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Metamath Proof Explorer

Theorem latm12

Description: A rearrangement of lattice meet. ( in12 analog.) (Contributed by NM, 8-Nov-2011)

Ref Expression
Hypotheses olmass.b 
|- B = ( Base ` K )
olmass.m 
|- ./\ = ( meet ` K )
Assertion latm12 
|- ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X ./\ ( Y ./\ Z ) ) = ( Y ./\ ( X ./\ 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 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 latmcom 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) = ( Y ./\ X ) )
8 4 5 6 7 syl3anc 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X ./\ Y ) = ( Y ./\ X ) )
9 8 oveq1d 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ Z ) = ( ( Y ./\ X ) ./\ Z ) )
10 1 2 latmassOLD 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Y ) ./\ Z ) = ( X ./\ ( Y ./\ Z ) ) )
11 simpr3 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
12 6 5 11 3jca 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y e. B /\ X e. B /\ Z e. B ) )
13 1 2 latmassOLD 
 |-  ( ( K e. OL /\ ( Y e. B /\ X e. B /\ Z e. B ) ) -> ( ( Y ./\ X ) ./\ Z ) = ( Y ./\ ( X ./\ Z ) ) )
14 12 13 syldan 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( Y ./\ X ) ./\ Z ) = ( Y ./\ ( X ./\ Z ) ) )
15 9 10 14 3eqtr3d 
 |-  ( ( K e. OL /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X ./\ ( Y ./\ Z ) ) = ( Y ./\ ( X ./\ Z ) ) )