Theorem ltmuldivsd

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

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

Proof

Step	Hyp	Ref	Expression
1		ltdivmulsd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		ltdivmulsd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		ltdivmulsd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		ltdivmulsd.4	⊢ ( 𝜑 → 0s <s 𝐶 )
5	4	gt0ne0sd	⊢ ( 𝜑 → 𝐶 ≠ 0s )
6	3 5	recsexd	⊢ ( 𝜑 → ∃ 𝑥 ∈ No ( 𝐶 ·s 𝑥 ) = 1s )
7	1 2 3 4 6	ltmuldivswd	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐶 ) <s 𝐵 ↔ 𝐴 <s ( 𝐵 /su 𝐶 ) ) )