Theorem addsgt0d

Description: The sum of two positive surreals is positive. (Contributed by Scott Fenton, 15-Apr-2025)

	Ref	Expression
Hypotheses	addsgt0d.1	$$⊢\phi \to A\in \mathrm{No}$$
	addsgt0d.2	$$⊢\phi \to B\in \mathrm{No}$$
	addsgt0d.3	$$⊢\phi \to 0_s{<}_{s}A$$
	addsgt0d.4	$$⊢\phi \to 0_s{<}_{s}B$$
Assertion	addsgt0d	$$⊢\phi \to 0_s{<}_s\left(A{+}_{s}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		addsgt0d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		addsgt0d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		addsgt0d.3	$$⊢\phi \to 0_s{<}_{s}A$$
4		addsgt0d.4	$$⊢\phi \to 0_s{<}_{s}B$$
5		0no	$$⊢0_s\in \mathrm{No}$$
6		addsrid	$$⊢0_s\in \mathrm{No}\to 0_s{+}_s{0}_s=0_s$$
7	5 6	ax-mp	$$⊢0_s{+}_s{0}_s=0_s$$
8	5	a1i	$$⊢\phi \to 0_s\in \mathrm{No}$$
9	8 8 1 2 3 4	lt2addsd	$$⊢\phi \to \left(0_s{+}_s{0}_s\right){<}_s\left(A{+}_{s}B\right)$$
10	7 9	eqbrtrrid	$$⊢\phi \to 0_s{<}_s\left(A{+}_{s}B\right)$$