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	|- ( ph -> A e. No )
	mulsgt0d.2	|- ( ph -> B e. No )
	mulsgt0d.3	|- ( ph -> 0s
	mulsgt0d.4	|- ( ph -> 0s
Assertion	mulsgt0d	|- ( ph -> 0s

Proof

Step	Hyp	Ref	Expression
1		mulsgt0d.1	|- ( ph -> A e. No )
2		mulsgt0d.2	|- ( ph -> B e. No )
3		mulsgt0d.3	|- ( ph -> 0s
4		mulsgt0d.4	|- ( ph -> 0s
5		mulsgt0	|- ( ( ( A e. No /\ 0s 0s
6	1 3 2 4 5	syl22anc	|- ( ph -> 0s