Theorem psercn 26567

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 15741	. . . . . 6 ⊢ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗) ∈ V
2	1	rgenw 3083	. . . . 5 ⊢ ∀𝑦 ∈ 𝑆 Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗) ∈ V
3		pserf.f	. . . . . 6 ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗))
4	3	fnmpt 6677	. . . . 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 6086	. . . . . . . . . . . 12 ⊢ (◡abs “ (0[,)𝑅)) ⊆ dom abs
8		absf 15391	. . . . . . . . . . . . 13 ⊢ abs:ℂ⟶ℝ
9	8	fdmi 6719	. . . . . . . . . . . 12 ⊢ dom abs = ℂ
10	7, 9	sseqtri 3986	. . . . . . . . . . 11 ⊢ (◡abs “ (0[,)𝑅)) ⊆ ℂ
11	6, 10	eqsstri 3984	. . . . . . . . . 10 ⊢ 𝑆 ⊆ ℂ
12	11	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
13	12	sselda 3938	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ)
14		0cn 11199	. . . . . . . . . . 11 ⊢ 0 ∈ ℂ
15		eqid 2763	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (abs ∘ − )
16	15	cnmetdval 24908	. . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
17	14, 13, 16	sylancr 598	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
18		abssub 15380	. . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
19	14, 13, 18	sylancr 598	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
20	13	subid1d 11559	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 − 0) = 𝑎)
21	20	fveq2d 6887	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘(𝑎 − 0)) = (abs‘𝑎))
22	17, 19, 21	3eqtrd 2802	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) = (abs‘𝑎))
23		breq2 5114	. . . . . . . . . . 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 5114	. . . . . . . . . . 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 489	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
26	25, 6	eleqtrdi 2873	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (◡abs “ (0[,)𝑅)))
27		ffn 6707	. . . . . . . . . . . . . . . . . 18 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
28		elpreima 7055	. . . . . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ (0[,)𝑅))
32		0re 11211	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
33		iccssxr 13458	. . . . . . . . . . . . . . . . 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 26558	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞))
38	37	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ (0[,]+∞))
39	33, 38	sselid 3936	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ*)
40		elico2 13438	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
41	32, 39, 40	sylancr 598	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < 𝑅)
45	13	abscld 15492	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ ℝ)
46		avglt1 12483	. . . . . . . . . . . . 13 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
47	45, 46	sylan 591	. . . . . . . . . . . 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 12146	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 1))
50	49	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ ¬ 𝑅 ∈ ℝ) → (abs‘𝑎) < ((abs‘𝑎) + 1))
51	23, 24, 48, 50	ifbothda 4527	. . . . . . . . . 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 5154	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀)
54	22, 53	eqbrtrd 5134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) < 𝑀)
55		cnxmet 24910	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
56	34, 3, 35, 36, 6, 52	psercnlem1 26566	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅))
57	56	simp1d 1160	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ+)
58	57	rpxrd 13062	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ*)
59		elbl 24526	. . . . . . . . 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 725	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀))
62	61	fvresd 6903	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) = (𝐹‘𝑎))
63	3	reseq1i 5976	. . . . . . . . . 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 26565	. . . . . . . . . . . . 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 3948	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆)
68	67	resmptd 6044	. . . . . . . . . 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 2810	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)))
70		eqid 2763	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗))
71	35	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐴:ℕ0⟶ℂ)
72		fveq2 6883	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑦 → (𝐺‘𝑘) = (𝐺‘𝑦))
73	72	seqeq3d 14047	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑦 → seq0( + , (𝐺‘𝑘)) = seq0( + , (𝐺‘𝑦)))
74	73	fveq1d 6885	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑦 → (seq0( + , (𝐺‘𝑘))‘𝑠) = (seq0( + , (𝐺‘𝑦))‘𝑠))
75	74	cbvmptv 5216	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑦))‘𝑠))
76		fveq2 6883	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑖 → (seq0( + , (𝐺‘𝑦))‘𝑠) = (seq0( + , (𝐺‘𝑦))‘𝑖))
77	76	mpteq2dv 5206	. . . . . . . . . . . 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 2810	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑖 → (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
79	78	cbvmptv 5216	. . . . . . . . . 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 13061	. . . . . . . . . 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 26564	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
83	69, 82	eqeltrd 2863	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
84		cncff 25033	. . . . . . . 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 2864	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹‘𝑎) ∈ ℂ)
88	87	ralrimiva 3157	. . . 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 594	. . 3 ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
91	67, 11	sstrdi 3950	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ)
92		ssid 3960	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
93		eqid 2763	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
94		eqid 2763	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
95	93	cnfldtopon 24920	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
96	95	toponrestid 23059	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
97	93, 94, 96	cncfcn 25050	. . . . . . . . 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 597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
99	83, 98	eleqtrd 2865	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
100	93	cnfldtop 24921	. . . . . . . . 9 ⊢ (TopOpen‘ℂfld) ∈ Top
101		unicntop 24923	. . . . . . . . . 10 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
102	101	restuni 23300	. . . . . . . . 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 598	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) = ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
104	61, 103	eleqtrd 2865	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
105		eqid 2763	. . . . . . . 8 ⊢ ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
106	105	cncnpi 23416	. . . . . . 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 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
108		cnex 11182	. . . . . . . . . . 11 ⊢ ℂ ∈ V
109	108, 11	ssexi 5294	. . . . . . . . . 10 ⊢ 𝑆 ∈ V
110	109	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑆 ∈ V)
111		restabs 23303	. . . . . . . . 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 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 6885	. . . . . 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 2866	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
116		resttop 23298	. . . . . . . 8 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
117	100, 109, 116	mp2an 704	. . . . . . 7 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top
118	117	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
119		dfss2 3924	. . . . . . . . . 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 24919	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
122	121	blopn 24638	. . . . . . . . . . 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 17482	. . . . . . . . . 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 2864	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
127		isopn3i 23220	. . . . . . . 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 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)) = (0(ball‘(abs ∘ − ))𝑀))
129	61, 128	eleqtrrd 2866	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)))
130	90	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐹:𝑆⟶ℂ)
131	101	restuni 23300	. . . . . . . 8 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆))
132	100, 11, 131	mp2an 704	. . . . . . 7 ⊢ 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆)
133	132, 101	cnprest 23427	. . . . . 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 851	. . . . 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 3157	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
137		resttopon 23299	. . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
138	95, 11, 137	mp2an 704	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)
139		cncnp 23418	. . . 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 704	. . 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 594	. 2 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
142		eqid 2763	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
143	93, 142, 96	cncfcn 25050	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
144	11, 92, 143	mp2an 704	. 2 ⊢ (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld))
145	141, 144	eleqtrrdi 2874	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416  Vcvv 3455   ∩ cin 3905   ⊆ wss 3906  ifcif 4488  ∪ cuni 4873   class class class wbr 5110   ↦ cmpt 5193  ^◡ccnv 5662  dom cdm 5663   ↾ cres 5665   “ cima 5666   ∘ ccom 5667   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412  supcsup 9401  ℂcc 11099  ℝcr 11100  0cc0 11101  1c1 11102   + caddc 11104   · cmul 11106  +∞cpnf 11241  ℝ^*cxr 11243   < clt 11244   ≤ cle 11245   − cmin 11442   / cdiv 11872  2c2 12296  ℕ₀cn0 12505  ℝ⁺crp 13017  [,)cico 13375  [,]cicc 13376  seqcseq 14039  ↑cexp 14099  abscabs 15287   ⇝ cli 15537  Σcsu 15739   ↾_t crest 17474  TopOpenctopn 17475  ∞Metcxmet 21488  ballcbl 21490  ℂ_fldccnfld 21503  Topctop 23031  TopOnctopon 23048  intcnt 23155   Cn ccn 23362   CnP ccnp 23363  –cn→ccncf 25016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-inf2 9611  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178  ax-pre-sup 11179  ax-addf 11180

