Metamath Proof Explorer


Theorem sge0pnfmpt

Description: If a term in the sum of nonnegative extended reals is +oo , then the value of the sum is +oo . (Contributed by Glauco Siliprandi, 3-Mar-2021)

Ref Expression
Hypotheses sge0pnfmpt.k 𝑘 𝜑
sge0pnfmpt.a ( 𝜑𝐴𝑉 )
sge0pnfmpt.b ( ( 𝜑𝑘𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) )
sge0pnfmpt.p ( 𝜑 → ∃ 𝑘𝐴 𝐵 = +∞ )
Assertion sge0pnfmpt ( 𝜑 → ( Σ^ ‘ ( 𝑘𝐴𝐵 ) ) = +∞ )

Proof

Step Hyp Ref Expression
1 sge0pnfmpt.k 𝑘 𝜑
2 sge0pnfmpt.a ( 𝜑𝐴𝑉 )
3 sge0pnfmpt.b ( ( 𝜑𝑘𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) )
4 sge0pnfmpt.p ( 𝜑 → ∃ 𝑘𝐴 𝐵 = +∞ )
5 eqid ( 𝑘𝐴𝐵 ) = ( 𝑘𝐴𝐵 )
6 1 3 5 fmptdf ( 𝜑 → ( 𝑘𝐴𝐵 ) : 𝐴 ⟶ ( 0 [,] +∞ ) )
7 eqcom ( 𝐵 = +∞ ↔ +∞ = 𝐵 )
8 7 rexbii ( ∃ 𝑘𝐴 𝐵 = +∞ ↔ ∃ 𝑘𝐴 +∞ = 𝐵 )
9 4 8 sylib ( 𝜑 → ∃ 𝑘𝐴 +∞ = 𝐵 )
10 pnfex +∞ ∈ V
11 10 a1i ( 𝜑 → +∞ ∈ V )
12 5 9 11 elrnmptd ( 𝜑 → +∞ ∈ ran ( 𝑘𝐴𝐵 ) )
13 2 6 12 sge0pnfval ( 𝜑 → ( Σ^ ‘ ( 𝑘𝐴𝐵 ) ) = +∞ )