Theorem ltmuls2d

Description: Multiplication of both sides of surreal less-than by a positive number. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Hypotheses	ltmuls12d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	ltmuls12d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	ltmuls12d.3	⊢ ( 𝜑 → 𝐶 ∈ No )
	ltmuls12d.4	⊢ ( 𝜑 → 0s <s 𝐶 )
Assertion	ltmuls2d	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐶 ·s 𝐴 ) <s ( 𝐶 ·s 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltmuls12d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		ltmuls12d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		ltmuls12d.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		ltmuls12d.4	⊢ ( 𝜑 → 0s <s 𝐶 )
5		ltmuls2	⊢ ( ( ( 𝐶 ∈ No ∧ 0s <s 𝐶 ) ∧ 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ( 𝐶 ·s 𝐴 ) <s ( 𝐶 ·s 𝐵 ) ) )
6	3 4 1 2 5	syl211anc	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐶 ·s 𝐴 ) <s ( 𝐶 ·s 𝐵 ) ) )