Theorem addsassd

Description: Surreal addition is associative. Part of theorem 3 of Conway p. 17. (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	addsassd	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) +s 𝐶 ) = ( 𝐴 +s ( 𝐵 +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		addsass	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 +s 𝐵 ) +s 𝐶 ) = ( 𝐴 +s ( 𝐵 +s 𝐶 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) +s 𝐶 ) = ( 𝐴 +s ( 𝐵 +s 𝐶 ) ) )