Description: The sum of nonnegative extended reals is nonnegative. (Contributed by Glauco Siliprandi, 17-Aug-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | sge0ge0.x | ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) | |
sge0ge0.f | ⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) | ||
Assertion | sge0ge0 | ⊢ ( 𝜑 → 0 ≤ ( Σ^ ‘ 𝐹 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0ge0.x | ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) | |
2 | sge0ge0.f | ⊢ ( 𝜑 → 𝐹 : 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 ) ) |