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Metamath Proof Explorer

Theorem latmass

Description: Lattice meet is associative. (Contributed by Stefan O'Rear, 30-Jan-2015)

Ref Expression
Hypotheses latmass.b 
|- B = ( Base ` K )
latmass.m 
|- ./\ = ( meet ` K )
Assertion latmass 
|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ Z ) = ( X ./\ ( Y ./\ Z ) ) )

Proof

Step Hyp Ref Expression
1 latmass.b 
 |-  B = ( Base ` K )
2 latmass.m 
 |-  ./\ = ( meet ` K )
3 eqid 
 |-  ( ODual ` K ) = ( ODual ` K )
4 3 odulat 
 |-  ( K e. Lat -> ( ODual ` K ) e. Lat )
5 3 1 odubas 
 |-  B = ( Base ` ( ODual ` K ) )
6 3 2 odujoin 
 |-  ./\ = ( join ` ( ODual ` K ) )
7 5 6 latjass 
 |-  ( ( ( ODual ` K ) e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ Z ) = ( X ./\ ( Y ./\ Z ) ) )
8 4 7 sylan 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ Z ) = ( X ./\ ( Y ./\ Z ) ) )