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Definition df-esum 30218
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 30217 . 2 class Σ*𝑘𝐴𝐵
5 cxrs 16207 . . . . 5 class *𝑠
6 cc0 9974 . . . . . 6 class 0
7 cpnf 10109 . . . . . 6 class +∞
8 cicc 12216 . . . . . 6 class [,]
96, 7, 8co 6690 . . . . 5 class (0[,]+∞)
10 cress 15905 . . . . 5 class s
115, 9, 10co 6690 . . . 4 class (ℝ*𝑠s (0[,]+∞))
123, 1, 2cmpt 4762 . . . 4 class (𝑘𝐴𝐵)
13 ctsu 21976 . . . 4 class tsums
1411, 12, 13co 6690 . . 3 class ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
1514cuni 4468 . 2 class ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
164, 15wceq 1523 1 wff Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
Colors of variables: wff setvar class
This definition is referenced by:  esumex  30219  esumcl  30220  esumeq12dvaf  30221  esumeq2  30226  nfesum1  30230  nfesum2  30231  cbvesum  30232  esumid  30234  esumval  30236  esumf1o  30240  esumsnf  30254  esumss  30262  esumpfinval  30265  esumpfinvalf  30266
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