Metamath Proof Explorer

Theorem 4atlem0a

Description: Lemma for 4at . (Contributed by NM, 10-Jul-2012)

Ref Expression
Hypotheses 4at.l 
|- .<_ = ( le ` K )
4at.j 
|- .\/ = ( join ` K )
4at.a 
|- A = ( Atoms ` K )
Assertion 4atlem0a 
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> -. R .<_ ( ( P .\/ Q ) .\/ S ) )

Proof

Step Hyp Ref Expression
1 4at.l 
 |-  .<_ = ( le ` K )
2 4at.j 
 |-  .\/ = ( join ` K )
3 4at.a 
 |-  A = ( Atoms ` K )
4 simprr 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> -. S .<_ ( ( P .\/ Q ) .\/ R ) )
5 simpl1 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> K e. HL )
6 simpl3l 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> R e. A )
7 simpl3r 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> S e. A )
8 simpl2l 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> P e. A )
9 simpl2r 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> Q e. A )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 2 3 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
12 5 8 9 11 syl3anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
13 simprl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> -. R .<_ ( P .\/ Q ) )
14 10 1 2 3 hlexch1 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( R .<_ ( ( P .\/ Q ) .\/ S ) -> S .<_ ( ( P .\/ Q ) .\/ R ) ) )
15 5 6 7 12 13 14 syl131anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( R .<_ ( ( P .\/ Q ) .\/ S ) -> S .<_ ( ( P .\/ Q ) .\/ R ) ) )
16 4 15 mtod 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> -. R .<_ ( ( P .\/ Q ) .\/ S ) )