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Theorem perfdvf 25419

Description: The derivative is a function, whenever it is defined relative to a perfect subset of the complex numbers. (Contributed by Mario Carneiro, 25-Dec-2016.)

**Hypothesis**
Ref	Expression
perfdvf.1	⊢ 𝐾 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
perfdvf	⊢ ((𝐾 ↾_t 𝑆) ∈ Perf → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)

Proof of Theorem perfdvf Dummy variables 𝑓 𝑠 𝑥 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dv 25383	. . . . . . . . . . . . . . . . . . . 20 ⊢ D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑_pm 𝑠) ↦ ∪ 𝑥 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥)))
2	1	dmmpossx 8051	. . . . . . . . . . . . . . . . . . 19 ⊢ dom D ⊆ ∪ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑_pm 𝑠))
3		simpl 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ⟨𝑆, 𝐹⟩ ∈ dom D )
4	2, 3	sselid 3980	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ⟨𝑆, 𝐹⟩ ∈ ∪ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑_pm 𝑠)))
5		oveq2 7416	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = 𝑆 → (ℂ ↑_pm 𝑠) = (ℂ ↑_pm 𝑆))
6	5	opeliunxp2 5838	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑆, 𝐹⟩ ∈ ∪ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑_pm 𝑠)) ↔ (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)))
7	4, 6	sylib 217	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)))
8	7	simprd 496	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
9		cnex 11190	. . . . . . . . . . . . . . . . 17 ⊢ ℂ ∈ V
10	7	simpld 495	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → 𝑆 ∈ 𝒫 ℂ)
11		elpm2g 8837	. . . . . . . . . . . . . . . . 17 ⊢ ((ℂ ∈ V ∧ 𝑆 ∈ 𝒫 ℂ) → (𝐹 ∈ (ℂ ↑_pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆)))
12	9, 10, 11	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝐹 ∈ (ℂ ↑_pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆)))
13	8, 12	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆))
14	13	simpld 495	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → 𝐹:dom 𝐹⟶ℂ)
15	14	adantr 481	. . . . . . . . . . . . 13 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) → 𝐹:dom 𝐹⟶ℂ)
16	2	sseli 3978	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → ⟨𝑆, 𝐹⟩ ∈ ∪ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑_pm 𝑠)))
17	16, 6	sylib 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)))
18	17	simprd 496	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → 𝐹 ∈ (ℂ ↑_pm 𝑆))
19	17	simpld 495	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → 𝑆 ∈ 𝒫 ℂ)
20	9, 19, 11	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → (𝐹 ∈ (ℂ ↑_pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆)))
21	18, 20	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆))
22	21	simprd 496	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → dom 𝐹 ⊆ 𝑆)
23	22	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → dom 𝐹 ⊆ 𝑆)
24	10	elpwid 4611	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → 𝑆 ⊆ ℂ)
25	23, 24	sstrd 3992	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → dom 𝐹 ⊆ ℂ)
26	25	adantr 481	. . . . . . . . . . . . 13 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) → dom 𝐹 ⊆ ℂ)
27		perfdvf.1	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
28	27	cnfldtopon 24298	. . . . . . . . . . . . . . . . 17 ⊢ 𝐾 ∈ (TopOn‘ℂ)
29		resttopon 22664	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
30	28, 24, 29	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
31		topontop 22414	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆) → (𝐾 ↾_t 𝑆) ∈ Top)
32	30, 31	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝐾 ↾_t 𝑆) ∈ Top)
33		toponuni 22415	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ∪ (𝐾 ↾_t 𝑆))
34	30, 33	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → 𝑆 = ∪ (𝐾 ↾_t 𝑆))
35	23, 34	sseqtrd 4022	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → dom 𝐹 ⊆ ∪ (𝐾 ↾_t 𝑆))
36		eqid 2732	. . . . . . . . . . . . . . . 16 ⊢ ∪ (𝐾 ↾_t 𝑆) = ∪ (𝐾 ↾_t 𝑆)
37	36	ntrss2 22560	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ↾_t 𝑆) ∈ Top ∧ dom 𝐹 ⊆ ∪ (𝐾 ↾_t 𝑆)) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ dom 𝐹)
38	32, 35, 37	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ dom 𝐹)
39	38	sselda 3982	. . . . . . . . . . . . 13 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) → 𝑥 ∈ dom 𝐹)
40	15, 26, 39	dvlem 25412	. . . . . . . . . . . 12 ⊢ ((((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) ∧ 𝑧 ∈ (dom 𝐹 ∖ {𝑥})) → (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)) ∈ ℂ)
41	40	fmpttd 7114	. . . . . . . . . . 11 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) → (𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))):(dom 𝐹 ∖ {𝑥})⟶ℂ)
