Theorem divsassd

Description: An associative law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025)

	Ref	Expression
Hypotheses	divsassd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	divsassd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	divsassd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
	divsassd.4	⊢ ( 𝜑 → 𝐶 ≠ 0s )
Assertion	divsassd	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐵 ) /su 𝐶 ) = ( 𝐴 ·s ( 𝐵 /su 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		divsassd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		divsassd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		divsassd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		divsassd.4	⊢ ( 𝜑 → 𝐶 ≠ 0s )
5	3 4	recsexd	⊢ ( 𝜑 → ∃ 𝑥 ∈ No ( 𝐶 ·s 𝑥 ) = 1s )
6	1 2 3 4 5	divsasswd	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐵 ) /su 𝐶 ) = ( 𝐴 ·s ( 𝐵 /su 𝐶 ) ) )