Theorem psercn 26352

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 15613	. . . . . 6 ⊢ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗) ∈ V
2	1	rgenw 3048	. . . . 5 ⊢ ∀𝑦 ∈ 𝑆 Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗) ∈ V
3		pserf.f	. . . . . 6 ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗))
4	3	fnmpt 6626	. . . . 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 6037	. . . . . . . . . . . 12 ⊢ (◡abs “ (0[,)𝑅)) ⊆ dom abs
8		absf 15263	. . . . . . . . . . . . 13 ⊢ abs:ℂ⟶ℝ
9	8	fdmi 6667	. . . . . . . . . . . 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 3937	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ)
14		0cn 11126	. . . . . . . . . . 11 ⊢ 0 ∈ ℂ
15		eqid 2729	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (abs ∘ − )
16	15	cnmetdval 24674	. . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
17	14, 13, 16	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
18		abssub 15252	. . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
19	14, 13, 18	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
20	13	subid1d 11482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 − 0) = 𝑎)
21	20	fveq2d 6830	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘(𝑎 − 0)) = (abs‘𝑎))
22	17, 19, 21	3eqtrd 2768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) = (abs‘𝑎))
23		breq2 5099	. . . . . . . . . . 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 5099	. . . . . . . . . . 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 2838	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (◡abs “ (0[,)𝑅)))
27		ffn 6656	. . . . . . . . . . . . . . . . . 18 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
28		elpreima 6996	. . . . . . . . . . . . . . . . . 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 11136	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
33		iccssxr 13351	. . . . . . . . . . . . . . . . 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 26342	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞))
38	37	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ (0[,]+∞))
39	33, 38	sselid 3935	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ*)
40		elico2 13331	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
41	32, 39, 40	sylancr 587	. . . . . . . . . . . . . . 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 1144	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑅)
44	43	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < 𝑅)
45	13	abscld 15364	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ ℝ)
46		avglt1 12380	. . . . . . . . . . . . 13 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
47	45, 46	sylan 580	. . . . . . . . . . . 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 12073	. . . . . . . . . . . 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 4517	. . . . . . . . . 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 5137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀)
54	22, 53	eqbrtrd 5117	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(abs ∘ − )𝑎) < 𝑀)
55		cnxmet 24676	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
56	34, 3, 35, 36, 6, 52	psercnlem1 26351	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅))
57	56	simp1d 1142	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ+)
58	57	rpxrd 12956	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ*)
59		elbl 24292	. . . . . . . . 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 1467	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ↔ (𝑎 ∈ ℂ ∧ (0(abs ∘ − )𝑎) < 𝑀)))
61	13, 54, 60	mpbir2and 713	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀))
62	61	fvresd 6846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) = (𝐹‘𝑎))
63	3	reseq1i 5930	. . . . . . . . . 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 26350	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ (◡abs “ (0[,]𝑀)) ∧ (◡abs “ (0[,]𝑀)) ⊆ 𝑆))
65	64	simp2d 1143	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ (◡abs “ (0[,]𝑀)))
66	64	simp3d 1144	. . . . . . . . . . . 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 5995	. . . . . . . . . 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 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)))
70		eqid 2729	. . . . . . . . . 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 6826	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑦 → (𝐺‘𝑘) = (𝐺‘𝑦))
73	72	seqeq3d 13934	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑦 → seq0( + , (𝐺‘𝑘)) = seq0( + , (𝐺‘𝑦)))
74	73	fveq1d 6828	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑦 → (seq0( + , (𝐺‘𝑘))‘𝑠) = (seq0( + , (𝐺‘𝑦))‘𝑠))
75	74	cbvmptv 5199	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑦))‘𝑠))
76		fveq2 6826	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑖 → (seq0( + , (𝐺‘𝑦))‘𝑠) = (seq0( + , (𝐺‘𝑦))‘𝑖))
77	76	mpteq2dv 5189	. . . . . . . . . . . 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 2776	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑖 → (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
79	78	cbvmptv 5199	. . . . . . . . . 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 12955	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ)
81	56	simp3d 1144	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 < 𝑅)
82	34, 70, 71, 36, 79, 80, 81, 65	psercn2 26348	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
83	69, 82	eqeltrd 2828	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
84		cncff 24802	. . . . . . . 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 7023	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) ∈ ℂ)
87	62, 86	eqeltrrd 2829	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹‘𝑎) ∈ ℂ)
88	87	ralrimiva 3121	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 (𝐹‘𝑎) ∈ ℂ)
89		ffnfv 7057	. . . 4 ⊢ (𝐹:𝑆⟶ℂ ↔ (𝐹 Fn 𝑆 ∧ ∀𝑎 ∈ 𝑆 (𝐹‘𝑎) ∈ ℂ))
90	5, 88, 89	sylanbrc 583	. . 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 2729	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
94		eqid 2729	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
95	93	cnfldtopon 24686	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
