Definition df-esum 34359

Description: Define a short-hand for the possibly infinite sum over the extended nonnegative reals. Σ^* is relying on the properties of the tsums, developed by Mario Carneiro. (Contributed by Thierry Arnoux, 21-Sep-2016.)

Assertion

Ref	Expression
df-esum	⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))

Detailed syntax breakdown of Definition df-esum

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3		vk	. . 3 setvar 𝑘
4	1, 2, 3	cesum 34358	. 2 class Σ*𝑘 ∈ 𝐴𝐵
5		cxrs 17550	. . . . 5 class ℝ*𝑠
6		cc0 11096	. . . . . 6 class 0
7		cpnf 11236	. . . . . 6 class +∞
8		cicc 13371	. . . . . 6 class [,]
9	6, 7, 8	co 7408	. . . . 5 class (0[,]+∞)
10		cress 17286	. . . . 5 class ↾s
11	5, 9, 10	co 7408	. . . 4 class (ℝ*𝑠 ↾s (0[,]+∞))
12	3, 1, 2	cmpt 5193	. . . 4 class (𝑘 ∈ 𝐴 ↦ 𝐵)
13		ctsu 24248	. . . 4 class tsums
14	11, 12, 13	co 7408	. . 3 class ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
15	14	cuni 4873	. 2 class ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
16	4, 15	wceq 1567	1 wff Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))

This definition is referenced by:  esumex  34360  esumcl  34361  esumeq12dvaf  34362  esumeq2  34367  nfesum1  34371  nfesum2  34372  cbvesum  34373  cbvesumv  34374  esumid  34375  esumval  34377  esumf1o  34381  esumsnf  34395  esumss  34403  esumpfinval  34406  esumpfinvalf  34407