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Theorem perfdvf 23418
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‘ℂfld)
Assertion
Ref Expression
perfdvf ((𝐾t 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.
StepHypRef Expression
1 df-dv 23382 . . . . . . . . . . . . . . . . . . . 20 D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)))
21dmmpt2ssx 7102 . . . . . . . . . . . . . . . . . . 19 dom D ⊆ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑pm 𝑠))
3 simpl 471 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ⟨𝑆, 𝐹⟩ ∈ dom D )
42, 3sseldi 3565 . . . . . . . . . . . . . . . . . 18 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ⟨𝑆, 𝐹⟩ ∈ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑pm 𝑠)))
5 oveq2 6535 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑆 → (ℂ ↑pm 𝑠) = (ℂ ↑pm 𝑆))
65opeliunxp2 5170 . . . . . . . . . . . . . . . . . 18 (⟨𝑆, 𝐹⟩ ∈ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑pm 𝑠)) ↔ (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)))
74, 6sylib 206 . . . . . . . . . . . . . . . . 17 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)))
87simprd 477 . . . . . . . . . . . . . . . 16 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → 𝐹 ∈ (ℂ ↑pm 𝑆))
9 cnex 9874 . . . . . . . . . . . . . . . . 17 ℂ ∈ V
107simpld 473 . . . . . . . . . . . . . . . . 17 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → 𝑆 ∈ 𝒫 ℂ)
11 elpm2g 7738 . . . . . . . . . . . . . . . . 17 ((ℂ ∈ V ∧ 𝑆 ∈ 𝒫 ℂ) → (𝐹 ∈ (ℂ ↑pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹𝑆)))
129, 10, 11sylancr 693 . . . . . . . . . . . . . . . 16 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝐹 ∈ (ℂ ↑pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹𝑆)))
138, 12mpbid 220 . . . . . . . . . . . . . . 15 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹𝑆))
1413simpld 473 . . . . . . . . . . . . . 14 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → 𝐹:dom 𝐹⟶ℂ)
1514adantr 479 . . . . . . . . . . . . 13 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) → 𝐹:dom 𝐹⟶ℂ)
162sseli 3563 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑆, 𝐹⟩ ∈ dom D → ⟨𝑆, 𝐹⟩ ∈ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑pm 𝑠)))
1716, 6sylib 206 . . . . . . . . . . . . . . . . . . 19 (⟨𝑆, 𝐹⟩ ∈ dom D → (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)))
1817simprd 477 . . . . . . . . . . . . . . . . . 18 (⟨𝑆, 𝐹⟩ ∈ dom D → 𝐹 ∈ (ℂ ↑pm 𝑆))
1917simpld 473 . . . . . . . . . . . . . . . . . . 19 (⟨𝑆, 𝐹⟩ ∈ dom D → 𝑆 ∈ 𝒫 ℂ)
209, 19, 11sylancr 693 . . . . . . . . . . . . . . . . . 18 (⟨𝑆, 𝐹⟩ ∈ dom D → (𝐹 ∈ (ℂ ↑pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹𝑆)))
2118, 20mpbid 220 . . . . . . . . . . . . . . . . 17 (⟨𝑆, 𝐹⟩ ∈ dom D → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹𝑆))
2221simprd 477 . . . . . . . . . . . . . . . 16 (⟨𝑆, 𝐹⟩ ∈ dom D → dom 𝐹𝑆)
2322adantr 479 . . . . . . . . . . . . . . 15 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → dom 𝐹𝑆)
2410elpwid 4117 . . . . . . . . . . . . . . 15 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → 𝑆 ⊆ ℂ)
2523, 24sstrd 3577 . . . . . . . . . . . . . 14 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → dom 𝐹 ⊆ ℂ)
2625adantr 479 . . . . . . . . . . . . 13 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) → dom 𝐹 ⊆ ℂ)
27 perfdvf.1 . . . . . . . . . . . . . . . . . 18 𝐾 = (TopOpen‘ℂfld)
2827cnfldtopon 22344 . . . . . . . . . . . . . . . . 17 𝐾 ∈ (TopOn‘ℂ)
29 resttopon 20723 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾t 𝑆) ∈ (TopOn‘𝑆))
3028, 24, 29sylancr 693 . . . . . . . . . . . . . . . 16 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝐾t 𝑆) ∈ (TopOn‘𝑆))
31 topontop 20489 . . . . . . . . . . . . . . . 16 ((𝐾t 𝑆) ∈ (TopOn‘𝑆) → (𝐾t 𝑆) ∈ Top)
3230, 31syl 17 . . . . . . . . . . . . . . 15 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝐾t 𝑆) ∈ Top)
33 toponuni 20490 . . . . . . . . . . . . . . . . 17 ((𝐾t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = (𝐾t 𝑆))
