Theorem adds42d

Description: Rearrangement of four terms in a surreal sum. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Hypotheses	adds4d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	adds4d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	adds4d.3	⊢ ( 𝜑 → 𝐶 ∈ No )
	adds4d.4	⊢ ( 𝜑 → 𝐷 ∈ No )
Assertion	adds42d	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) +s ( 𝐶 +s 𝐷 ) ) = ( ( 𝐴 +s 𝐶 ) +s ( 𝐷 +s 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		adds4d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		adds4d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		adds4d.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		adds4d.4	⊢ ( 𝜑 → 𝐷 ∈ No )
5	1 2 3 4	adds4d	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) +s ( 𝐶 +s 𝐷 ) ) = ( ( 𝐴 +s 𝐶 ) +s ( 𝐵 +s 𝐷 ) ) )
6	2 4	addscomd	⊢ ( 𝜑 → ( 𝐵 +s 𝐷 ) = ( 𝐷 +s 𝐵 ) )
7	6	oveq2d	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐶 ) +s ( 𝐵 +s 𝐷 ) ) = ( ( 𝐴 +s 𝐶 ) +s ( 𝐷 +s 𝐵 ) ) )
8	5 7	eqtrd	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) +s ( 𝐶 +s 𝐷 ) ) = ( ( 𝐴 +s 𝐶 ) +s ( 𝐷 +s 𝐵 ) ) )