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Metamath Proof Explorer

Theorem hlatj12

Description: Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 for atoms. (Contributed by NM, 4-Jun-2012)

Ref Expression
Hypotheses hlatjcom.j 
|- .\/ = ( join ` K )
hlatjcom.a 
|- A = ( Atoms ` K )
Assertion hlatj12 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .\/ ( Q .\/ R ) ) = ( Q .\/ ( P .\/ R ) ) )

Proof

Step Hyp Ref Expression
1 hlatjcom.j 
 |-  .\/ = ( join ` K )
2 hlatjcom.a 
 |-  A = ( Atoms ` K )
3 1 2 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
4 3 3adant3r3 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
5 4 oveq1d 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( Q .\/ P ) .\/ R ) )
6 1 2 hlatjass 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( P .\/ ( Q .\/ R ) ) )
7 simpl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> K e. HL )
8 simpr2 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> Q e. A )
9 simpr1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> P e. A )
10 simpr3 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. A )
11 1 2 hlatjass 
 |-  ( ( K e. HL /\ ( Q e. A /\ P e. A /\ R e. A ) ) -> ( ( Q .\/ P ) .\/ R ) = ( Q .\/ ( P .\/ R ) ) )
12 7 8 9 10 11 syl13anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( Q .\/ P ) .\/ R ) = ( Q .\/ ( P .\/ R ) ) )
13 5 6 12 3eqtr3d 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .\/ ( Q .\/ R ) ) = ( Q .\/ ( P .\/ R ) ) )