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	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	sn-addgt0d.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	sn-addgt0d.1	⊢ ( 𝜑 → 0 < 𝐴 )
	sn-addgt0d.2	⊢ ( 𝜑 → 0 < 𝐵 )
Assertion	sn-addgt0d	⊢ ( 𝜑 → 0 < ( 𝐴 + 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sn-addgt0d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		sn-addgt0d.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		sn-addgt0d.1	⊢ ( 𝜑 → 0 < 𝐴 )
4		sn-addgt0d.2	⊢ ( 𝜑 → 0 < 𝐵 )
5		0red	⊢ ( 𝜑 → 0 ∈ ℝ )
6	1 2	readdcld	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ∈ ℝ )
7		sn-ltaddpos	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 < 𝐵 ↔ 𝐴 < ( 𝐴 + 𝐵 ) ) )
8	2 1 7	syl2anc	⊢ ( 𝜑 → ( 0 < 𝐵 ↔ 𝐴 < ( 𝐴 + 𝐵 ) ) )
9	4 8	mpbid	⊢ ( 𝜑 → 𝐴 < ( 𝐴 + 𝐵 ) )
10	5 1 6 3 9	lttrd	⊢ ( 𝜑 → 0 < ( 𝐴 + 𝐵 ) )