Definition df-ana 26740

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 26738	. 2 class Ana
2		vs	. . 3 setvar 𝑠
3		cr 11238	. . . 4 class ℝ
4		cc 11237	. . . 4 class ℂ
5	3, 4	cpr 4590	. . 3 class {ℝ, ℂ}
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1569	. . . . . 6 class 𝑥
8		vf	. . . . . . . . . 10 setvar 𝑓
9	8	cv 1569	. . . . . . . . 9 class 𝑓
10		cpnf 11379	. . . . . . . . . 10 class +∞
11	2	cv 1569	. . . . . . . . . . 11 class 𝑠
12		ctayl 26737	. . . . . . . . . . 11 class Tayl
13	11, 9, 12	co 7451	. . . . . . . . . 10 class (𝑠 Tayl 𝑓)
14	10, 7, 13	co 7451	. . . . . . . . 9 class (+∞(𝑠 Tayl 𝑓)𝑥)
15	9, 14	cin 4092	. . . . . . . 8 class (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥))
16	15	cdm 5658	. . . . . . 7 class dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥))
17		ccnfld 21729	. . . . . . . . . 10 class ℂfld
18		ctopn 17642	. . . . . . . . . 10 class TopOpen
19	17, 18	cfv 6566	. . . . . . . . 9 class (TopOpen‘ℂfld)
20		crest 17641	. . . . . . . . 9 class ↾t
21	19, 11, 20	co 7451	. . . . . . . 8 class ((TopOpen‘ℂfld) ↾t 𝑠)
22		cnt 23390	. . . . . . . 8 class int
23	21, 22	cfv 6566	. . . . . . 7 class (int‘((TopOpen‘ℂfld) ↾t 𝑠))
24	16, 23	cfv 6566	. . . . . 6 class ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))
25	7, 24	wcel 2147	. . . . 5 wff 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))
26	9	cdm 5658	. . . . 5 class dom 𝑓
27	25, 6, 26	wral 3080	. . . 4 wff ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))
28		cpm 8886	. . . . 5 class ↑pm
29	4, 11, 28	co 7451	. . . 4 class (ℂ ↑pm 𝑠)
30	27, 8, 29	crab 3416	. . 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 5192	. 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 1570	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 used by: (None)