Metamath Proof Explorer

Theorem 3atlem6

Description: Lemma for 3at . (Contributed by NM, 23-Jun-2012)

Ref Expression
Hypotheses 3at.l 
|- .<_ = ( le ` K )
3at.j 
|- .\/ = ( join ` K )
3at.a 
|- A = ( Atoms ` K )
Assertion 3atlem6 
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )

Proof

Step Hyp Ref Expression
1 3at.l 
 |-  .<_ = ( le ` K )
2 3at.j 
 |-  .\/ = ( join ` K )
3 3at.a 
 |-  A = ( Atoms ` K )
4 simp11 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> K e. HL )
5 simp121 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> P e. A )
6 simp122 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> Q e. A )
7 simp123 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> R e. A )
8 2 3 hlatj32 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( P .\/ R ) .\/ Q ) )
9 4 5 6 7 8 syl13anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( P .\/ R ) .\/ Q ) )
10 5 7 6 3jca 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( P e. A /\ R e. A /\ Q e. A ) )
11 simp13 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( S e. A /\ T e. A /\ U e. A ) )
12 simp21 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> -. R .<_ ( P .\/ Q ) )
13 simp22 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> P =/= Q )
14 13 necomd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> Q =/= P )
15 1 2 3 hlatexch1 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ P e. A ) /\ Q =/= P ) -> ( Q .<_ ( P .\/ R ) -> R .<_ ( P .\/ Q ) ) )
16 4 6 7 5 14 15 syl131anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( Q .<_ ( P .\/ R ) -> R .<_ ( P .\/ Q ) ) )
17 12 16 mtod 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> -. Q .<_ ( P .\/ R ) )
18 4 hllatd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> K e. Lat )
19 eqid 
 |-  ( Base ` K ) = ( Base ` K )
20 19 3 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
21 7 20 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> R e. ( Base ` K ) )
22 19 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
23 5 22 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> P e. ( Base ` K ) )
24 19 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
25 6 24 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> Q e. ( Base ` K ) )
26 19 1 2 latnlej1l 
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= P )
27 26 necomd 
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> P =/= R )
28 18 21 23 25 12 27 syl131anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> P =/= R )
29 simp23 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> Q .<_ ( P .\/ U ) )
30 simp133 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> U e. A )
31 1 2 3 hlatexchb1 
 |-  ( ( K e. HL /\ ( Q e. A /\ U e. A /\ P e. A ) /\ Q =/= P ) -> ( Q .<_ ( P .\/ U ) <-> ( P .\/ Q ) = ( P .\/ U ) ) )
32 4 6 30 5 14 31 syl131anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( Q .<_ ( P .\/ U ) <-> ( P .\/ Q ) = ( P .\/ U ) ) )
33 29 32 mpbid 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( P .\/ Q ) = ( P .\/ U ) )
34 33 breq2d 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( R .<_ ( P .\/ Q ) <-> R .<_ ( P .\/ U ) ) )
35 12 34 mtbid 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> -. R .<_ ( P .\/ U ) )
36 simp3 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) )
37 9 36 eqbrtrrd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ R ) .\/ Q ) .<_ ( ( S .\/ T ) .\/ U ) )
38 1 2 3 3atlem5 
 |-  ( ( ( K e. HL /\ ( P e. A /\ R e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. Q .<_ ( P .\/ R ) /\ P =/= R /\ -. R .<_ ( P .\/ U ) ) /\ ( ( P .\/ R ) .\/ Q ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ R ) .\/ Q ) = ( ( S .\/ T ) .\/ U ) )
39 4 10 11 17 28 35 37 38 syl331anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ R ) .\/ Q ) = ( ( S .\/ T ) .\/ U ) )
40 9 39 eqtrd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )