Definition df-esum 34707

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 34706	. 2 class Σ*𝑘 ∈ 𝐴𝐵
5		cxrs 17678	. . . . 5 class ℝ*𝑠
6		cc0 11204	. . . . . 6 class 0
7		cpnf 11344	. . . . . 6 class +∞
8		cicc 13483	. . . . . 6 class [,]
9	6, 7, 8	co 7421	. . . . 5 class (0[,]+∞)
10		cress 17414	. . . . 5 class ↾s
11	5, 9, 10	co 7421	. . . 4 class (ℝ*𝑠 ↾s (0[,]+∞))
12	3, 1, 2	cmpt 5185	. . . 4 class (𝑘 ∈ 𝐴 ↦ 𝐵)
13		ctsu 24451	. . . 4 class tsums
14	11, 12, 13	co 7421	. . 3 class ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
15	14	cuni 4866	. 2 class ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
16	4, 15	wceq 1569	1 wff Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))

This definition is used by:  esumex  34708  esumcl  34709  esumeq12dvaf  34710  esumeq2  34715  nfesum1  34719  nfesum2  34720  cbvesum  34721  cbvesumv  34722  esumid  34723  esumval  34725  esumf1o  34729  esumsnf  34743  esumss  34751  esumpfinval  34754  esumpfinvalf  34755