Metamath Proof Explorer

Theorem latjjdir

Description: Lattice join distributes over itself. (Contributed by NM, 2-Aug-2012)

Ref Expression
Hypotheses latjass.b 
|- B = ( Base ` K )
latjass.j 
|- .\/ = ( join ` K )
Assertion latjjdir 
|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ Y ) .\/ Z ) = ( ( X .\/ Z ) .\/ ( Y .\/ Z ) ) )

Proof

Step Hyp Ref Expression
1 latjass.b 
 |-  B = ( Base ` K )
2 latjass.j 
 |-  .\/ = ( join ` K )
3 1 2 latjidm 
 |-  ( ( K e. Lat /\ Z e. B ) -> ( Z .\/ Z ) = Z )
4 3 3ad2antr3 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Z .\/ Z ) = Z )
5 4 oveq2d 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ Y ) .\/ ( Z .\/ Z ) ) = ( ( X .\/ Y ) .\/ Z ) )
6 simpl 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. Lat )
7 simpr1 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )
8 simpr2 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
9 simpr3 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
10 1 2 latj4 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ Z e. B ) ) -> ( ( X .\/ Y ) .\/ ( Z .\/ Z ) ) = ( ( X .\/ Z ) .\/ ( Y .\/ Z ) ) )
11 6 7 8 9 9 10 syl122anc 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ Y ) .\/ ( Z .\/ Z ) ) = ( ( X .\/ Z ) .\/ ( Y .\/ Z ) ) )
12 5 11 eqtr3d 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ Y ) .\/ Z ) = ( ( X .\/ Z ) .\/ ( Y .\/ Z ) ) )