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8158  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-er 8695  df-map 8827  df-pm 8828  df-ixp 8897  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-fsupp 9323  df-fi 9372  df-sup 9403  df-inf 9404  df-oi 9473  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-div 11873  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-z 12593  df-dec 12713  df-uz 12864  df-q 12974  df-rp 13018  df-xneg 13138  df-xadd 13139  df-xmul 13140  df-ico 13379  df-icc 13380  df-fz 13537  df-fzo 13685  df-fl 13827  df-seq 14040  df-exp 14100  df-hash 14369  df-cj 15152  df-re 15153  df-im 15154  df-sqrt 15288  df-abs 15289  df-limsup 15524  df-clim 15541  df-rlim 15542  df-sum 15740  df-struct 17208  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-mulr 17325  df-starv 17326  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-unif 17334  df-hom 17335  df-cco 17336  df-rest 17476  df-topn 17477  df-0g 17495  df-gsum 17496  df-topgen 17497  df-pt 17498  df-prds 17501  df-xrs 17557  df-qtop 17562  df-imas 17563  df-xps 17565  df-mre 17639  df-mrc 17640  df-acs 17642  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-submnd 18843  df-mulg 19135  df-cntz 19388  df-cmn 19853  df-psmet 21495  df-xmet 21496  df-met 21497  df-bl 21498  df-mopn 21499  df-cnfld 21504  df-top 23032  df-topon 23049  df-topsp 23071  df-bases 23084  df-ntr 23158  df-cn 23365  df-cnp 23366  df-tx 23700  df-hmeo 23893  df-xms 24458  df-ms 24459  df-tms 24460  df-cncf 25018  df-ulm 26518

This theorem is referenced by:  pserdvlem2  26569  pserdv  26570  abelth  26582  logtayl  26803