Metamath Proof Explorer

Theorem add32

Description: Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999)

Ref Expression
Assertion add32 
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )

Proof

Step Hyp Ref Expression
1 addcom 
 |-  ( ( B e. CC /\ C e. CC ) -> ( B + C ) = ( C + B ) )
2 1 oveq2d 
 |-  ( ( B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( A + ( C + B ) ) )
3 2 3adant1 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( A + ( C + B ) ) )
4 addass 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
5 addass 
 |-  ( ( A e. CC /\ C e. CC /\ B e. CC ) -> ( ( A + C ) + B ) = ( A + ( C + B ) ) )
6 5 3com23 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) + B ) = ( A + ( C + B ) ) )
7 3 4 6 3eqtr4d 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )