Theorem divsmuld

Description: Relationship between surreal division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025)

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

Proof

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