Metamath Proof Explorer

Theorem chj12

Description: A rearrangement of Hilbert lattice join. (Contributed by NM, 15-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion chj12 
|- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A vH ( B vH C ) ) = ( B vH ( A vH C ) ) )

Proof

Step Hyp Ref Expression
1 chjcom 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A vH B ) = ( B vH A ) )
2 1 3adant3 
 |-  ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A vH B ) = ( B vH A ) )
3 2 oveq1d 
 |-  ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( A vH B ) vH C ) = ( ( B vH A ) vH C ) )
4 chjass 
 |-  ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( A vH B ) vH C ) = ( A vH ( B vH C ) ) )
5 chjass 
 |-  ( ( B e. CH /\ A e. CH /\ C e. CH ) -> ( ( B vH A ) vH C ) = ( B vH ( A vH C ) ) )
6 5 3com12 
 |-  ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( B vH A ) vH C ) = ( B vH ( A vH C ) ) )
7 3 4 6 3eqtr3d 
 |-  ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A vH ( B vH C ) ) = ( B vH ( A vH C ) ) )