Theorem sn-addgt0d

Description: The sum of positive numbers is positive. Proof of addgt0d without ax-mulcom . (Contributed by SN, 25-Jan-2025)

	Ref	Expression
Hypotheses	sn-addgt0d.a	$$⊢\phi \to A\in ℝ$$
	sn-addgt0d.b	$$⊢\phi \to B\in ℝ$$
	sn-addgt0d.1	$$⊢\phi \to 0<A$$
	sn-addgt0d.2	$$⊢\phi \to 0<B$$
Assertion	sn-addgt0d	$$⊢\phi \to 0<A+B$$

Proof

Step	Hyp	Ref	Expression
1		sn-addgt0d.a	$$⊢\phi \to A\in ℝ$$
2		sn-addgt0d.b	$$⊢\phi \to B\in ℝ$$
3		sn-addgt0d.1	$$⊢\phi \to 0<A$$
4		sn-addgt0d.2	$$⊢\phi \to 0<B$$
5		0red	$$⊢\phi \to 0\in ℝ$$
6	1 2	readdcld	$$⊢\phi \to A+B\in ℝ$$
7		sn-ltaddpos	$$⊢\left(B\in ℝ\wedge A\in ℝ\right)\to \left(0<B\leftrightarrow A<A+B\right)$$
8	2 1 7	syl2anc	$$⊢\phi \to \left(0<B\leftrightarrow A<A+B\right)$$
9	4 8	mpbid	$$⊢\phi \to A<A+B$$
10	5 1 6 3 9	lttrd	$$⊢\phi \to 0<A+B$$