42	26	ssdifssd 4142	. . . . . . . . . . 11 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) → (dom 𝐹 ∖ {𝑥}) ⊆ ℂ)
43	36	ntrss3 22563	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ↾_t 𝑆) ∈ Top ∧ dom 𝐹 ⊆ ∪ (𝐾 ↾_t 𝑆)) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ ∪ (𝐾 ↾_t 𝑆))
44	32, 35, 43	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ ∪ (𝐾 ↾_t 𝑆))
45	44, 34	sseqtrrd 4023	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ 𝑆)
46		restabs 22668	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ 𝑆 ∧ 𝑆 ∈ 𝒫 ℂ) → ((𝐾 ↾_t 𝑆) ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) = (𝐾 ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)))
47	28, 45, 10, 46	mp3an2i 1466	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((𝐾 ↾_t 𝑆) ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) = (𝐾 ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)))
48		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝐾 ↾_t 𝑆) ∈ Perf)
49	36	ntropn 22552	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ↾_t 𝑆) ∈ Top ∧ dom 𝐹 ⊆ ∪ (𝐾 ↾_t 𝑆)) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ∈ (𝐾 ↾_t 𝑆))
50	32, 35, 49	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ∈ (𝐾 ↾_t 𝑆))
51		eqid 2732	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ↾_t 𝑆) ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) = ((𝐾 ↾_t 𝑆) ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹))
52	36, 51	perfopn 22688	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ↾_t 𝑆) ∈ Perf ∧ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ∈ (𝐾 ↾_t 𝑆)) → ((𝐾 ↾_t 𝑆) ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) ∈ Perf)
53	48, 50, 52	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((𝐾 ↾_t 𝑆) ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) ∈ Perf)
54	47, 53	eqeltrrd 2834	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝐾 ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) ∈ Perf)
55	27	cnfldtop 24299	. . . . . . . . . . . . . . . 16 ⊢ 𝐾 ∈ Top
56	45, 24	sstrd 3992	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ ℂ)
57	28	toponunii 22417	. . . . . . . . . . . . . . . . 17 ⊢ ℂ = ∪ 𝐾
58		eqid 2732	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) = (𝐾 ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹))
59	57, 58	restperf 22687	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Top ∧ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ ℂ) → ((𝐾 ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) ∈ Perf ↔ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ ((limPt‘𝐾)‘((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹))))
60	55, 56, 59	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((𝐾 ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) ∈ Perf ↔ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ ((limPt‘𝐾)‘((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹))))
61	54, 60	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ ((limPt‘𝐾)‘((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)))
62	57	lpss3 22647	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Top ∧ dom 𝐹 ⊆ ℂ ∧ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ dom 𝐹) → ((limPt‘𝐾)‘((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) ⊆ ((limPt‘𝐾)‘dom 𝐹))
63	55, 25, 38, 62	mp3an2i 1466	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((limPt‘𝐾)‘((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) ⊆ ((limPt‘𝐾)‘dom 𝐹))
64	61, 63	sstrd 3992	. . . . . . . . . . . . 13 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ ((limPt‘𝐾)‘dom 𝐹))
65	64	sselda 3982	. . . . . . . . . . . 12 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) → 𝑥 ∈ ((limPt‘𝐾)‘dom 𝐹))
66	57	lpdifsn 22646	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ dom 𝐹 ⊆ ℂ) → (𝑥 ∈ ((limPt‘𝐾)‘dom 𝐹) ↔ 𝑥 ∈ ((limPt‘𝐾)‘(dom 𝐹 ∖ {𝑥}))))
67	55, 26, 66	sylancr 587	. . . . . . . . . . . 12 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) → (𝑥 ∈ ((limPt‘𝐾)‘dom 𝐹) ↔ 𝑥 ∈ ((limPt‘𝐾)‘(dom 𝐹 ∖ {𝑥}))))
68	65, 67	mpbid 231	. . . . . . . . . . 11 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) → 𝑥 ∈ ((limPt‘𝐾)‘(dom 𝐹 ∖ {𝑥})))
69	41, 42, 68, 27	limcmo 25398	. . . . . . . . . 10 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) → ∃*𝑦 𝑦 ∈ ((𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥))
70	69	ex 413	. . . . . . . . 9 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) → ∃*𝑦 𝑦 ∈ ((𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥)))
71		moanimv 2615	. . . . . . . . 9 ⊢ (∃𝑦(𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ∧ 𝑦 ∈ ((𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥)) ↔ (𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) → ∃𝑦 𝑦 ∈ ((𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥)))
