Definition df-ana 26692

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 26690	. 2 class Ana
2		vs	. . 3 setvar 𝑠
3		cr 11206	. . . 4 class ℝ
4		cc 11205	. . . 4 class ℂ
5	3, 4	cpr 4586	. . 3 class {ℝ, ℂ}
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1568	. . . . . 6 class 𝑥
8		vf	. . . . . . . . . 10 setvar 𝑓
9	8	cv 1568	. . . . . . . . 9 class 𝑓
10		cpnf 11347	. . . . . . . . . 10 class +∞
11	2	cv 1568	. . . . . . . . . . 11 class 𝑠
12		ctayl 26689	. . . . . . . . . . 11 class Tayl
13	11, 9, 12	co 7424	. . . . . . . . . 10 class (𝑠 Tayl 𝑓)
14	10, 7, 13	co 7424	. . . . . . . . 9 class (+∞(𝑠 Tayl 𝑓)𝑥)
15	9, 14	cin 4088	. . . . . . . 8 class (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥))
16	15	cdm 5651	. . . . . . 7 class dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥))
17		ccnfld 21687	. . . . . . . . . 10 class ℂfld
18		ctopn 17601	. . . . . . . . . 10 class TopOpen
19	17, 18	cfv 6543	. . . . . . . . 9 class (TopOpen‘ℂfld)
20		crest 17600	. . . . . . . . 9 class ↾t
21	19, 11, 20	co 7424	. . . . . . . 8 class ((TopOpen‘ℂfld) ↾t 𝑠)
22		cnt 23344	. . . . . . . 8 class int
23	21, 22	cfv 6543	. . . . . . 7 class (int‘((TopOpen‘ℂfld) ↾t 𝑠))
24	16, 23	cfv 6543	. . . . . 6 class ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))
25	7, 24	wcel 2144	. . . . 5 wff 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))
26	9	cdm 5651	. . . . 5 class dom 𝑓
27	25, 6, 26	wral 3076	. . . 4 wff ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))
28		cpm 8856	. . . . 5 class ↑pm
29	4, 11, 28	co 7424	. . . 4 class (ℂ ↑pm 𝑠)
30	27, 8, 29	crab 3412	. . 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 5185	. 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 1569	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)