Theorem sltdivmuld

Description: Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		sltdivmuld.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		sltdivmuld.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		sltdivmuld.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		sltdivmuld.4	⊢ ( 𝜑 → 0s <s 𝐶 )
5	4	sgt0ne0d	⊢ ( 𝜑 → 𝐶 ≠ 0s )
6	3 5	recsexd	⊢ ( 𝜑 → ∃ 𝑥 ∈ No ( 𝐶 ·s 𝑥 ) = 1s )
7	1 2 3 4 6	sltdivmulwd	⊢ ( 𝜑 → ( ( 𝐴 /su 𝐶 ) <s 𝐵 ↔ 𝐴 <s ( 𝐶 ·s 𝐵 ) ) )