Definition df-ana 26239

Description: Define the set of analytic functions, which are functions such that the Taylor series of the function at each point converges to the function in some neighborhood of the point. (Contributed by Mario Carneiro, 31-Dec-2016.)

Assertion

Ref	Expression
df-ana	⊢ Ana = (𝑠 ∈ {ℝ, ℂ} ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))})

Distinct variable group:   𝑓,𝑠,𝑥

Detailed syntax breakdown of Definition df-ana

Step	Hyp	Ref	Expression
1		cana 26237	. 2 class Ana
2		vs	. . 3 setvar 𝑠
3		cr 11043	. . . 4 class ℝ
4		cc 11042	. . . 4 class ℂ
5	3, 4	cpr 4587	. . 3 class {ℝ, ℂ}
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1539	. . . . . 6 class 𝑥
8		vf	. . . . . . . . . 10 setvar 𝑓
9	8	cv 1539	. . . . . . . . 9 class 𝑓
10		cpnf 11181	. . . . . . . . . 10 class +∞
11	2	cv 1539	. . . . . . . . . . 11 class 𝑠
12		ctayl 26236	. . . . . . . . . . 11 class Tayl
13	11, 9, 12	co 7369	. . . . . . . . . 10 class (𝑠 Tayl 𝑓)
14	10, 7, 13	co 7369	. . . . . . . . 9 class (+∞(𝑠 Tayl 𝑓)𝑥)
15	9, 14	cin 3910	. . . . . . . 8 class (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥))
16	15	cdm 5631	. . . . . . 7 class dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥))
17		ccnfld 21240	. . . . . . . . . 10 class ℂfld
18		ctopn 17360	. . . . . . . . . 10 class TopOpen
19	17, 18	cfv 6499	. . . . . . . . 9 class (TopOpen‘ℂfld)
20		crest 17359	. . . . . . . . 9 class ↾t
21	19, 11, 20	co 7369	. . . . . . . 8 class ((TopOpen‘ℂfld) ↾t 𝑠)
22		cnt 22880	. . . . . . . 8 class int
23	21, 22	cfv 6499	. . . . . . 7 class (int‘((TopOpen‘ℂfld) ↾t 𝑠))
24	16, 23	cfv 6499	. . . . . 6 class ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))
25	7, 24	wcel 2109	. . . . 5 wff 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))
26	9	cdm 5631	. . . . 5 class dom 𝑓
27	25, 6, 26	wral 3044	. . . 4 wff ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))
28		cpm 8777	. . . . 5 class ↑pm
29	4, 11, 28	co 7369	. . . 4 class (ℂ ↑pm 𝑠)
30	27, 8, 29	crab 3402	. . 3 class {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))}
31	2, 5, 30	cmpt 5183	. 2 class (𝑠 ∈ {ℝ, ℂ} ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))})
32	1, 31	wceq 1540	1 wff Ana = (𝑠 ∈ {ℝ, ℂ} ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))})

This definition is referenced by: (None)