Theorem adds4d

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	adds4d	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) +s ( 𝐶 +s 𝐷 ) ) = ( ( 𝐴 +s 𝐶 ) +s ( 𝐵 +s 𝐷 ) ) )

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	adds32d	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) +s 𝐶 ) = ( ( 𝐴 +s 𝐶 ) +s 𝐵 ) )
6	5	oveq1d	⊢ ( 𝜑 → ( ( ( 𝐴 +s 𝐵 ) +s 𝐶 ) +s 𝐷 ) = ( ( ( 𝐴 +s 𝐶 ) +s 𝐵 ) +s 𝐷 ) )
7	1 2	addscld	⊢ ( 𝜑 → ( 𝐴 +s 𝐵 ) ∈ No )
8	7 3 4	addsassd	⊢ ( 𝜑 → ( ( ( 𝐴 +s 𝐵 ) +s 𝐶 ) +s 𝐷 ) = ( ( 𝐴 +s 𝐵 ) +s ( 𝐶 +s 𝐷 ) ) )
9	1 3	addscld	⊢ ( 𝜑 → ( 𝐴 +s 𝐶 ) ∈ No )
10	9 2 4	addsassd	⊢ ( 𝜑 → ( ( ( 𝐴 +s 𝐶 ) +s 𝐵 ) +s 𝐷 ) = ( ( 𝐴 +s 𝐶 ) +s ( 𝐵 +s 𝐷 ) ) )
11	6 8 10	3eqtr3d	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) +s ( 𝐶 +s 𝐷 ) ) = ( ( 𝐴 +s 𝐶 ) +s ( 𝐵 +s 𝐷 ) ) )