Definition df-esum 34538

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 34537	. 2 class Σ*𝑘 ∈ 𝐴𝐵
5		cxrs 17586	. . . . 5 class ℝ*𝑠
6		cc0 11124	. . . . . 6 class 0
7		cpnf 11264	. . . . . 6 class +∞
8		cicc 13401	. . . . . 6 class [,]
9	6, 7, 8	co 7413	. . . . 5 class (0[,]+∞)
10		cress 17322	. . . . 5 class ↾s
11	5, 9, 10	co 7413	. . . 4 class (ℝ*𝑠 ↾s (0[,]+∞))
12	3, 1, 2	cmpt 5186	. . . 4 class (𝑘 ∈ 𝐴 ↦ 𝐵)
13		ctsu 24352	. . . 4 class tsums
14	11, 12, 13	co 7413	. . 3 class ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
15	14	cuni 4867	. 2 class ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
16	4, 15	wceq 1570	1 wff Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))

This definition is used by:  esumex  34539  esumcl  34540  esumeq12dvaf  34541  esumeq2  34546  nfesum1  34550  nfesum2  34551  cbvesum  34552  cbvesumv  34553  esumid  34554  esumval  34556  esumf1o  34560  esumsnf  34574  esumss  34582  esumpfinval  34585  esumpfinvalf  34586