Metamath Proof Explorer

Theorem 3atlem7

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 3atlem7 
|- ( ( ( 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 ) /\ ( ( 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 simpl1 
 |-  ( ( ( ( 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 ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ Q .<_ ( P .\/ U ) ) -> ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) )
5 simpl2l 
 |-  ( ( ( ( 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 ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ Q .<_ ( P .\/ U ) ) -> -. R .<_ ( P .\/ Q ) )
6 simpl2r 
 |-  ( ( ( ( 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 ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ Q .<_ ( P .\/ U ) ) -> P =/= Q )
7 simpr 
 |-  ( ( ( ( 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 ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ Q .<_ ( P .\/ U ) ) -> Q .<_ ( P .\/ U ) )
8 simpl3 
 |-  ( ( ( ( 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 ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ Q .<_ ( P .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) )
9 1 2 3 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 ) )
10 4 5 6 7 8 9 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 ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ Q .<_ ( P .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )
11 simpl1 
 |-  ( ( ( ( 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 ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) -> ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) )
12 simpl2l 
 |-  ( ( ( ( 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 ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) -> -. R .<_ ( P .\/ Q ) )
13 simpl2r 
 |-  ( ( ( ( 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 ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) -> P =/= Q )
14 simpr 
 |-  ( ( ( ( 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 ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) -> -. Q .<_ ( P .\/ U ) )
15 simpl3 
 |-  ( ( ( ( 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 ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) )
16 1 2 3 3atlem5 
 |-  ( ( ( 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 ) )
17 11 12 13 14 15 16 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 ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )
18 10 17 pm2.61dan 
 |-  ( ( ( 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 ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )