Metamath Proof Explorer

Theorem latj12

Description: Swap 1st and 2nd members of lattice join. ( chj12 analog.) (Contributed by NM, 4-Jun-2012)

Ref Expression
Hypotheses latjass.b 
|- B = ( Base ` K )
latjass.j 
|- .\/ = ( join ` K )
Assertion latj12 
|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .\/ ( Y .\/ Z ) ) = ( Y .\/ ( X .\/ Z ) ) )

Proof

Step Hyp Ref Expression
1 latjass.b 
 |-  B = ( Base ` K )
2 latjass.j 
 |-  .\/ = ( join ` K )
3 1 2 latjcom 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) = ( Y .\/ X ) )
4 3 3adant3r3 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .\/ Y ) = ( Y .\/ X ) )
5 4 oveq1d 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ Y ) .\/ Z ) = ( ( Y .\/ X ) .\/ Z ) )
6 1 2 latjass 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ Y ) .\/ Z ) = ( X .\/ ( Y .\/ Z ) ) )
7 simpl 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. Lat )
8 simpr2 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
9 simpr1 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )
10 simpr3 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
11 1 2 latjass 
 |-  ( ( K e. Lat /\ ( Y e. B /\ X e. B /\ Z e. B ) ) -> ( ( Y .\/ X ) .\/ Z ) = ( Y .\/ ( X .\/ Z ) ) )
12 7 8 9 10 11 syl13anc 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( Y .\/ X ) .\/ Z ) = ( Y .\/ ( X .\/ Z ) ) )
13 5 6 12 3eqtr3d 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .\/ ( Y .\/ Z ) ) = ( Y .\/ ( X .\/ Z ) ) )