Theorem mulsgt0d

Description: The product of two positive surreals is positive. Theorem 9 of Conway p. 20. (Contributed by Scott Fenton, 6-Mar-2025)

	Ref	Expression
Hypotheses	mulsgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	mulsgt0d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	mulsgt0d.3	⊢ ( 𝜑 → 0s <s 𝐴 )
	mulsgt0d.4	⊢ ( 𝜑 → 0s <s 𝐵 )
Assertion	mulsgt0d	⊢ ( 𝜑 → 0s <s ( 𝐴 ·s 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		mulsgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		mulsgt0d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		mulsgt0d.3	⊢ ( 𝜑 → 0s <s 𝐴 )
4		mulsgt0d.4	⊢ ( 𝜑 → 0s <s 𝐵 )
5		mulsgt0	⊢ ( ( ( 𝐴 ∈ No ∧ 0s <s 𝐴 ) ∧ ( 𝐵 ∈ No ∧ 0s <s 𝐵 ) ) → 0s <s ( 𝐴 ·s 𝐵 ) )
6	1 3 2 4 5	syl22anc	⊢ ( 𝜑 → 0s <s ( 𝐴 ·s 𝐵 ) )