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Metamath Proof Explorer

Theorem latmrot

Description: Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012)

Ref Expression
Hypotheses olmass.b 
|- B = ( Base ` K )
olmass.m 
|- ./\ = ( meet ` K )
Assertion latmrot 
|- ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ Z ) = ( ( Z ./\ X ) ./\ 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 3 adantr 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. Lat )
5 simpr1 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )
6 simpr2 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
7 1 2 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) e. B )
8 4 5 6 7 syl3anc 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X ./\ Y ) e. B )
9 simpr3 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
10 1 2 latmcom 
 |-  ( ( K e. Lat /\ ( X ./\ Y ) e. B /\ Z e. B ) -> ( ( X ./\ Y ) ./\ Z ) = ( Z ./\ ( X ./\ Y ) ) )
11 4 8 9 10 syl3anc 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ Z ) = ( Z ./\ ( X ./\ Y ) ) )
12 simpl 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. OL )
13 1 2 latmassOLD 
 |-  ( ( K e. OL /\ ( Z e. B /\ X e. B /\ Y e. B ) ) -> ( ( Z ./\ X ) ./\ Y ) = ( Z ./\ ( X ./\ Y ) ) )
14 12 9 5 6 13 syl13anc 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( Z ./\ X ) ./\ Y ) = ( Z ./\ ( X ./\ Y ) ) )
15 11 14 eqtr4d 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ Z ) = ( ( Z ./\ X ) ./\ Y ) )