96	95	toponrestid 22824	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
97	93, 94, 96	cncfcn 24819	. . . . . . . . 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 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
99	83, 98	eleqtrd 2830	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
100	93	cnfldtop 24687	. . . . . . . . 9 ⊢ (TopOpen‘ℂfld) ∈ Top
101		unicntop 24689	. . . . . . . . . 10 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
102	101	restuni 23065	. . . . . . . . 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 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) = ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
104	61, 103	eleqtrd 2830	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
105		eqid 2729	. . . . . . . 8 ⊢ ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ∪ ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
106	105	cncnpi 23181	. . . . . . 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 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
108		cnex 11109	. . . . . . . . . . 11 ⊢ ℂ ∈ V
109	108, 11	ssexi 5264	. . . . . . . . . 10 ⊢ 𝑆 ∈ V
110	109	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑆 ∈ V)
111		restabs 23068	. . . . . . . . 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 1468	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
113	112	oveq1d 7368	. . . . . . 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 6828	. . . . . 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 2831	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
116		resttop 23063	. . . . . . . 8 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
117	100, 109, 116	mp2an 692	. . . . . . 7 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top
118	117	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
119		dfss2 3923	. . . . . . . . . 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 24685	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
122	121	blopn 24404	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
123	55, 14, 58, 122	mp3an12i 1467	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
124		elrestr 17350	. . . . . . . . . 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 1467	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
126	120, 125	eqeltrrd 2829	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
127		isopn3i 22985	. . . . . . . 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 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)) = (0(ball‘(abs ∘ − ))𝑀))
129	61, 128	eleqtrrd 2831	. . . . . 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 23065	. . . . . . . 8 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆))
132	100, 11, 131	mp2an 692	. . . . . . 7 ⊢ 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆)
133	132, 101	cnprest 23192	. . . . . 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 838	. . . . 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 3121	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
137		resttopon 23064	. . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
138	95, 11, 137	mp2an 692	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)
139		cncnp 23183	. . . 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 692	. . 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 583	. 2 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
142		eqid 2729	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
143	93, 142, 96	cncfcn 24819	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
144	11, 92, 143	mp2an 692	. 2 ⊢ (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld))
145	141, 144	eleqtrrdi 2839	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3396  Vcvv 3438   ∩ cin 3904   ⊆ wss 3905  ifcif 4478  ∪ cuni 4861   class class class wbr 5095   ↦ cmpt 5176  ^◡ccnv 5622  dom cdm 5623   ↾ cres 5625   “ cima 5626   ∘ ccom 5627   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  supcsup 9349  ℂcc 11026  ℝcr 11027  0cc0 11028  1c1 11029   + caddc 11031   · cmul 11033  +∞cpnf 11165  ℝ^*cxr 11167   < clt 11168   ≤ cle 11169   − cmin 11365   / cdiv 11795  2c2 12201  ℕ₀cn0 12402  ℝ⁺crp 12911  [,)cico 13268  [,]cicc 13269  seqcseq 13926  ↑cexp 13986  abscabs 15159   ⇝ cli 15409  Σcsu 15611   ↾_t crest 17342  TopOpenctopn 17343  ∞Metcxmet 21264  ballcbl 21266  ℂ_fldccnfld 21279  Topctop 22796  TopOnctopon 22813  intcnt 22920   Cn ccn 23127   CnP ccnp 23128  –cn→ccncf 24785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106  ax-addf 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-of 7617  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8632  df-map 8762  df-pm 8763  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-fsupp 9271  df-fi 9320  df-sup 9351  df-inf 9352  df-oi 9421  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-div 11796  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-n0 12403  df-z 12490  df-dec 12610  df-uz 12754  df-q 12868  df-rp 12912  df-xneg 13032  df-xadd 13033  df-xmul 13034  df-ico 13272  df-icc 13273  df-fz 13429  df-fzo 13576  df-fl 13714  df-seq 13927  df-exp 13987  df-hash 14256  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-limsup 15396  df-clim 15413  df-rlim 15414  df-sum 15612  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-mulr 17193  df-starv 17194  df-sca 17195  df-vsca 17196  df-ip 17197  df-tset 17198  df-ple 17199  df-ds 17201  df-unif 17202  df-hom 17203  df-cco 17204  df-rest 17344  df-topn 17345  df-0g 17363  df-gsum 17364  df-topgen 17365  df-pt 17366  df-prds 17369  df-xrs 17424  df-qtop 17429  df-imas 17430  df-xps 17432  df-mre 17506  df-mrc 17507  df-acs 17509  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-submnd 18676  df-mulg 18965  df-cntz 19214  df-cmn 19679  df-psmet 21271  df-xmet 21272  df-met 21273  df-bl 21274  df-mopn 21275  df-cnfld 21280  df-top 22797  df-topon 22814  df-topsp 22836  df-bases 22849  df-ntr 22923  df-cn 23130  df-cnp 23131  df-tx 23465  df-hmeo 23658  df-xms 24224  df-ms 24225  df-tms 24226  df-cncf 24787  df-ulm 26302

This theorem is referenced by:  pserdvlem2  26354  pserdv  26355  abelth  26367  logtayl  26585