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Metamath Proof Explorer

Theorem latjjdi

Description: Lattice join distributes over itself. (Contributed by NM, 30-Jul-2012)

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

Proof

Step Hyp Ref Expression
1 latjass.b 
 |-  B = ( Base ` K )
2 latjass.j 
 |-  .\/ = ( join ` K )
3 simpr1 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )
4 1 2 latjidm 
 |-  ( ( K e. Lat /\ X e. B ) -> ( X .\/ X ) = X )
5 3 4 syldan 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .\/ X ) = X )
6 5 oveq1d 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ X ) .\/ ( Y .\/ Z ) ) = ( X .\/ ( Y .\/ Z ) ) )
7 simpl 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. Lat )
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 /\ X e. B ) /\ ( Y e. B /\ Z e. B ) ) -> ( ( X .\/ X ) .\/ ( Y .\/ Z ) ) = ( ( X .\/ Y ) .\/ ( X .\/ Z ) ) )
11 7 3 3 8 9 10 syl122anc 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ X ) .\/ ( Y .\/ Z ) ) = ( ( X .\/ Y ) .\/ ( X .\/ Z ) ) )
12 6 11 eqtr3d 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .\/ ( Y .\/ Z ) ) = ( ( X .\/ Y ) .\/ ( X .\/ Z ) ) )