Definition df-esum 33095

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*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 33094	. 2 class Σ*𝑘 ∈ 𝐴𝐵
5		cxrs 17448	. . . . 5 class ℝ*𝑠
6		cc0 11112	. . . . . 6 class 0
7		cpnf 11247	. . . . . 6 class +∞
8		cicc 13329	. . . . . 6 class [,]
9	6, 7, 8	co 7411	. . . . 5 class (0[,]+∞)
10		cress 17175	. . . . 5 class ↾s
11	5, 9, 10	co 7411	. . . 4 class (ℝ*𝑠 ↾s (0[,]+∞))
12	3, 1, 2	cmpt 5231	. . . 4 class (𝑘 ∈ 𝐴 ↦ 𝐵)
13		ctsu 23637	. . . 4 class tsums
14	11, 12, 13	co 7411	. . 3 class ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
15	14	cuni 4908	. 2 class ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
16	4, 15	wceq 1541	1 wff Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))

This definition is referenced by:  esumex  33096  esumcl  33097  esumeq12dvaf  33098  esumeq2  33103  nfesum1  33107  nfesum2  33108  cbvesum  33109  esumid  33111  esumval  33113  esumf1o  33117  esumsnf  33131  esumss  33139  esumpfinval  33142  esumpfinvalf  33143