |-  ( ( ( 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 
 |-  ( ( Q .\/ R ) = ( P .\/ R ) <-> ( P .\/ R ) = ( Q .\/ R ) )
57 55 56 syl6ib 
 |-  ( ( ( 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 ) ) )
58 51 57 syld 
 |-  ( ( ( 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 ) ) )
59 58 necon3ad 
 |-  ( ( ( 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 ) ) )
60 40 59 mpd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q =/= R ) -> -. Q .<_ ( P .\/ R ) )
61 1 2 3 4 2llnma1 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ R ) ) -> ( ( R .\/ P ) ./\ ( R .\/ Q ) ) = R )
62 36 37 38 39 60 61 syl131anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) /\ Q =/= R ) -> ( ( R .\/ P ) ./\ ( R .\/ Q ) ) = R )
63 35 62 pm2.61dane 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> ( ( R .\/ P ) ./\ ( R .\/ Q ) ) = R )
64 13 63 eqtrd 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .\/ R ) =/= ( Q .\/ R ) ) -> ( ( P .\/ R ) ./\ ( Q .\/ R ) ) = R )