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	$$⊢\phi \to A\in \mathrm{No}$$
	mulsgt0d.2	$$⊢\phi \to B\in \mathrm{No}$$
	mulsgt0d.3	$$⊢\phi \to 0_s{<}_{s}A$$
	mulsgt0d.4	$$⊢\phi \to 0_s{<}_{s}B$$
Assertion	mulsgt0d	$$⊢\phi \to 0_s{<}_s\left(A{\cdot }_{s}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		mulsgt0d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		mulsgt0d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		mulsgt0d.3	$$⊢\phi \to 0_s{<}_{s}A$$
4		mulsgt0d.4	$$⊢\phi \to 0_s{<}_{s}B$$
5		mulsgt0	$$⊢\left(\left(A\in \mathrm{No}\wedge 0_s{<}_{s}A\right)\wedge \left(B\in \mathrm{No}\wedge 0_s{<}_{s}B\right)\right)\to 0_s{<}_s\left(A{\cdot }_{s}B\right)$$
6	1 3 2 4 5	syl22anc	$$⊢\phi \to 0_s{<}_s\left(A{\cdot }_{s}B\right)$$