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Metamath Proof Explorer

Theorem 3atlem4

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 3atlem4 
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ R ) )

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 ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> K e. HL )
5 simp12 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> ( P e. A /\ Q e. A /\ R e. A ) )
6 simp13l 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> S e. A )
7 simp13r 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> T e. A )
8 simp123 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> R e. A )
9 6 7 8 3jca 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> ( S e. A /\ T e. A /\ R e. A ) )
10 simp2l 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> -. R .<_ ( P .\/ Q ) )
11 4 hllatd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> K e. Lat )
12 eqid 
 |-  ( Base ` K ) = ( Base ` K )
13 12 3 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
14 8 13 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> R e. ( Base ` K ) )
15 simp121 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> P e. A )
16 12 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
17 15 16 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> P e. ( Base ` K ) )
18 simp122 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> Q e. A )
19 12 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
20 18 19 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> Q e. ( Base ` K ) )
21 12 1 2 latnlej1l 
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= P )
22 11 14 17 20 10 21 syl131anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> R =/= P )
23 22 necomd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> P =/= R )
24 simp2r 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> P =/= Q )
25 24 necomd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> Q =/= P )
26 1 2 3 hlatexch1 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ P e. A ) /\ Q =/= P ) -> ( Q .<_ ( P .\/ R ) -> R .<_ ( P .\/ Q ) ) )
27 4 18 8 15 25 26 syl131anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> ( Q .<_ ( P .\/ R ) -> R .<_ ( P .\/ Q ) ) )
28 10 27 mtod 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> -. Q .<_ ( P .\/ R ) )
29 simp3 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) )
30 1 2 3 3atlem3 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ R e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= R /\ -. Q .<_ ( P .\/ R ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ R ) )
31 4 5 9 10 23 28 29 30 syl331anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ R ) )