Theorem adds32d

Description: Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Scott Fenton, 22-Jan-2025)

	Ref	Expression
Hypotheses	addsassd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	addsassd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	addsassd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
Assertion	adds32d	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) +s 𝐶 ) = ( ( 𝐴 +s 𝐶 ) +s 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		addsassd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		addsassd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		addsassd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4	2 3	addscomd	⊢ ( 𝜑 → ( 𝐵 +s 𝐶 ) = ( 𝐶 +s 𝐵 ) )
5	4	oveq2d	⊢ ( 𝜑 → ( 𝐴 +s ( 𝐵 +s 𝐶 ) ) = ( 𝐴 +s ( 𝐶 +s 𝐵 ) ) )
6	1 2 3	addsassd	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) +s 𝐶 ) = ( 𝐴 +s ( 𝐵 +s 𝐶 ) ) )
7	1 3 2	addsassd	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐶 ) +s 𝐵 ) = ( 𝐴 +s ( 𝐶 +s 𝐵 ) ) )
8	5 6 7	3eqtr4d	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) +s 𝐶 ) = ( ( 𝐴 +s 𝐶 ) +s 𝐵 ) )