Theorem sn-addlt0d

Description: The sum of negative numbers is negative. (Contributed by SN, 25-Jan-2025)

	Ref	Expression
Hypotheses	sn-addlt0d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	sn-addlt0d.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	sn-addlt0d.1	⊢ ( 𝜑 → 𝐴 < 0 )
	sn-addlt0d.2	⊢ ( 𝜑 → 𝐵 < 0 )
Assertion	sn-addlt0d	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) < 0 )

Proof

Step	Hyp	Ref	Expression
1		sn-addlt0d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		sn-addlt0d.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		sn-addlt0d.1	⊢ ( 𝜑 → 𝐴 < 0 )
4		sn-addlt0d.2	⊢ ( 𝜑 → 𝐵 < 0 )
5	1 2	readdcld	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ∈ ℝ )
6		0red	⊢ ( 𝜑 → 0 ∈ ℝ )
7		sn-ltaddneg	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 𝐵 < 0 ↔ ( 𝐴 + 𝐵 ) < 𝐴 ) )
8	2 1 7	syl2anc	⊢ ( 𝜑 → ( 𝐵 < 0 ↔ ( 𝐴 + 𝐵 ) < 𝐴 ) )
9	4 8	mpbid	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) < 𝐴 )
10	5 1 6 9 3	lttrd	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) < 0 )