Definition df-ana 26683

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 26681	. 2 class Ana
2		vs	. . 3 setvar 𝑠
3		cr 11199	. . . 4 class ℝ
4		cc 11198	. . . 4 class ℂ
5	3, 4	cpr 4586	. . 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 11340	. . . . . . . . . 10 class +∞
11	2	cv 1569	. . . . . . . . . . 11 class 𝑠
12		ctayl 26680	. . . . . . . . . . 11 class Tayl
13	11, 9, 12	co 7420	. . . . . . . . . 10 class (𝑠 Tayl 𝑓)
14	10, 7, 13	co 7420	. . . . . . . . 9 class (+∞(𝑠 Tayl 𝑓)𝑥)
15	9, 14	cin 3898	. . . . . . . 8 class (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥))
16	15	cdm 5651	. . . . . . 7 class dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥))
17		ccnfld 21678	. . . . . . . . . 10 class ℂfld
18		ctopn 17592	. . . . . . . . . 10 class TopOpen
19	17, 18	cfv 6538	. . . . . . . . 9 class (TopOpen‘ℂfld)
20		crest 17591	. . . . . . . . 9 class ↾t
21	19, 11, 20	co 7420	. . . . . . . 8 class ((TopOpen‘ℂfld) ↾t 𝑠)
22		cnt 23335	. . . . . . . 8 class int
23	21, 22	cfv 6538	. . . . . . 7 class (int‘((TopOpen‘ℂfld) ↾t 𝑠))
24	16, 23	cfv 6538	. . . . . 6 class ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))
25	7, 24	wcel 2145	. . . . 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 3077	. . . 4 wff ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))
28		cpm 8848	. . . . 5 class ↑pm
29	4, 11, 28	co 7420	. . . 4 class (ℂ ↑pm 𝑠)
30	27, 8, 29	crab 3413	. . 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 5186	. 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)