Theorem psercn 26278

Description: An infinite series converges to a continuous function on the open disk of radius 𝑅, where 𝑅 is the radius of convergence of the series. (Contributed by Mario Carneiro, 4-Mar-2015.)

Hypotheses

Ref	Expression
pserf.g	⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))
pserf.f	⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗))
pserf.a	⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)
pserf.r	⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < )
psercn.s	⊢ 𝑆 = (◡abs “ (0[,)𝑅))
psercn.m	⊢ 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))

Assertion

Ref	Expression
psercn	⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

Distinct variable groups:   𝑗,𝑎,𝑛,𝑟,𝑥,𝑦,𝐴   𝑗,𝑀,𝑦   𝑗,𝐺,𝑟,𝑦   𝑆,𝑎,𝑗,𝑦   𝐹,𝑎   𝜑,𝑎,𝑗,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑛,𝑟)   𝑅(𝑥,𝑦,𝑗,𝑛,𝑟,𝑎)   𝑆(𝑥,𝑛,𝑟)   𝐹(𝑥,𝑦,𝑗,𝑛,𝑟)   𝐺(𝑥,𝑛,𝑎)   𝑀(𝑥,𝑛,𝑟,𝑎)

Proof of Theorem psercn

Dummy variables 𝑘 𝑠 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sumex 15641	. . . . . 6 ⊢ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗) ∈ V
2	1	rgenw 3064	. . . . 5 ⊢ ∀𝑦 ∈ 𝑆 Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗) ∈ V
3		pserf.f	. . . . . 6 ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗))
4	3	fnmpt 6690	. . . . 5 ⊢ (∀𝑦 ∈ 𝑆 Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗) ∈ V → 𝐹 Fn 𝑆)
5	2, 4	mp1i 13	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝑆)
6		psercn.s	. . . . . . . . . . 11 ⊢ 𝑆 = (◡abs “ (0[,)𝑅))
7		cnvimass 6080	. . . . . . . . . . . 12 ⊢ (◡abs “ (0[,)𝑅)) ⊆ dom abs
8		absf 15291	. . . . . . . . . . . . 13 ⊢ abs:ℂ⟶ℝ
9	8	fdmi 6729	. . . . . . . . . . . 12 ⊢ dom abs = ℂ
10	7, 9	sseqtri 4018	. . . . . . . . . . 11 ⊢ (◡abs “ (0[,)𝑅)) ⊆ ℂ
11	6, 10	eqsstri 4016	. . . . . . . . . 10 ⊢ 𝑆 ⊆ ℂ
12	11	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
13	12	sselda 3982	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ)
14		0cn 11213	. . . . . . . . . . 11 ⊢ 0 ∈ ℂ
15		eqid 2731	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (abs ∘ − )
16	15	cnmetdval 24607	. . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
17	14, 13, 16	sylancr 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
18		abssub 15280	. . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
19	14, 13, 18	sylancr 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
20	13	subid1d 11567	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 − 0) = 𝑎)
21	20	fveq2d 6895	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘(𝑎 − 0)) = (abs‘𝑎))
22	17, 19, 21	3eqtrd 2775	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) = (abs‘𝑎))
23		breq2 5152	. . . . . . . . . . 11 ⊢ ((((abs‘𝑎) + 𝑅) / 2) = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) → ((abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2) ↔ (abs‘𝑎) < if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))))
24		breq2 5152	. . . . . . . . . . 11 ⊢ (((abs‘𝑎) + 1) = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) → ((abs‘𝑎) < ((abs‘𝑎) + 1) ↔ (abs‘𝑎) < if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))))
25		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
26	25, 6	eleqtrdi 2842	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (◡abs “ (0[,)𝑅)))
27		ffn 6717	. . . . . . . . . . . . . . . . . 18 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
28		elpreima 7059	. . . . . . . . . . . . . . . . . 18 ⊢ (abs Fn ℂ → (𝑎 ∈ (◡abs “ (0[,)𝑅)) ↔ (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅))))
29	8, 27, 28	mp2b 10	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (◡abs “ (0[,)𝑅)) ↔ (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅)))
30	26, 29	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅)))
31	30	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ (0[,)𝑅))
32		0re 11223	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
33		iccssxr 13414	. . . . . . . . . . . . . . . . 17 ⊢ (0[,]+∞) ⊆ ℝ*
34		pserf.g	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))
35		pserf.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)
36		pserf.r	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < )
37	34, 35, 36	radcnvcl 26268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞))
38	37	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ (0[,]+∞))
39	33, 38	sselid 3980	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ*)
40		elico2 13395	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
41	32, 39, 40	sylancr 586	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
42	31, 41	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅))
43	42	simp3d 1143	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑅)
44	43	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < 𝑅)
45	13	abscld 15390	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ ℝ)
46		avglt1 12457	. . . . . . . . . . . . 13 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
47	45, 46	sylan 579	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
48	44, 47	mpbid 231	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2))
49	45	ltp1d 12151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 1))
50	49	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ ¬ 𝑅 ∈ ℝ) → (abs‘𝑎) < ((abs‘𝑎) + 1))
51	23, 24, 48, 50	ifbothda 4566	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)))
52		psercn.m	. . . . . . . . . 10 ⊢ 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))
53	51, 52	breqtrrdi 5190	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀)
54	22, 53	eqbrtrd 5170	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) < 𝑀)
55		cnxmet 24609	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
56	34, 3, 35, 36, 6, 52	psercnlem1 26277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅))
57	56	simp1d 1141	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ+)
58	57	rpxrd 13024	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ*)
59		elbl 24214	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ↔ (𝑎 ∈ ℂ ∧ (0(abs ∘ − )𝑎) < 𝑀)))
60	55, 14, 58, 59	mp3an12i 1464	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ↔ (𝑎 ∈ ℂ ∧ (0(abs ∘ − )𝑎) < 𝑀)))
61	13, 54, 60	mpbir2and 710	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀))
62	61	fvresd 6911	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) = (𝐹‘𝑎))
63	3	reseq1i 5977	. . . . . . . . . 10 ⊢ (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = ((𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) ↾ (0(ball‘(abs ∘ − ))𝑀))
64	34, 3, 35, 36, 6, 56	psercnlem2 26276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ (◡abs “ (0[,]𝑀)) ∧ (◡abs “ (0[,]𝑀)) ⊆ 𝑆))
65	64	simp2d 1142	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ (◡abs “ (0[,]𝑀)))
66	64	simp3d 1143	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (◡abs “ (0[,]𝑀)) ⊆ 𝑆)
67	65, 66	sstrd 3992	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆)
68	67	resmptd 6040	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) ↾ (0(ball‘(abs ∘ − ))𝑀)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)))
69	63, 68	eqtrid 2783	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)))
70		eqid 2731	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗))
71	35	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐴:ℕ0⟶ℂ)
72		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑦 → (𝐺‘𝑘) = (𝐺‘𝑦))
73	72	seqeq3d 13981	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑦 → seq0( + , (𝐺‘𝑘)) = seq0( + , (𝐺‘𝑦)))
74	73	fveq1d 6893	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑦 → (seq0( + , (𝐺‘𝑘))‘𝑠) = (seq0( + , (𝐺‘𝑦))‘𝑠))
75	74	cbvmptv 5261	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑦))‘𝑠))
76		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑖 → (seq0( + , (𝐺‘𝑦))‘𝑠) = (seq0( + , (𝐺‘𝑦))‘𝑖))
77	76	mpteq2dv 5250	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑖 → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑦))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
78	75, 77	eqtrid 2783	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑖 → (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
79	78	cbvmptv 5261	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℕ0 ↦ (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑘))‘𝑠))) = (𝑖 ∈ ℕ0 ↦ (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
80	57	rpred 13023	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ)
81	56	simp3d 1143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 < 𝑅)
82	34, 70, 71, 36, 79, 80, 81, 65	psercn2 26274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
83	69, 82	eqeltrd 2832	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
84		cncff 24733	. . . . . . . 8 ⊢ ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)):(0(ball‘(abs ∘ − ))𝑀)⟶ℂ)
85	83, 84	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)):(0(ball‘(abs ∘ − ))𝑀)⟶ℂ)
86	85, 61	ffvelcdmd 7087	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) ∈ ℂ)
87	62, 86	eqeltrrd 2833	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹‘𝑎) ∈ ℂ)
88	87	ralrimiva 3145	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 (𝐹‘𝑎) ∈ ℂ)
89		ffnfv 7120	. . . 4 ⊢ (𝐹:𝑆⟶ℂ ↔ (𝐹 Fn 𝑆 ∧ ∀𝑎 ∈ 𝑆 (𝐹‘𝑎) ∈ ℂ))
90	5, 88, 89	sylanbrc 582	. . 3 ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
91	67, 11	sstrdi 3994	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ)
92		ssid 4004	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
93		eqid 2731	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
94		eqid 2731	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
95	93	cnfldtopon 24619	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
96	95	toponrestid 22743	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
97	93, 94, 96	cncfcn 24750	. . . . . . . . 9 ⊢ (((0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
98	91, 92, 97	sylancl 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
99	83, 98	eleqtrd 2834	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
100	93	cnfldtop 24620	. . . . . . . . 9 ⊢ (TopOpen‘ℂfld) ∈ Top
101		unicntop 24622	. . . . . . . . . 10 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
102	101	restuni 22986	. . . . . . . . 9 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ) → (0(ball‘(abs ∘ − ))𝑀) = ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
103	100, 91, 102	sylancr 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) = ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
104	61, 103	eleqtrd 2834	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
105		eqid 2731	. . . . . . . 8 ⊢ ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
106	105	cncnpi 23102	. . . . . . 7 ⊢ (((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)) ∧ 𝑎 ∈ ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
107	99, 104, 106	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
108		cnex 11197	. . . . . . . . . . 11 ⊢ ℂ ∈ V
109	108, 11	ssexi 5322	. . . . . . . . . 10 ⊢ 𝑆 ∈ V
110	109	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑆 ∈ V)
111		restabs 22989	. . . . . . . . 9 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆 ∧ 𝑆 ∈ V) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
112	100, 67, 110, 111	mp3an2i 1465	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
113	112	oveq1d 7427	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld)) = (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld)))
114	113	fveq1d 6893	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎) = ((((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
115	107, 114	eleqtrrd 2835	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
116		resttop 22984	. . . . . . . 8 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
117	100, 109, 116	mp2an 689	. . . . . . 7 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top
118	117	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
119		df-ss 3965	. . . . . . . . . 10 ⊢ ((0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆 ↔ ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) = (0(ball‘(abs ∘ − ))𝑀))
120	67, 119	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) = (0(ball‘(abs ∘ − ))𝑀))
121	93	cnfldtopn 24618	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
122	121	blopn 24329	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
123	55, 14, 58, 122	mp3an12i 1464	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
124		elrestr 17381	. . . . . . . . . 10 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V ∧ (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld)) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
125	100, 109, 123, 124	mp3an12i 1464	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
126	120, 125	eqeltrrd 2833	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
127		isopn3i 22906	. . . . . . . 8 ⊢ ((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ (0(ball‘(abs ∘ − ))𝑀) ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)) = (0(ball‘(abs ∘ − ))𝑀))
128	117, 126, 127	sylancr 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)) = (0(ball‘(abs ∘ − ))𝑀))
129	61, 128	eleqtrrd 2835	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)))
130	90	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐹:𝑆⟶ℂ)
131	101	restuni 22986	. . . . . . . 8 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆))
132	100, 11, 131	mp2an 689	. . . . . . 7 ⊢ 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆)
133	132, 101	cnprest 23113	. . . . . 6 ⊢ (((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆) ∧ (𝑎 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)) ∧ 𝐹:𝑆⟶ℂ)) → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎) ↔ (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎)))
134	118, 67, 129, 130, 133	syl22anc 836	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎) ↔ (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎)))
135	115, 134	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
136	135	ralrimiva 3145	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
137		resttopon 22985	. . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
138	95, 11, 137	mp2an 689	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)
139		cncnp 23104	. . . 4 ⊢ ((((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) → (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑎 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))))
140	138, 95, 139	mp2an 689	. . 3 ⊢ (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑎 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎)))
141	90, 136, 140	sylanbrc 582	. 2 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
142		eqid 2731	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
143	93, 142, 96	cncfcn 24750	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
144	11, 92, 143	mp2an 689	. 2 ⊢ (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld))
145	141, 144	eleqtrrdi 2843	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3060  {crab 3431  Vcvv 3473   ∩ cin 3947   ⊆ wss 3948  ifcif 4528  ∪ cuni 4908   class class class wbr 5148   ↦ cmpt 5231  ^◡ccnv 5675  dom cdm 5676   ↾ cres 5678   “ cima 5679   ∘ ccom 5680   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7412  supcsup 9441  ℂcc 11114  ℝcr 11115  0cc0 11116  1c1 11117   + caddc 11119   · cmul 11121  +∞cpnf 11252  ℝ^*cxr 11254   < clt 11255   ≤ cle 11256   − cmin 11451   / cdiv 11878  2c2 12274  ℕ₀cn0 12479  ℝ⁺crp 12981  [,)cico 13333  [,]cicc 13334  seqcseq 13973  ↑cexp 14034  abscabs 15188   ⇝ cli 15435  Σcsu 15639   ↾_t crest 17373  TopOpenctopn 17374  ∞Metcxmet 21218  ballcbl 21220  ℂ_fldccnfld 21233  Topctop 22715  TopOnctopon 22732  intcnt 22841   Cn ccn 23048   CnP ccnp 23049  –cn→ccncf 24716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194  ax-addf 11195

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8152  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-er 8709  df-map 8828  df-pm 8829  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9368  df-fi 9412  df-sup 9443  df-inf 9444  df-oi 9511  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-q 12940  df-rp 12982  df-xneg 13099  df-xadd 13100  df-xmul 13101  df-ico 13337  df-icc 13338  df-fz 13492  df-fzo 13635  df-fl 13764  df-seq 13974  df-exp 14035  df-hash 14298  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-limsup 15422  df-clim 15439  df-rlim 15440  df-sum 15640  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-starv 17219  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-unif 17227  df-hom 17228  df-cco 17229  df-rest 17375  df-topn 17376  df-0g 17394  df-gsum 17395  df-topgen 17396  df-pt 17397  df-prds 17400  df-xrs 17455  df-qtop 17460  df-imas 17461  df-xps 17463  df-mre 17537  df-mrc 17538  df-acs 17540  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-submnd 18712  df-mulg 18994  df-cntz 19229  df-cmn 19698  df-psmet 21225  df-xmet 21226  df-met 21227  df-bl 21228  df-mopn 21229  df-cnfld 21234  df-top 22716  df-topon 22733  df-topsp 22755  df-bases 22769  df-ntr 22844  df-cn 23051  df-cnp 23052  df-tx 23386  df-hmeo 23579  df-xms 24146  df-ms 24147  df-tms 24148  df-cncf 24718  df-ulm 26228

This theorem is referenced by:  pserdvlem2  26280  pserdv  26281  abelth  26293  logtayl  26508