Metamath Proof Explorer

Theorem nadd32

Description: Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Scott Fenton, 20-Jan-2025)

Ref Expression
Assertion nadd32 
|- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( A +no B ) +no C ) = ( ( A +no C ) +no B ) )

Proof

Step Hyp Ref Expression
1 naddcom 
 |-  ( ( B e. On /\ C e. On ) -> ( B +no C ) = ( C +no B ) )
2 1 3adant1 
 |-  ( ( A e. On /\ B e. On /\ C e. On ) -> ( B +no C ) = ( C +no B ) )
3 2 oveq2d 
 |-  ( ( A e. On /\ B e. On /\ C e. On ) -> ( A +no ( B +no C ) ) = ( A +no ( C +no B ) ) )
4 naddass 
 |-  ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( A +no B ) +no C ) = ( A +no ( B +no C ) ) )
5 naddass 
 |-  ( ( A e. On /\ C e. On /\ B e. On ) -> ( ( A +no C ) +no B ) = ( A +no ( C +no B ) ) )
6 5 3com23 
 |-  ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( A +no C ) +no B ) = ( A +no ( C +no B ) ) )
7 3 4 6 3eqtr4d 
 |-  ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( A +no B ) +no C ) = ( ( A +no C ) +no B ) )