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Definition df-esum 31395
Description: Define a short-hand for the possibly infinite sum over the extended nonnegative reals. Σ* is relying on the properties of the tsums, developped by Mario Carneiro. (Contributed by Thierry Arnoux, 21-Sep-2016.)
Assertion
Ref Expression
df-esum Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))

Detailed syntax breakdown of Definition df-esum
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cB . . 3 class 𝐵
3 vk . . 3 setvar 𝑘
41, 2, 3cesum 31394 . 2 class Σ*𝑘𝐴𝐵
5 cxrs 16768 . . . . 5 class *𝑠
6 cc0 10530 . . . . . 6 class 0
7 cpnf 10665 . . . . . 6 class +∞
8 cicc 12733 . . . . . 6 class [,]
96, 7, 8co 7139 . . . . 5 class (0[,]+∞)
10 cress 16479 . . . . 5 class s
115, 9, 10co 7139 . . . 4 class (ℝ*𝑠s (0[,]+∞))
123, 1, 2cmpt 5113 . . . 4 class (𝑘𝐴𝐵)
13 ctsu 22734 . . . 4 class tsums
1411, 12, 13co 7139 . . 3 class ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
1514cuni 4803 . 2 class ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
164, 15wceq 1538 1 wff Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
Colors of variables: wff setvar class
This definition is referenced by:  esumex  31396  esumcl  31397  esumeq12dvaf  31398  esumeq2  31403  nfesum1  31407  nfesum2  31408  cbvesum  31409  esumid  31411  esumval  31413  esumf1o  31417  esumsnf  31431  esumss  31439  esumpfinval  31442  esumpfinvalf  31443
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