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Metamath Proof Explorer

Theorem latm32

Description: A rearrangement of lattice meet. ( in12 analog.) (Contributed by NM, 13-Nov-2012)

Ref Expression
Hypotheses olmass.b 
|- B = ( Base ` K )
olmass.m 
|- ./\ = ( meet ` K )
Assertion latm32 
|- ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ Z ) = ( ( X ./\ Z ) ./\ Y ) )

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 1 2 latmcom 
 |-  ( ( K e. Lat /\ Y e. B /\ Z e. B ) -> ( Y ./\ Z ) = ( Z ./\ Y ) )
5 3 4 syl3an1 
 |-  ( ( K e. OL /\ Y e. B /\ Z e. B ) -> ( Y ./\ Z ) = ( Z ./\ Y ) )
6 5 3adant3r1 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y ./\ Z ) = ( Z ./\ Y ) )
7 6 oveq2d 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X ./\ ( Y ./\ Z ) ) = ( X ./\ ( Z ./\ Y ) ) )
8 1 2 latmassOLD 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ 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 simpr3 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
12 simpr2 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
13 1 2 latmassOLD 
 |-  ( ( K e. OL /\ ( X e. B /\ Z e. B /\ Y e. B ) ) -> ( ( X ./\ Z ) ./\ Y ) = ( X ./\ ( Z ./\ Y ) ) )
14 9 10 11 12 13 syl13anc 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Z ) ./\ Y ) = ( X ./\ ( Z ./\ Y ) ) )
15 7 8 14 3eqtr4d 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ Z ) = ( ( X ./\ Z ) ./\ Y ) )