Definition df-liminf 43300

Description: Define the inferior limit of a sequence of extended real numbers. (Contributed by GS, 2-Jan-2022.)

Assertion

Ref	Expression
df-liminf	⊢ lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))

Distinct variable group:   𝑥,𝑘

Detailed syntax breakdown of Definition df-liminf

Step	Hyp	Ref	Expression
1		clsi 43299	. 2 class lim inf
2		vx	. . 3 setvar 𝑥
3		cvv 3433	. . 3 class V
4		vk	. . . . . 6 setvar 𝑘
5		cr 10879	. . . . . 6 class ℝ
6	2	cv 1538	. . . . . . . . 9 class 𝑥
7	4	cv 1538	. . . . . . . . . 10 class 𝑘
8		cpnf 11015	. . . . . . . . . 10 class +∞
9		cico 13090	. . . . . . . . . 10 class [,)
10	7, 8, 9	co 7284	. . . . . . . . 9 class (𝑘[,)+∞)
11	6, 10	cima 5593	. . . . . . . 8 class (𝑥 “ (𝑘[,)+∞))
12		cxr 11017	. . . . . . . 8 class ℝ*
13	11, 12	cin 3887	. . . . . . 7 class ((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*)
14		clt 11018	. . . . . . 7 class <
15	13, 12, 14	cinf 9209	. . . . . 6 class inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )
16	4, 5, 15	cmpt 5158	. . . . 5 class (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
17	16	crn 5591	. . . 4 class ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
18	17, 12, 14	csup 9208	. . 3 class sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )
19	2, 3, 18	cmpt 5158	. 2 class (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
20	1, 19	wceq 1539	1 wff lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))

This definition is referenced by:  liminfval  43307