72	70, 71	sylibr 233	. . . . . . . 8 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ∃*𝑦(𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ∧ 𝑦 ∈ ((𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥)))
73		eqid 2732	. . . . . . . . . 10 ⊢ (𝐾 ↾_t 𝑆) = (𝐾 ↾_t 𝑆)
74		eqid 2732	. . . . . . . . . 10 ⊢ (𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) = (𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)))
75	73, 27, 74, 24, 14, 23	eldv 25414	. . . . . . . . 9 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝑥(𝑆 D 𝐹)𝑦 ↔ (𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ∧ 𝑦 ∈ ((𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥))))
76	75	mobidv 2543	. . . . . . . 8 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (∃𝑦 𝑥(𝑆 D 𝐹)𝑦 ↔ ∃𝑦(𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ∧ 𝑦 ∈ ((𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥))))
77	72, 76	mpbird 256	. . . . . . 7 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ∃*𝑦 𝑥(𝑆 D 𝐹)𝑦)
78	77	alrimiv 1930	. . . . . 6 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ∀𝑥∃*𝑦 𝑥(𝑆 D 𝐹)𝑦)
79		reldv 25386	. . . . . . 7 ⊢ Rel (𝑆 D 𝐹)
80		dffun6 6556	. . . . . . 7 ⊢ (Fun (𝑆 D 𝐹) ↔ (Rel (𝑆 D 𝐹) ∧ ∀𝑥∃*𝑦 𝑥(𝑆 D 𝐹)𝑦))
81	79, 80	mpbiran 707	. . . . . 6 ⊢ (Fun (𝑆 D 𝐹) ↔ ∀𝑥∃*𝑦 𝑥(𝑆 D 𝐹)𝑦)
82	78, 81	sylibr 233	. . . . 5 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → Fun (𝑆 D 𝐹))
83	82	funfnd 6579	. . . 4 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝑆 D 𝐹) Fn dom (𝑆 D 𝐹))
84		vex 3478	. . . . . . 7 ⊢ 𝑦 ∈ V
85	84	elrn 5893	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑆 D 𝐹) ↔ ∃𝑥 𝑥(𝑆 D 𝐹)𝑦)
86	24, 14, 23	dvcl 25415	. . . . . . . 8 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥(𝑆 D 𝐹)𝑦) → 𝑦 ∈ ℂ)
87	86	ex 413	. . . . . . 7 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝑥(𝑆 D 𝐹)𝑦 → 𝑦 ∈ ℂ))
88	87	exlimdv 1936	. . . . . 6 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (∃𝑥 𝑥(𝑆 D 𝐹)𝑦 → 𝑦 ∈ ℂ))
89	85, 88	biimtrid 241	. . . . 5 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝑦 ∈ ran (𝑆 D 𝐹) → 𝑦 ∈ ℂ))
90	89	ssrdv 3988	. . . 4 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ran (𝑆 D 𝐹) ⊆ ℂ)
91		df-f 6547	. . . 4 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ ((𝑆 D 𝐹) Fn dom (𝑆 D 𝐹) ∧ ran (𝑆 D 𝐹) ⊆ ℂ))
92	83, 90, 91	sylanbrc 583	. . 3 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
93	92	ex 413	. 2 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → ((𝐾 ↾_t 𝑆) ∈ Perf → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ))
94		f0 6772	. . . 4 ⊢ ∅:∅⟶ℂ
95		df-ov 7411	. . . . . 6 ⊢ (𝑆 D 𝐹) = ( D ‘⟨𝑆, 𝐹⟩)
96		ndmfv 6926	. . . . . 6 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → ( D ‘⟨𝑆, 𝐹⟩) = ∅)
97	95, 96	eqtrid 2784	. . . . 5 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → (𝑆 D 𝐹) = ∅)
98	97	dmeqd 5905	. . . . . 6 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) = dom ∅)
99		dm0 5920	. . . . . 6 ⊢ dom ∅ = ∅
100	98, 99	eqtrdi 2788	. . . . 5 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) = ∅)
101	97, 100	feq12d 6705	. . . 4 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ ∅:∅⟶ℂ))
102	94, 101	mpbiri 257	. . 3 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
103	102	a1d 25	. 2 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → ((𝐾 ↾_t 𝑆) ∈ Perf → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ))
104	93, 103	pm2.61i 182	1 ⊢ ((𝐾 ↾_t 𝑆) ∈ Perf → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∃*wmo 2532 Vcvv 3474 ∖ cdif 3945 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {csn 4628 ⟨cop 4634 ∪ cuni 4908 ∪ ciun 4997 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 dom cdm 5676 ran crn 5677 Rel wrel 5681 Fun wfun 6537 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ↑_pm cpm 8820 ℂcc 11107 − cmin 11443 / cdiv 11870 ↾_t crest 17365 TopOpenctopn 17366 ℂ_fldccnfld 20943 Topctop 22394 TopOnctopon 22411 intcnt 22520 limPtclp 22637 Perfcperf 22638 lim_ℂ climc 25378 D cdv 25379

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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-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-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-inf 9437 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-icc 13330 df-fz 13484 df-seq 13966 df-exp 14027 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-struct 17079 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-rest 17367 df-topn 17368 df-topgen 17388 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-lp 22639 df-perf 22640 df-cnp 22731 df-haus 22818 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-xms 23825 df-ms 23826 df-limc 25382 df-dv 25383

This theorem is referenced by: dvfg 25422

W3C validator