Theorem psercn 26669

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 15779	. . . . . 6 ⊢ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗) ∈ V
2	1	rgenw 3082	. . . . 5 ⊢ ∀𝑦 ∈ 𝑆 Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗) ∈ V
3		pserf.f	. . . . . 6 ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗))
4	3	fnmpt 6676	. . . . 5 ⊢ (∀𝑦 ∈ 𝑆 Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗) ∈ V → 𝐹 Fn 𝑆)
5	2, 4	mp1i 14	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝑆)
6		psercn.s	. . . . . . . . . . 11 ⊢ 𝑆 = (◡abs “ (0[,)𝑅))
7		cnvimass 6082	. . . . . . . . . . . 12 ⊢ (◡abs “ (0[,)𝑅)) ⊆ dom abs
8		absf 15429	. . . . . . . . . . . . 13 ⊢ abs:ℂ⟶ℝ
9	8	fdmi 6718	. . . . . . . . . . . 12 ⊢ dom abs = ℂ
10	7, 9	sseqtri 3982	. . . . . . . . . . 11 ⊢ (◡abs “ (0[,)𝑅)) ⊆ ℂ
11	6, 10	eqsstri 3980	. . . . . . . . . 10 ⊢ 𝑆 ⊆ ℂ
12	11	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
13	12	sselda 3934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ)
14		0cn 11226	. . . . . . . . . . 11 ⊢ 0 ∈ ℂ
15		eqid 2762	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (abs ∘ − )
16	15	cnmetdval 25002	. . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
17	14, 13, 16	sylancr 599	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
18		abssub 15418	. . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
19	14, 13, 18	sylancr 599	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
20	13	subid1d 11586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 − 0) = 𝑎)
21	20	fveq2d 6886	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘(𝑎 − 0)) = (abs‘𝑎))
22	17, 19, 21	3eqtrd 2801	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) = (abs‘𝑎))
23		breq2 5111	. . . . . . . . . . 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 5111	. . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
26	25, 6	eleqtrdi 2872	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (◡abs “ (0[,)𝑅)))
27		ffn 6706	. . . . . . . . . . . . . . . . . 18 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
28		elpreima 7054	. . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅)))
31	30	simprd 501	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ (0[,)𝑅))
32		0re 11238	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
33		iccssxr 13487	. . . . . . . . . . . . . . . . 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 26660	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞))
38	37	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ (0[,]+∞))
39	33, 38	sselid 3932	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ*)
40		elico2 13467	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
41	32, 39, 40	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
42	31, 41	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅))
43	42	simp3d 1162	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑅)
44	43	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < 𝑅)
45	13	abscld 15530	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ ℝ)
46		avglt1 12510	. . . . . . . . . . . . 13 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
47	45, 46	sylan 592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
48	44, 47	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2))
49	45	ltp1d 12173	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 1))
50	49	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ ¬ 𝑅 ∈ ℝ) → (abs‘𝑎) < ((abs‘𝑎) + 1))
51	23, 24, 48, 50	ifbothda 4524	. . . . . . . . . 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 5151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀)
54	22, 53	eqbrtrd 5131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) < 𝑀)
55		cnxmet 25004	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
56	34, 3, 35, 36, 6, 52	psercnlem1 26668	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅))
57	56	simp1d 1160	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ+)
58	57	rpxrd 13091	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ*)
59		elbl 24620	. . . . . . . . 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 1494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ↔ (𝑎 ∈ ℂ ∧ (0(abs ∘ − )𝑎) < 𝑀)))
61	13, 54, 60	mpbir2and 726	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀))
62	61	fvresd 6902	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) = (𝐹‘𝑎))
63	3	reseq1i 5972	. . . . . . . . . 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 26667	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ (◡abs “ (0[,]𝑀)) ∧ (◡abs “ (0[,]𝑀)) ⊆ 𝑆))
65	64	simp2d 1161	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ (◡abs “ (0[,]𝑀)))
66	64	simp3d 1162	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (◡abs “ (0[,]𝑀)) ⊆ 𝑆)
67	65, 66	sstrd 3944	. . . . . . . . . . 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 2809	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)))
70		eqid 2762	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗))
71	35	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐴:ℕ0⟶ℂ)
72		fveq2 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑦 → (𝐺‘𝑘) = (𝐺‘𝑦))
73	72	seqeq3d 14077	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑦 → seq0( + , (𝐺‘𝑘)) = seq0( + , (𝐺‘𝑦)))
74	73	fveq1d 6884	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑦 → (seq0( + , (𝐺‘𝑘))‘𝑠) = (seq0( + , (𝐺‘𝑦))‘𝑠))
75	74	cbvmptv 5213	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑦))‘𝑠))
76		fveq2 6882	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑖 → (seq0( + , (𝐺‘𝑦))‘𝑠) = (seq0( + , (𝐺‘𝑦))‘𝑖))
77	76	mpteq2dv 5203	. . . . . . . . . . . 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 2809	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑖 → (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
79	78	cbvmptv 5213	. . . . . . . . . 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 13090	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ)
81	56	simp3d 1162	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 < 𝑅)
82	34, 70, 71, 36, 79, 80, 81, 65	psercn2 26666	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
83	69, 82	eqeltrd 2862	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
84		cncff 25127	. . . . . . . 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 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)):(0(ball‘(abs ∘ − ))𝑀)⟶ℂ)
86	85, 61	ffvelcdmd 7082	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) ∈ ℂ)
87	62, 86	eqeltrrd 2863	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹‘𝑎) ∈ ℂ)
88	87	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 (𝐹‘𝑎) ∈ ℂ)
89		ffnfv 7116	. . . 4 ⊢ (𝐹:𝑆⟶ℂ ↔ (𝐹 Fn 𝑆 ∧ ∀𝑎 ∈ 𝑆 (𝐹‘𝑎) ∈ ℂ))
90	5, 88, 89	sylanbrc 595	. . 3 ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
91	67, 11	sstrdi 3946	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ)
92		ssid 3956	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
93		eqid 2762	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
94		eqid 2762	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
95	93	cnfldtopon 25014	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
96	95	toponrestid 23152	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
97	93, 94, 96	cncfcn 25144	. . . . . . . . 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 598	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
99	83, 98	eleqtrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
100	93	cnfldtop 25015	. . . . . . . . 9 ⊢ (TopOpen‘ℂfld) ∈ Top
101		unicntop 25017	. . . . . . . . . 10 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
102	101	restuni 23393	. . . . . . . . 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 599	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) = ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
104	61, 103	eleqtrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
105		eqid 2762	. . . . . . . 8 ⊢ ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
106	105	cncnpi 23509	. . . . . . 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 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
108		cnex 11209	. . . . . . . . . . 11 ⊢ ℂ ∈ V
109	108, 11	ssexi 5291	. . . . . . . . . 10 ⊢ 𝑆 ∈ V
110	109	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑆 ∈ V)
111		restabs 23396	. . . . . . . . 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 1495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
113	112	oveq1d 7432	. . . . . . 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 6884	. . . . . 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 2865	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
116		resttop 23391	. . . . . . . 8 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
117	100, 109, 116	mp2an 705	. . . . . . 7 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top
118	117	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
119		dfss2 3920	. . . . . . . . . 10 ⊢ ((0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆 ↔ ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) = (0(ball‘(abs ∘ − ))𝑀))
120	67, 119	sylib 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) = (0(ball‘(abs ∘ − ))𝑀))
121	93	cnfldtopn 25013	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
122	121	blopn 24732	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
123	55, 14, 58, 122	mp3an12i 1494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
124		elrestr 17519	. . . . . . . . . 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 1494	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
126	120, 125	eqeltrrd 2863	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
127		isopn3i 23313	. . . . . . . 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 599	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)) = (0(ball‘(abs ∘ − ))𝑀))
129	61, 128	eleqtrrd 2865	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)))
130	90	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐹:𝑆⟶ℂ)
131	101	restuni 23393	. . . . . . . 8 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆))
132	100, 11, 131	mp2an 705	. . . . . . 7 ⊢ 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆)
133	132, 101	cnprest 23520	. . . . . 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 852	. . . . 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 260	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
136	135	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
137		resttopon 23392	. . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
138	95, 11, 137	mp2an 705	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)
139		cncnp 23511	. . . 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 705	. . 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 595	. 2 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
142		eqid 2762	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
143	93, 142, 96	cncfcn 25144	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
144	11, 92, 143	mp2an 705	. 2 ⊢ (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld))
145	141, 144	eleqtrrdi 2873	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  ifcif 4485  ∪ cuni 4870   class class class wbr 5107   ↦ cmpt 5190  ^◡ccnv 5658  dom cdm 5659   ↾ cres 5661   “ cima 5662   ∘ ccom 5663   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417  supcsup 9414  ℂcc 11126  ℝcr 11127  0cc0 11128  1c1 11129   + caddc 11131   · cmul 11133  +∞cpnf 11268  ℝ^*cxr 11270   < clt 11271   ≤ cle 11272   − cmin 11469   / cdiv 11899  2c2 12323  ℕ₀cn0 12532  ℝ⁺crp 13046  [,)cico 13404  [,]cicc 13405  seqcseq 14069  ↑cexp 14129  abscabs 15325   ⇝ cli 15575  Σcsu 15777   ↾_t crest 17511  TopOpenctopn 17512  ∞Metcxmet 21576  ballcbl 21578  ℂ_fldccnfld 21591  Topctop 23124  TopOnctopon 23141  intcnt 23248   Cn ccn 23455   CnP ccnp 23456  –cn→ccncf 25110

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-inf2 9624  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-pre-sup 11206  ax-addf 11207

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8163  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-fsupp 9336  df-fi 9385  df-sup 9416  df-inf 9417  df-oi 9486  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-9 12338  df-n0 12533  df-z 12620  df-dec 12741  df-uz 12892  df-q 13002  df-rp 13047  df-xneg 13167  df-xadd 13168  df-xmul 13169  df-ico 13408  df-icc 13409  df-fz 13566  df-fzo 13714  df-fl 13857  df-seq 14070  df-exp 14130  df-hash 14399  df-cj 15190  df-re 15191  df-im 15192  df-sqrt 15326  df-abs 15327  df-limsup 15562  df-clim 15579  df-rlim 15580  df-sum 15778  df-struct 17245  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-mulr 17362  df-starv 17363  df-sca 17364  df-vsca 17365  df-ip 17366  df-tset 17367  df-ple 17368  df-ds 17370  df-unif 17371  df-hom 17372  df-cco 17373  df-rest 17513  df-topn 17514  df-0g 17532  df-gsum 17533  df-topgen 17534  df-pt 17535  df-prds 17538  df-xrs 17594  df-qtop 17599  df-imas 17600  df-xps 17602  df-mre 17676  df-mrc 17677  df-acs 17679  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-submnd 18898  df-mulg 19197  df-cntz 19450  df-cmn 19915  df-psmet 21583  df-xmet 21584  df-met 21585  df-bl 21586  df-mopn 21587  df-cnfld 21592  df-top 23125  df-topon 23142  df-topsp 23164  df-bases 23177  df-ntr 23251  df-cn 23458  df-cnp 23459  df-tx 23794  df-hmeo 23987  df-xms 24552  df-ms 24553  df-tms 24554  df-cncf 25112  df-ulm 26620

This theorem is used by:  pserdvlem2  26671  pserdv  26672  abelth  26684  logtayl  26905