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Definition df-esum 29223
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 29222 . 2 class Σ*𝑘𝐴𝐵
5 cxrs 15929 . . . . 5 class *𝑠
6 cc0 9792 . . . . . 6 class 0
7 cpnf 9927 . . . . . 6 class +∞
8 cicc 12005 . . . . . 6 class [,]
96, 7, 8co 6527 . . . . 5 class (0[,]+∞)
10 cress 15642 . . . . 5 class s
115, 9, 10co 6527 . . . 4 class (ℝ*𝑠s (0[,]+∞))
123, 1, 2cmpt 4637 . . . 4 class (𝑘𝐴𝐵)
13 ctsu 21681 . . . 4 class tsums
1411, 12, 13co 6527 . . 3 class ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
1514cuni 4366 . 2 class ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
164, 15wceq 1474 1 wff Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
Colors of variables: wff setvar class
This definition is referenced by:  esumex  29224  esumcl  29225  esumeq12dvaf  29226  esumeq2  29231  nfesum1  29235  nfesum2  29236  cbvesum  29237  esumid  29239  esumval  29241  esumf1o  29245  esumsnf  29259  esumss  29267  esumpfinval  29270  esumpfinvalf  29271
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