Theorem adds12d

Description: Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Scott Fenton, 9-Mar-2025)

	Ref	Expression
Hypotheses	addsassd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	addsassd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	addsassd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
Assertion	adds12d	⊢ ( 𝜑 → ( 𝐴 +s ( 𝐵 +s 𝐶 ) ) = ( 𝐵 +s ( 𝐴 +s 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		addsassd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		addsassd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		addsassd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4	1 2	addscomd	⊢ ( 𝜑 → ( 𝐴 +s 𝐵 ) = ( 𝐵 +s 𝐴 ) )
5	4	oveq1d	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) +s 𝐶 ) = ( ( 𝐵 +s 𝐴 ) +s 𝐶 ) )
6	1 2 3	addsassd	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) +s 𝐶 ) = ( 𝐴 +s ( 𝐵 +s 𝐶 ) ) )
7	2 1 3	addsassd	⊢ ( 𝜑 → ( ( 𝐵 +s 𝐴 ) +s 𝐶 ) = ( 𝐵 +s ( 𝐴 +s 𝐶 ) ) )
8	5 6 7	3eqtr3d	⊢ ( 𝜑 → ( 𝐴 +s ( 𝐵 +s 𝐶 ) ) = ( 𝐵 +s ( 𝐴 +s 𝐶 ) ) )