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Metamath Proof Explorer

Theorem 2atm

Description: An atom majorized by two different atom joins (which could be atoms or lines) is equal to their intersection. (Contributed by NM, 30-Jun-2013)

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

Proof

Step Hyp Ref Expression
1 2atm.l 
 |-  .<_ = ( le ` K )
2 2atm.j 
 |-  .\/ = ( join ` K )
3 2atm.m 
 |-  ./\ = ( meet ` K )
4 2atm.a 
 |-  A = ( Atoms ` K )
5 simp31 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> T .<_ ( P .\/ Q ) )
6 simp32 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> T .<_ ( R .\/ S ) )
7 simp11 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> K e. HL )
8 7 hllatd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> K e. Lat )
9 simp23 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> T e. A )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 4 atbase 
 |-  ( T e. A -> T e. ( Base ` K ) )
12 9 11 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> T e. ( Base ` K ) )
13 simp12 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> P e. A )
14 10 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
15 13 14 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> P e. ( Base ` K ) )
16 simp13 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> Q e. A )
17 10 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
18 16 17 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> Q e. ( Base ` K ) )
19 10 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
20 8 15 18 19 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
21 simp21 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> R e. A )
22 simp22 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> S e. A )
23 10 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) )
24 7 21 22 23 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( R .\/ S ) e. ( Base ` K ) )
25 10 1 3 latlem12 
 |-  ( ( K e. Lat /\ ( T e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( R .\/ S ) e. ( Base ` K ) ) ) -> ( ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) ) <-> T .<_ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) ) )
26 8 12 20 24 25 syl13anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) ) <-> T .<_ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) ) )
27 5 6 26 mpbi2and 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> T .<_ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) )
28 hlatl 
 |-  ( K e. HL -> K e. AtLat )
29 7 28 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> K e. AtLat )
30 10 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( R .\/ S ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. ( Base ` K ) )
31 8 20 24 30 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. ( Base ` K ) )
32 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
33 10 1 32 4 atlen0 
 |-  ( ( ( K e. AtLat /\ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. ( Base ` K ) /\ T e. A ) /\ T .<_ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) ) -> ( ( P .\/ Q ) ./\ ( R .\/ S ) ) =/= ( 0. ` K ) )
34 29 31 9 27 33 syl31anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( P .\/ Q ) ./\ ( R .\/ S ) ) =/= ( 0. ` K ) )
35 34 neneqd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> -. ( ( P .\/ Q ) ./\ ( R .\/ S ) ) = ( 0. ` K ) )
36 simp33 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( P .\/ Q ) =/= ( R .\/ S ) )
37 2 3 32 4 2atmat0 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. A \/ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) = ( 0. ` K ) ) )
38 7 13 16 21 22 36 37 syl33anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. A \/ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) = ( 0. ` K ) ) )
39 38 ord 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( -. ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. A -> ( ( P .\/ Q ) ./\ ( R .\/ S ) ) = ( 0. ` K ) ) )
40 35 39 mt3d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. A )
41 1 4 atcmp 
 |-  ( ( K e. AtLat /\ T e. A /\ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. A ) -> ( T .<_ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) <-> T = ( ( P .\/ Q ) ./\ ( R .\/ S ) ) ) )
42 29 9 40 41 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( T .<_ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) <-> T = ( ( P .\/ Q ) ./\ ( R .\/ S ) ) ) )
43 27 42 mpbid 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> T = ( ( P .\/ Q ) ./\ ( R .\/ S ) ) )