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Metamath Proof Explorer

Theorem dlatjmdi

Description: In a distributive lattice, joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015)

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

Proof

Step Hyp Ref Expression
1 dlatjmdi.b 
 |-  B = ( Base ` K )
2 dlatjmdi.j 
 |-  .\/ = ( join ` K )
3 dlatjmdi.m 
 |-  ./\ = ( meet ` K )
4 eqid 
 |-  ( ODual ` K ) = ( ODual ` K )
5 4 odudlatb 
 |-  ( K e. DLat -> ( K e. DLat <-> ( ODual ` K ) e. DLat ) )
6 5 ibi 
 |-  ( K e. DLat -> ( ODual ` K ) e. DLat )
7 4 1 odubas 
 |-  B = ( Base ` ( ODual ` K ) )
8 4 3 odujoin 
 |-  ./\ = ( join ` ( ODual ` K ) )
9 4 2 odumeet 
 |-  .\/ = ( meet ` ( ODual ` K ) )
10 7 8 9 dlatmjdi 
 |-  ( ( ( ODual ` K ) e. DLat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .\/ ( Y ./\ Z ) ) = ( ( X .\/ Y ) ./\ ( X .\/ Z ) ) )
11 6 10 sylan 
 |-  ( ( K e. DLat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .\/ ( Y ./\ Z ) ) = ( ( X .\/ Y ) ./\ ( X .\/ Z ) ) )