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Theorem perfdvf 25420

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 25384	. . . . . . . . . . . . . . . . . . . 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 8052	. . . . . . . . . . . . . . . . . . 19 ⊢ dom D ⊆ ∪ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑_pm 𝑠))
3		simpl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ⟨𝑆, 𝐹⟩ ∈ dom D )
4	2, 3	sselid 3981	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ⟨𝑆, 𝐹⟩ ∈ ∪ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑_pm 𝑠)))
5		oveq2 7417	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = 𝑆 → (ℂ ↑_pm 𝑠) = (ℂ ↑_pm 𝑆))
6	5	opeliunxp2 5839	. . . . . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
9		cnex 11191	. . . . . . . . . . . . . . . . 17 ⊢ ℂ ∈ V
10	7	simpld 496	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → 𝑆 ∈ 𝒫 ℂ)
11		elpm2g 8838	. . . . . . . . . . . . . . . . 17 ⊢ ((ℂ ∈ V ∧ 𝑆 ∈ 𝒫 ℂ) → (𝐹 ∈ (ℂ ↑_pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆)))
12	9, 10, 11	sylancr 588	. . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → 𝐹:dom 𝐹⟶ℂ)
15	14	adantr 482	. . . . . . . . . . . . 13 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) → 𝐹:dom 𝐹⟶ℂ)
16	2	sseli 3979	. . . . . . . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → 𝐹 ∈ (ℂ ↑_pm 𝑆))
19	17	simpld 496	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → 𝑆 ∈ 𝒫 ℂ)
20	9, 19, 11	sylancr 588	. . . . . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → dom 𝐹 ⊆ 𝑆)
23	22	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → dom 𝐹 ⊆ 𝑆)
24	10	elpwid 4612	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → 𝑆 ⊆ ℂ)
25	23, 24	sstrd 3993	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → dom 𝐹 ⊆ ℂ)
26	25	adantr 482	. . . . . . . . . . . . 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 24299	. . . . . . . . . . . . . . . . 17 ⊢ 𝐾 ∈ (TopOn‘ℂ)
29		resttopon 22665	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
30	28, 24, 29	sylancr 588	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
31		topontop 22415	. . . . . . . . . . . . . . . 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 22416	. . . . . . . . . . . . . . . . 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 4023	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → dom 𝐹 ⊆ ∪ (𝐾 ↾_t 𝑆))
36		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ ∪ (𝐾 ↾_t 𝑆) = ∪ (𝐾 ↾_t 𝑆)
37	36	ntrss2 22561	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ↾_t 𝑆) ∈ Top ∧ dom 𝐹 ⊆ ∪ (𝐾 ↾_t 𝑆)) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ dom 𝐹)
38	32, 35, 37	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ dom 𝐹)
39	38	sselda 3983	. . . . . . . . . . . . 13 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) → 𝑥 ∈ dom 𝐹)
40	15, 26, 39	dvlem 25413	. . . . . . . . . . . 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 7115	. . . . . . . . . . 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 4143	. . . . . . . . . . 11 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) → (dom 𝐹 ∖ {𝑥}) ⊆ ℂ)
43	36	ntrss3 22564	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ↾_t 𝑆) ∈ Top ∧ dom 𝐹 ⊆ ∪ (𝐾 ↾_t 𝑆)) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ ∪ (𝐾 ↾_t 𝑆))
44	32, 35, 43	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ ∪ (𝐾 ↾_t 𝑆))
45	44, 34	sseqtrrd 4024	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ 𝑆)
46		restabs 22669	. . . . . . . . . . . . . . . . 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 1467	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝐾 ↾_t 𝑆) ∈ Perf)
49	36	ntropn 22553	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ↾_t 𝑆) ∈ Top ∧ dom 𝐹 ⊆ ∪ (𝐾 ↾_t 𝑆)) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ∈ (𝐾 ↾_t 𝑆))
50	32, 35, 49	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ∈ (𝐾 ↾_t 𝑆))
51		eqid 2733	. . . . . . . . . . . . . . . . . 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 22689	. . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((𝐾 ↾_t 𝑆) ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) ∈ Perf)
54	47, 53	eqeltrrd 2835	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝐾 ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) ∈ Perf)
55	27	cnfldtop 24300	. . . . . . . . . . . . . . . 16 ⊢ 𝐾 ∈ Top
56	45, 24	sstrd 3993	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ ℂ)
57	28	toponunii 22418	. . . . . . . . . . . . . . . . 17 ⊢ ℂ = ∪ 𝐾
58		eqid 2733	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) = (𝐾 ↾_t ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹))
59	57, 58	restperf 22688	. . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . 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 22648	. . . . . . . . . . . . . . 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 1467	. . . . . . . . . . . . . 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 3993	. . . . . . . . . . . . 13 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹) ⊆ ((limPt‘𝐾)‘dom 𝐹))
65	64	sselda 3983	. . . . . . . . . . . 12 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾 ↾_t 𝑆))‘dom 𝐹)) → 𝑥 ∈ ((limPt‘𝐾)‘dom 𝐹))
66	57	lpdifsn 22647	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ dom 𝐹 ⊆ ℂ) → (𝑥 ∈ ((limPt‘𝐾)‘dom 𝐹) ↔ 𝑥 ∈ ((limPt‘𝐾)‘(dom 𝐹 ∖ {𝑥}))))
67	55, 26, 66	sylancr 588	. . . . . . . . . . . 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 25399	. . . . . . . . . 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 414	. . . . . . . . 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 2616	. . . . . . . . 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 2733	. . . . . . . . . 10 ⊢ (𝐾 ↾_t 𝑆) = (𝐾 ↾_t 𝑆)
74		eqid 2733	. . . . . . . . . 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 25415	. . . . . . . . 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 2544	. . . . . . . 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 257	. . . . . . 7 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ∃*𝑦 𝑥(𝑆 D 𝐹)𝑦)
78	77	alrimiv 1931	. . . . . 6 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ∀𝑥∃*𝑦 𝑥(𝑆 D 𝐹)𝑦)
79		reldv 25387	. . . . . . 7 ⊢ Rel (𝑆 D 𝐹)
80		dffun6 6557	. . . . . . 7 ⊢ (Fun (𝑆 D 𝐹) ↔ (Rel (𝑆 D 𝐹) ∧ ∀𝑥∃*𝑦 𝑥(𝑆 D 𝐹)𝑦))
81	79, 80	mpbiran 708	. . . . . 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 6580	. . . 4 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝑆 D 𝐹) Fn dom (𝑆 D 𝐹))
84		vex 3479	. . . . . . 7 ⊢ 𝑦 ∈ V
85	84	elrn 5894	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑆 D 𝐹) ↔ ∃𝑥 𝑥(𝑆 D 𝐹)𝑦)
86	24, 14, 23	dvcl 25416	. . . . . . . 8 ⊢ (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) ∧ 𝑥(𝑆 D 𝐹)𝑦) → 𝑦 ∈ ℂ)
87	86	ex 414	. . . . . . 7 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝑥(𝑆 D 𝐹)𝑦 → 𝑦 ∈ ℂ))
88	87	exlimdv 1937	. . . . . 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 3989	. . . 4 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → ran (𝑆 D 𝐹) ⊆ ℂ)
91		df-f 6548	. . . 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 584	. . 3 ⊢ ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾 ↾_t 𝑆) ∈ Perf) → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
93	92	ex 414	. 2 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → ((𝐾 ↾_t 𝑆) ∈ Perf → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ))
94		f0 6773	. . . 4 ⊢ ∅:∅⟶ℂ
95		df-ov 7412	. . . . . 6 ⊢ (𝑆 D 𝐹) = ( D ‘⟨𝑆, 𝐹⟩)
96		ndmfv 6927	. . . . . 6 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → ( D ‘⟨𝑆, 𝐹⟩) = ∅)
97	95, 96	eqtrid 2785	. . . . 5 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → (𝑆 D 𝐹) = ∅)
98	97	dmeqd 5906	. . . . . 6 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) = dom ∅)
99		dm0 5921	. . . . . 6 ⊢ dom ∅ = ∅
100	98, 99	eqtrdi 2789	. . . . 5 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) = ∅)
101	97, 100	feq12d 6706	. . . 4 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ ∅:∅⟶ℂ))
102	94, 101	mpbiri 258	. . 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 397 ∀wal 1540 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃*wmo 2533 Vcvv 3475 ∖ cdif 3946 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 {csn 4629 ⟨cop 4635 ∪ cuni 4909 ∪ ciun 4998 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 dom cdm 5677 ran crn 5678 Rel wrel 5682 Fun wfun 6538 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ↑_pm cpm 8821 ℂcc 11108 − cmin 11444 / cdiv 11871 ↾_t crest 17366 TopOpenctopn 17367 ℂ_fldccnfld 20944 Topctop 22395 TopOnctopon 22412 intcnt 22521 limPtclp 22638 Perfcperf 22639 lim_ℂ climc 25379 D cdv 25380

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-icc 13331 df-fz 13485 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-struct 17080 df-slot 17115 df-ndx 17127 df-base 17145 df-plusg 17210 df-mulr 17211 df-starv 17212 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-rest 17368 df-topn 17369 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cnp 22732 df-haus 22819 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-limc 25383 df-dv 25384

This theorem is referenced by: dvfg 25423

W3C validator