Metamath Proof Explorer


Theorem sge0val

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

Ref Expression
Assertion sge0val X V F : X 0 +∞ sum^ F = if +∞ ran F +∞ sup ran y 𝒫 X Fin w y F w * <

Proof

Step Hyp Ref Expression
1 df-sumge0 sum^ = x V if +∞ ran x +∞ sup ran y 𝒫 dom x Fin w y x w * <
2 1 a1i X V F : X 0 +∞ sum^ = x V if +∞ ran x +∞ sup ran y 𝒫 dom x Fin w y x w * <
3 rneq x = F ran x = ran F
4 3 eleq2d x = F +∞ ran x +∞ ran F
5 4 adantl X V F : X 0 +∞ x = F +∞ ran x +∞ ran F
6 dmeq x = F dom x = dom F
7 6 adantl F : X 0 +∞ x = F dom x = dom F
8 fdm F : X 0 +∞ dom F = X
9 8 adantr F : X 0 +∞ x = F dom F = X
10 7 9 eqtrd F : X 0 +∞ x = F dom x = X
11 10 pweqd F : X 0 +∞ x = F 𝒫 dom x = 𝒫 X
12 11 ineq1d F : X 0 +∞ x = F 𝒫 dom x Fin = 𝒫 X Fin
13 12 mpteq1d F : X 0 +∞ x = F y 𝒫 dom x Fin w y x w = y 𝒫 X Fin w y x w
14 13 adantll X V F : X 0 +∞ x = F y 𝒫 dom x Fin w y x w = y 𝒫 X Fin w y x w
15 fveq1 x = F x w = F w
16 15 sumeq2sdv x = F w y x w = w y F w
17 16 mpteq2dv x = F y 𝒫 X Fin w y x w = y 𝒫 X Fin w y F w
18 17 adantl X V F : X 0 +∞ x = F y 𝒫 X Fin w y x w = y 𝒫 X Fin w y F w
19 14 18 eqtrd X V F : X 0 +∞ x = F y 𝒫 dom x Fin w y x w = y 𝒫 X Fin w y F w
20 19 rneqd X V F : X 0 +∞ x = F ran y 𝒫 dom x Fin w y x w = ran y 𝒫 X Fin w y F w
21 20 supeq1d X V F : X 0 +∞ x = F sup ran y 𝒫 dom x Fin w y x w * < = sup ran y 𝒫 X Fin w y F w * <
22 5 21 ifbieq2d X V F : X 0 +∞ x = F if +∞ ran x +∞ sup ran y 𝒫 dom x Fin w y x w * < = if +∞ ran F +∞ sup ran y 𝒫 X Fin w y F w * <
23 simpr X V F : X 0 +∞ F : X 0 +∞
24 simpl X V F : X 0 +∞ X V
25 23 24 fexd X V F : X 0 +∞ F V
26 pnfxr +∞ *
27 26 a1i X V F : X 0 +∞ +∞ *
28 xrltso < Or *
29 28 supex sup ran y 𝒫 X Fin w y F w * < V
30 29 a1i X V F : X 0 +∞ sup ran y 𝒫 X Fin w y F w * < V
31 27 30 ifexd X V F : X 0 +∞ if +∞ ran F +∞ sup ran y 𝒫 X Fin w y F w * < V
32 2 22 25 31 fvmptd X V F : X 0 +∞ sum^ F = if +∞ ran F +∞ sup ran y 𝒫 X Fin w y F w * <