Metamath Proof Explorer

Theorem latjrot

Description: Rotate lattice join of 3 classes. (Contributed by NM, 23-Jul-2012)

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

Proof

Step Hyp Ref Expression
1 latjass.b 
 |-  B = ( Base ` K )
2 latjass.j 
 |-  .\/ = ( join ` K )
3 1 2 latj31 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ Y ) .\/ Z ) = ( ( Z .\/ Y ) .\/ X ) )
4 simpl 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. Lat )
5 simpr3 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
6 simpr2 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
7 simpr1 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )
8 1 2 latj32 
 |-  ( ( K e. Lat /\ ( Z e. B /\ Y e. B /\ X e. B ) ) -> ( ( Z .\/ Y ) .\/ X ) = ( ( Z .\/ X ) .\/ Y ) )
9 4 5 6 7 8 syl13anc 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( Z .\/ Y ) .\/ X ) = ( ( Z .\/ X ) .\/ Y ) )
10 3 9 eqtrd 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ Y ) .\/ Z ) = ( ( Z .\/ X ) .\/ Y ) )