Definition df-esum 33992

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 33991	. 2 class Σ*𝑘 ∈ 𝐴𝐵
5		cxrs 17560	. . . . 5 class ℝ*𝑠
6		cc0 11184	. . . . . 6 class 0
7		cpnf 11321	. . . . . 6 class +∞
8		cicc 13410	. . . . . 6 class [,]
9	6, 7, 8	co 7448	. . . . 5 class (0[,]+∞)
10		cress 17287	. . . . 5 class ↾s
11	5, 9, 10	co 7448	. . . 4 class (ℝ*𝑠 ↾s (0[,]+∞))
12	3, 1, 2	cmpt 5249	. . . 4 class (𝑘 ∈ 𝐴 ↦ 𝐵)
13		ctsu 24155	. . . 4 class tsums
14	11, 12, 13	co 7448	. . 3 class ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
15	14	cuni 4931	. 2 class ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
16	4, 15	wceq 1537	1 wff Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))

This definition is referenced by:  esumex  33993  esumcl  33994  esumeq12dvaf  33995  esumeq2  34000  nfesum1  34004  nfesum2  34005  cbvesum  34006  cbvesumv  34007  esumid  34008  esumval  34010  esumf1o  34014  esumsnf  34028  esumss  34036  esumpfinval  34039  esumpfinvalf  34040