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 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 +no 𝐵 ) +no 𝐶 ) = ( ( 𝐴 +no 𝐶 ) +no 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		naddcom	⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐵 +no 𝐶 ) = ( 𝐶 +no 𝐵 ) )
2	1	3adant1	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐵 +no 𝐶 ) = ( 𝐶 +no 𝐵 ) )
3	2	oveq2d	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 +no ( 𝐵 +no 𝐶 ) ) = ( 𝐴 +no ( 𝐶 +no 𝐵 ) ) )
4		naddass	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 +no 𝐵 ) +no 𝐶 ) = ( 𝐴 +no ( 𝐵 +no 𝐶 ) ) )
5		naddass	⊢ ( ( 𝐴 ∈ On ∧ 𝐶 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +no 𝐶 ) +no 𝐵 ) = ( 𝐴 +no ( 𝐶 +no 𝐵 ) ) )
6	5	3com23	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 +no 𝐶 ) +no 𝐵 ) = ( 𝐴 +no ( 𝐶 +no 𝐵 ) ) )
7	3 4 6	3eqtr4d	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 +no 𝐵 ) +no 𝐶 ) = ( ( 𝐴 +no 𝐶 ) +no 𝐵 ) )