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Metamath Proof Explorer

Theorem 4atlem4b

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 4atlem4b 
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( Q .\/ ( ( P .\/ 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 simpl2 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> P e. A )
6 simpl3 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> Q e. A )
7 simprl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> R e. A )
8 simprr 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> S e. A )
9 2 3 hlatj4 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ R ) .\/ ( Q .\/ S ) ) )
10 4 5 6 7 8 9 syl122anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ R ) .\/ ( Q .\/ S ) ) )
11 4 hllatd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> K e. Lat )
12 eqid 
 |-  ( Base ` K ) = ( Base ` K )
13 12 2 3 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ R e. A ) -> ( P .\/ R ) e. ( Base ` K ) )
14 4 5 7 13 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( P .\/ R ) e. ( Base ` K ) )
15 12 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
16 6 15 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> Q e. ( Base ` K ) )
17 12 3 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
18 17 ad2antll 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> S e. ( Base ` K ) )
19 12 2 latj12 
 |-  ( ( K e. Lat /\ ( ( P .\/ R ) e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( ( P .\/ R ) .\/ ( Q .\/ S ) ) = ( Q .\/ ( ( P .\/ R ) .\/ S ) ) )
20 11 14 16 18 19 syl13anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ R ) .\/ ( Q .\/ S ) ) = ( Q .\/ ( ( P .\/ R ) .\/ S ) ) )
21 10 20 eqtrd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( Q .\/ ( ( P .\/ R ) .\/ S ) ) )