Metamath Proof Explorer

Theorem 4atlem4a

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 4atlem4a 
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( P .\/ ( ( Q .\/ R ) .\/ S ) ) )

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 simpl2 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> P e. A )
7 eqid 
 |-  ( Base ` K ) = ( Base ` K )
8 7 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
9 6 8 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> P e. ( Base ` K ) )
10 simpl3 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> Q e. A )
11 7 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
12 10 11 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> Q e. ( Base ` K ) )
13 simprl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> R e. A )
14 simprr 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> S e. A )
15 7 2 3 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) )
16 4 13 14 15 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( R .\/ S ) e. ( Base ` K ) )
17 7 2 latjass 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ ( R .\/ S ) e. ( Base ` K ) ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( P .\/ ( Q .\/ ( R .\/ S ) ) ) )
18 5 9 12 16 17 syl13anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( P .\/ ( Q .\/ ( R .\/ S ) ) ) )
19 2 3 hlatjass 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( ( Q .\/ R ) .\/ S ) = ( Q .\/ ( R .\/ S ) ) )
20 4 10 13 14 19 syl13anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( Q .\/ R ) .\/ S ) = ( Q .\/ ( R .\/ S ) ) )
21 20 oveq2d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( P .\/ ( ( Q .\/ R ) .\/ S ) ) = ( P .\/ ( Q .\/ ( R .\/ S ) ) ) )
22 18 21 eqtr4d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( P .\/ ( ( Q .\/ R ) .\/ S ) ) )