Metamath Proof Explorer

Theorem latj32

Description: Swap 2nd and 3rd members of lattice join. Lemma 2.2 in MegPav2002 p. 362. (Contributed by NM, 2-Dec-2011)

Ref Expression
Hypotheses latjass.b 
|- B = ( Base ` K )
latjass.j 
|- .\/ = ( join ` K )
Assertion latj32 
|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ Y ) .\/ Z ) = ( ( X .\/ Z ) .\/ Y ) )

Proof

Step Hyp Ref Expression
1 latjass.b 
 |-  B = ( Base ` K )
2 latjass.j 
 |-  .\/ = ( join ` K )
3 1 2 latjcom 
 |-  ( ( K e. Lat /\ Y e. B /\ Z e. B ) -> ( Y .\/ Z ) = ( Z .\/ Y ) )
4 3 3adant3r1 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y .\/ Z ) = ( Z .\/ Y ) )
5 4 oveq2d 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .\/ ( Y .\/ Z ) ) = ( X .\/ ( Z .\/ Y ) ) )
6 1 2 latjass 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ Y ) .\/ Z ) = ( X .\/ ( Y .\/ Z ) ) )
7 simpr1 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )
8 simpr3 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
9 simpr2 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
10 7 8 9 3jca 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X e. B /\ Z e. B /\ Y e. B ) )
11 1 2 latjass 
 |-  ( ( K e. Lat /\ ( X e. B /\ Z e. B /\ Y e. B ) ) -> ( ( X .\/ Z ) .\/ Y ) = ( X .\/ ( Z .\/ Y ) ) )
12 10 11 syldan 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ Z ) .\/ Y ) = ( X .\/ ( Z .\/ Y ) ) )
13 5 6 12 3eqtr4d 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ Y ) .\/ Z ) = ( ( X .\/ Z ) .\/ Y ) )