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Metamath Proof Explorer

Theorem 4atlem4d

Description: Lemma for 4at . Frequently used associative law. (Contributed by NM, 9-Jul-2012)

Ref Expression
Hypotheses 4at.l 
|- .<_ = ( le ` K )
4at.j 
|- .\/ = ( join ` K )
4at.a 
|- A = ( Atoms ` K )
Assertion 4atlem4d 
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( S .\/ ( ( P .\/ Q ) .\/ R ) ) )

Proof

Step Hyp Ref Expression
1 4at.l 
 |-  .<_ = ( le ` K )
2 4at.j 
 |-  .\/ = ( join ` K )
3 4at.a 
 |-  A = ( Atoms ` K )
4 simpl1 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> K e. HL )
5 4 hllatd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> K e. Lat )
6 eqid 
 |-  ( Base ` K ) = ( Base ` K )
7 6 2 3 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
8 7 adantr 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
9 6 3 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
10 9 ad2antrl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> R e. ( Base ` K ) )
11 6 3 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
12 11 ad2antll 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> S e. ( Base ` K ) )
13 6 2 latjass 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) = ( ( P .\/ Q ) .\/ ( R .\/ S ) ) )
14 5 8 10 12 13 syl13anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) = ( ( P .\/ Q ) .\/ ( R .\/ S ) ) )
15 6 2 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) )
16 5 8 10 15 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) )
17 6 2 latjcom 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) = ( S .\/ ( ( P .\/ Q ) .\/ R ) ) )
18 5 16 12 17 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) = ( S .\/ ( ( P .\/ Q ) .\/ R ) ) )
19 14 18 eqtr3d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( S .\/ ( ( P .\/ Q ) .\/ R ) ) )