Metamath Proof Explorer

Theorem hlatj4

Description: Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 for atoms. (Contributed by NM, 9-Aug-2012)

Ref Expression
Hypotheses hlatjcom.j 
|- .\/ = ( join ` K )
hlatjcom.a 
|- A = ( Atoms ` K )
Assertion hlatj4 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ R ) .\/ ( Q .\/ S ) ) )

Proof

Step Hyp Ref Expression
1 hlatjcom.j 
 |-  .\/ = ( join ` K )
2 hlatjcom.a 
 |-  A = ( Atoms ` K )
3 hllat 
 |-  ( K e. HL -> K e. Lat )
4 3 3ad2ant1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> K e. Lat )
5 simp2l 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> P e. A )
6 eqid 
 |-  ( Base ` K ) = ( Base ` K )
7 6 2 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
8 5 7 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> P e. ( Base ` K ) )
9 simp2r 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> Q e. A )
10 6 2 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
11 9 10 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> Q e. ( Base ` K ) )
12 simp3l 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> R e. A )
13 6 2 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
14 12 13 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> R e. ( Base ` K ) )
15 simp3r 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> S e. A )
16 6 2 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
17 15 16 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> S e. ( Base ` K ) )
18 6 1 latj4 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ ( R e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ R ) .\/ ( Q .\/ S ) ) )
19 4 8 11 14 17 18 syl122anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ R ) .\/ ( Q .\/ S ) ) )