Metamath Proof Explorer


Theorem sge0ge0

Description: The sum of nonnegative extended reals is nonnegative. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses sge0ge0.x ( 𝜑𝑋𝑉 )
sge0ge0.f ( 𝜑𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) )
Assertion sge0ge0 ( 𝜑 → 0 ≤ ( Σ^𝐹 ) )

Proof

Step Hyp Ref Expression
1 sge0ge0.x ( 𝜑𝑋𝑉 )
2 sge0ge0.f ( 𝜑𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) )
3 0xr 0 ∈ ℝ*
4 3 a1i ( 𝜑 → 0 ∈ ℝ* )
5 pnfxr +∞ ∈ ℝ*
6 5 a1i ( 𝜑 → +∞ ∈ ℝ* )
7 1 2 sge0cl ( 𝜑 → ( Σ^𝐹 ) ∈ ( 0 [,] +∞ ) )
8 iccgelb ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ( Σ^𝐹 ) ∈ ( 0 [,] +∞ ) ) → 0 ≤ ( Σ^𝐹 ) )
9 4 6 7 8 syl3anc ( 𝜑 → 0 ≤ ( Σ^𝐹 ) )