Metamath Proof Explorer


Theorem sge0pnffigtmpt

Description: If the generalized sum of nonnegative reals is +oo , then any real number can be dominated by finite subsums. (Contributed by Glauco Siliprandi, 21-Nov-2020)

Ref Expression
Hypotheses sge0pnffigtmpt.k k φ
sge0pnffigtmpt.a φ A V
sge0pnffigtmpt.b φ k A B 0 +∞
sge0pnffigtmpt.p φ sum^ k A B = +∞
sge0pnffigtmpt.y φ Y
Assertion sge0pnffigtmpt φ x 𝒫 A Fin Y < sum^ k x B

Proof

Step Hyp Ref Expression
1 sge0pnffigtmpt.k k φ
2 sge0pnffigtmpt.a φ A V
3 sge0pnffigtmpt.b φ k A B 0 +∞
4 sge0pnffigtmpt.p φ sum^ k A B = +∞
5 sge0pnffigtmpt.y φ Y
6 eqid k A B = k A B
7 1 3 6 fmptdf φ k A B : A 0 +∞
8 2 7 4 5 sge0pnffigt φ x 𝒫 A Fin Y < sum^ k A B x
9 simpr x 𝒫 A Fin Y < sum^ k A B x Y < sum^ k A B x
10 elpwinss x 𝒫 A Fin x A
11 10 adantr x 𝒫 A Fin Y < sum^ k A B x x A
12 11 resmptd x 𝒫 A Fin Y < sum^ k A B x k A B x = k x B
13 12 fveq2d x 𝒫 A Fin Y < sum^ k A B x sum^ k A B x = sum^ k x B
14 9 13 breqtrd x 𝒫 A Fin Y < sum^ k A B x Y < sum^ k x B
15 14 ex x 𝒫 A Fin Y < sum^ k A B x Y < sum^ k x B
16 15 adantl φ x 𝒫 A Fin Y < sum^ k A B x Y < sum^ k x B
17 16 reximdva φ x 𝒫 A Fin Y < sum^ k A B x x 𝒫 A Fin Y < sum^ k x B
18 8 17 mpd φ x 𝒫 A Fin Y < sum^ k x B