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Metamath Proof Explorer

Theorem 3atlem3

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 3atlem3 
|- ( ( ( 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 =/= U /\ -. 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 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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ P .<_ ( T .\/ U ) ) -> ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) )
5 simpl21 
 |-  ( ( ( ( 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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ P .<_ ( T .\/ U ) ) -> -. R .<_ ( P .\/ Q ) )
6 simpl22 
 |-  ( ( ( ( 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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ P .<_ ( T .\/ U ) ) -> P =/= U )
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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ P .<_ ( T .\/ U ) ) -> P .<_ ( T .\/ U ) )
8 6 7 jca 
 |-  ( ( ( ( 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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ P .<_ ( T .\/ U ) ) -> ( P =/= U /\ P .<_ ( T .\/ U ) ) )
9 simpl23 
 |-  ( ( ( ( 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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ P .<_ ( T .\/ U ) ) -> -. Q .<_ ( P .\/ U ) )
10 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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ P .<_ ( T .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) )
11 1 2 3 3atlem2 
 |-  ( ( ( 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 =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )
12 4 5 8 9 10 11 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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ P .<_ ( T .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )
13 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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ -. P .<_ ( T .\/ U ) ) -> ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) )
14 simpl21 
 |-  ( ( ( ( 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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ -. P .<_ ( T .\/ U ) ) -> -. R .<_ ( P .\/ Q ) )
15 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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ -. P .<_ ( T .\/ U ) ) -> -. P .<_ ( T .\/ U ) )
16 simpl23 
 |-  ( ( ( ( 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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ -. P .<_ ( T .\/ U ) ) -> -. Q .<_ ( P .\/ U ) )
17 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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ -. P .<_ ( T .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) )
18 1 2 3 3atlem1 
 |-  ( ( ( 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 .<_ ( T .\/ U ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )
19 13 14 15 16 17 18 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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) /\ -. P .<_ ( T .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )
20 12 19 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 =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )