Theorem psercn 26396

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 15615	. . . . . 6 ⊢ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗) ∈ V
2	1	rgenw 3056	. . . . 5 ⊢ ∀𝑦 ∈ 𝑆 Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗) ∈ V
3		pserf.f	. . . . . 6 ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗))
4	3	fnmpt 6633	. . . . 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 6042	. . . . . . . . . . . 12 ⊢ (◡abs “ (0[,)𝑅)) ⊆ dom abs
8		absf 15265	. . . . . . . . . . . . 13 ⊢ abs:ℂ⟶ℝ
9	8	fdmi 6674	. . . . . . . . . . . 12 ⊢ dom abs = ℂ
10	7, 9	sseqtri 3983	. . . . . . . . . . 11 ⊢ (◡abs “ (0[,)𝑅)) ⊆ ℂ
11	6, 10	eqsstri 3981	. . . . . . . . . 10 ⊢ 𝑆 ⊆ ℂ
12	11	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
13	12	sselda 3934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ)
14		0cn 11128	. . . . . . . . . . 11 ⊢ 0 ∈ ℂ
15		eqid 2737	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (abs ∘ − )
16	15	cnmetdval 24718	. . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
17	14, 13, 16	sylancr 588	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
18		abssub 15254	. . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
19	14, 13, 18	sylancr 588	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
20	13	subid1d 11485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 − 0) = 𝑎)
21	20	fveq2d 6839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘(𝑎 − 0)) = (abs‘𝑎))
22	17, 19, 21	3eqtrd 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) = (abs‘𝑎))
23		breq2 5103	. . . . . . . . . . 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 5103	. . . . . . . . . . 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 2847	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (◡abs “ (0[,)𝑅)))
27		ffn 6663	. . . . . . . . . . . . . . . . . 18 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
28		elpreima 7005	. . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅)))
31	30	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ (0[,)𝑅))
32		0re 11138	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
33		iccssxr 13350	. . . . . . . . . . . . . . . . 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 26386	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞))
38	37	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ (0[,]+∞))
39	33, 38	sselid 3932	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ*)
40		elico2 13330	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
41	32, 39, 40	sylancr 588	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
42	31, 41	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅))
43	42	simp3d 1145	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑅)
44	43	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < 𝑅)
45	13	abscld 15366	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ ℝ)
46		avglt1 12383	. . . . . . . . . . . . 13 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
47	45, 46	sylan 581	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
48	44, 47	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2))
49	45	ltp1d 12076	. . . . . . . . . . . 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 4519	. . . . . . . . . 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 5141	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀)
54	22, 53	eqbrtrd 5121	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) < 𝑀)
55		cnxmet 24720	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
56	34, 3, 35, 36, 6, 52	psercnlem1 26395	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅))
57	56	simp1d 1143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ+)
58	57	rpxrd 12954	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ*)
59		elbl 24336	. . . . . . . . 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 1468	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ↔ (𝑎 ∈ ℂ ∧ (0(abs ∘ − )𝑎) < 𝑀)))
61	13, 54, 60	mpbir2and 714	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀))
62	61	fvresd 6855	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) = (𝐹‘𝑎))
63	3	reseq1i 5935	. . . . . . . . . 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 26394	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ (◡abs “ (0[,]𝑀)) ∧ (◡abs “ (0[,]𝑀)) ⊆ 𝑆))
65	64	simp2d 1144	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ (◡abs “ (0[,]𝑀)))
66	64	simp3d 1145	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (◡abs “ (0[,]𝑀)) ⊆ 𝑆)
67	65, 66	sstrd 3945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆)
68	67	resmptd 6000	. . . . . . . . . 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 2784	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)))
70		eqid 2737	. . . . . . . . . 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 6835	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑦 → (𝐺‘𝑘) = (𝐺‘𝑦))
73	72	seqeq3d 13936	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑦 → seq0( + , (𝐺‘𝑘)) = seq0( + , (𝐺‘𝑦)))
74	73	fveq1d 6837	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑦 → (seq0( + , (𝐺‘𝑘))‘𝑠) = (seq0( + , (𝐺‘𝑦))‘𝑠))
75	74	cbvmptv 5203	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑦))‘𝑠))
76		fveq2 6835	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑖 → (seq0( + , (𝐺‘𝑦))‘𝑠) = (seq0( + , (𝐺‘𝑦))‘𝑖))
77	76	mpteq2dv 5193	. . . . . . . . . . . 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 2784	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑖 → (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
79	78	cbvmptv 5203	. . . . . . . . . 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 12953	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ)
81	56	simp3d 1145	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 < 𝑅)
82	34, 70, 71, 36, 79, 80, 81, 65	psercn2 26392	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
83	69, 82	eqeltrd 2837	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
84		cncff 24846	. . . . . . . 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 7032	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) ∈ ℂ)
87	62, 86	eqeltrrd 2838	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹‘𝑎) ∈ ℂ)
88	87	ralrimiva 3129	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 (𝐹‘𝑎) ∈ ℂ)
89		ffnfv 7066	. . . 4 ⊢ (𝐹:𝑆⟶ℂ ↔ (𝐹 Fn 𝑆 ∧ ∀𝑎 ∈ 𝑆 (𝐹‘𝑎) ∈ ℂ))
90	5, 88, 89	sylanbrc 584	. . 3 ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
91	67, 11	sstrdi 3947	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ)
92		ssid 3957	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
93		eqid 2737	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
94		eqid 2737	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
95	93	cnfldtopon 24730	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
96	95	toponrestid 22869	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
97	93, 94, 96	cncfcn 24863	. . . . . . . . 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 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
99	83, 98	eleqtrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
100	93	cnfldtop 24731	. . . . . . . . 9 ⊢ (TopOpen‘ℂfld) ∈ Top
101		unicntop 24733	. . . . . . . . . 10 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
102	101	restuni 23110	. . . . . . . . 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 588	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) = ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
104	61, 103	eleqtrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
105		eqid 2737	. . . . . . . 8 ⊢ ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
106	105	cncnpi 23226	. . . . . . 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 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
108		cnex 11111	. . . . . . . . . . 11 ⊢ ℂ ∈ V
109	108, 11	ssexi 5268	. . . . . . . . . 10 ⊢ 𝑆 ∈ V
110	109	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑆 ∈ V)
111		restabs 23113	. . . . . . . . 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 1469	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
113	112	oveq1d 7375	. . . . . . 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 6837	. . . . . 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 2840	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
116		resttop 23108	. . . . . . . 8 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
117	100, 109, 116	mp2an 693	. . . . . . 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 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) = (0(ball‘(abs ∘ − ))𝑀))
121	93	cnfldtopn 24729	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
122	121	blopn 24448	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
123	55, 14, 58, 122	mp3an12i 1468	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
124		elrestr 17352	. . . . . . . . . 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 1468	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
126	120, 125	eqeltrrd 2838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
127		isopn3i 23030	. . . . . . . 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 588	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)) = (0(ball‘(abs ∘ − ))𝑀))
129	61, 128	eleqtrrd 2840	. . . . . 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 23110	. . . . . . . 8 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆))
132	100, 11, 131	mp2an 693	. . . . . . 7 ⊢ 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆)
133	132, 101	cnprest 23237	. . . . . 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 839	. . . . 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 3129	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
137		resttopon 23109	. . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
138	95, 11, 137	mp2an 693	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)
139		cncnp 23228	. . . 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 693	. . 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 584	. 2 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
142		eqid 2737	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
143	93, 142, 96	cncfcn 24863	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
144	11, 92, 143	mp2an 693	. 2 ⊢ (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld))
145	141, 144	eleqtrrdi 2848	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3400  Vcvv 3441   ∩ cin 3901   ⊆ wss 3902  ifcif 4480  ∪ cuni 4864   class class class wbr 5099   ↦ cmpt 5180  ^◡ccnv 5624  dom cdm 5625   ↾ cres 5627   “ cima 5628   ∘ ccom 5629   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360  supcsup 9347  ℂcc 11028  ℝcr 11029  0cc0 11030  1c1 11031   + caddc 11033   · cmul 11035  +∞cpnf 11167  ℝ^*cxr 11169   < clt 11170   ≤ cle 11171   − cmin 11368   / cdiv 11798  2c2 12204  ℕ₀cn0 12405  ℝ⁺crp 12909  [,)cico 13267  [,]cicc 13268  seqcseq 13928  ↑cexp 13988  abscabs 15161   ⇝ cli 15411  Σcsu 15613   ↾_t crest 17344  TopOpenctopn 17345  ∞Metcxmet 21298  ballcbl 21300  ℂ_fldccnfld 21313  Topctop 22841  TopOnctopon 22858  intcnt 22965   Cn ccn 23172   CnP ccnp 23173  –cn→ccncf 24829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108  ax-addf 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-fi 9318  df-sup 9349  df-inf 9350  df-oi 9419  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12406  df-z 12493  df-dec 12612  df-uz 12756  df-q 12866  df-rp 12910  df-xneg 13030  df-xadd 13031  df-xmul 13032  df-ico 13271  df-icc 13272  df-fz 13428  df-fzo 13575  df-fl 13716  df-seq 13929  df-exp 13989  df-hash 14258  df-cj 15026  df-re 15027  df-im 15028  df-sqrt 15162  df-abs 15163  df-limsup 15398  df-clim 15415  df-rlim 15416  df-sum 15614  df-struct 17078  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17141  df-ress 17162  df-plusg 17194  df-mulr 17195  df-starv 17196  df-sca 17197  df-vsca 17198  df-ip 17199  df-tset 17200  df-ple 17201  df-ds 17203  df-unif 17204  df-hom 17205  df-cco 17206  df-rest 17346  df-topn 17347  df-0g 17365  df-gsum 17366  df-topgen 17367  df-pt 17368  df-prds 17371  df-xrs 17427  df-qtop 17432  df-imas 17433  df-xps 17435  df-mre 17509  df-mrc 17510  df-acs 17512  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-submnd 18713  df-mulg 19002  df-cntz 19250  df-cmn 19715  df-psmet 21305  df-xmet 21306  df-met 21307  df-bl 21308  df-mopn 21309  df-cnfld 21314  df-top 22842  df-topon 22859  df-topsp 22881  df-bases 22894  df-ntr 22968  df-cn 23175  df-cnp 23176  df-tx 23510  df-hmeo 23703  df-xms 24268  df-ms 24269  df-tms 24270  df-cncf 24831  df-ulm 26346

This theorem is referenced by:  pserdvlem2  26398  pserdv  26399  abelth  26411  logtayl  26629