Metamath Proof Explorer

Theorem latmmdir

Description: Lattice meet distributes over itself. ( inindir analog.) (Contributed by NM, 6-Jun-2012)

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

Proof

Step Hyp Ref Expression
1 olmass.b 
 |-  B = ( Base ` K )
2 olmass.m 
 |-  ./\ = ( meet ` K )
3 ollat 
 |-  ( K e. OL -> K e. Lat )
4 3 adantr 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. Lat )
5 simpr3 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
6 1 2 latmidm 
 |-  ( ( K e. Lat /\ Z e. B ) -> ( Z ./\ Z ) = Z )
7 4 5 6 syl2anc 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Z ./\ Z ) = Z )
8 7 oveq2d 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ ( Z ./\ Z ) ) = ( ( X ./\ Y ) ./\ Z ) )
9 simpl 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. OL )
10 simpr1 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )
11 simpr2 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
12 1 2 latm4 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ ( Z ./\ Z ) ) = ( ( X ./\ Z ) ./\ ( Y ./\ Z ) ) )
13 9 10 11 5 5 12 syl122anc 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ ( Z ./\ Z ) ) = ( ( X ./\ Z ) ./\ ( Y ./\ Z ) ) )
14 8 13 eqtr3d 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ Z ) = ( ( X ./\ Z ) ./\ ( Y ./\ Z ) ) )