3430, 33syl 17 . . . . . . . . . . . . . . . 16 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → 𝑆 = (𝐾t 𝑆))
3523, 34sseqtrd 3603 . . . . . . . . . . . . . . 15 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → dom 𝐹 (𝐾t 𝑆))
36 eqid 2609 . . . . . . . . . . . . . . . 16 (𝐾t 𝑆) = (𝐾t 𝑆)
3736ntrss2 20619 . . . . . . . . . . . . . . 15 (((𝐾t 𝑆) ∈ Top ∧ dom 𝐹 (𝐾t 𝑆)) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ dom 𝐹)
3832, 35, 37syl2anc 690 . . . . . . . . . . . . . 14 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ dom 𝐹)
3938sselda 3567 . . . . . . . . . . . . 13 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) → 𝑥 ∈ dom 𝐹)
4015, 26, 39dvlem 23411 . . . . . . . . . . . 12 ((((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) ∧ 𝑧 ∈ (dom 𝐹 ∖ {𝑥})) → (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥)) ∈ ℂ)
41 eqid 2609 . . . . . . . . . . . 12 (𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥))) = (𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥)))
4240, 41fmptd 6277 . . . . . . . . . . 11 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) → (𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥))):(dom 𝐹 ∖ {𝑥})⟶ℂ)
4326ssdifssd 3709 . . . . . . . . . . 11 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) → (dom 𝐹 ∖ {𝑥}) ⊆ ℂ)
4428a1i 11 . . . . . . . . . . . . . . . . 17 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → 𝐾 ∈ (TopOn‘ℂ))
4536ntrss3 20622 . . . . . . . . . . . . . . . . . . 19 (((𝐾t 𝑆) ∈ Top ∧ dom 𝐹 (𝐾t 𝑆)) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ (𝐾t 𝑆))
4632, 35, 45syl2anc 690 . . . . . . . . . . . . . . . . . 18 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ (𝐾t 𝑆))
4746, 34sseqtr4d 3604 . . . . . . . . . . . . . . . . 17 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ 𝑆)
48 restabs 20727 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ (TopOn‘ℂ) ∧ ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ 𝑆𝑆 ∈ 𝒫 ℂ) → ((𝐾t 𝑆) ↾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) = (𝐾t ((int‘(𝐾t 𝑆))‘dom 𝐹)))
4944, 47, 10, 48syl3anc 1317 . . . . . . . . . . . . . . . 16 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((𝐾t 𝑆) ↾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) = (𝐾t ((int‘(𝐾t 𝑆))‘dom 𝐹)))
50 simpr 475 . . . . . . . . . . . . . . . . 17 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝐾t 𝑆) ∈ Perf)
5136ntropn 20611 . . . . . . . . . . . . . . . . . 18 (((𝐾t 𝑆) ∈ Top ∧ dom 𝐹 (𝐾t 𝑆)) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ∈ (𝐾t 𝑆))
5232, 35, 51syl2anc 690 . . . . . . . . . . . . . . . . 17 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ∈ (𝐾t 𝑆))
53 eqid 2609 . . . . . . . . . . . . . . . . . 18 ((𝐾t 𝑆) ↾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) = ((𝐾t 𝑆) ↾t ((int‘(𝐾t 𝑆))‘dom 𝐹))
5436, 53perfopn 20747 . . . . . . . . . . . . . . . . 17 (((𝐾t 𝑆) ∈ Perf ∧ ((int‘(𝐾t 𝑆))‘dom 𝐹) ∈ (𝐾t 𝑆)) → ((𝐾t 𝑆) ↾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) ∈ Perf)
5550, 52, 54syl2anc 690 . . . . . . . . . . . . . . . 16 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((𝐾t 𝑆) ↾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) ∈ Perf)
5649, 55eqeltrrd 2688 . . . . . . . . . . . . . . 15 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝐾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) ∈ Perf)
5727cnfldtop 22345 . . . . . . . . . . . . . . . 16 𝐾 ∈ Top
5847, 24sstrd 3577 . . . . . . . . . . . . . . . 16 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ ℂ)
5928toponunii 20495 . . . . . . . . . . . . . . . . 17 ℂ = 𝐾
60 eqid 2609 . . . . . . . . . . . . . . . . 17 (𝐾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) = (𝐾t ((int‘(𝐾t 𝑆))‘dom 𝐹))
6159, 60restperf 20746 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Top ∧ ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ ℂ) → ((𝐾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) ∈ Perf ↔ ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ ((limPt‘𝐾)‘((int‘(𝐾t 𝑆))‘dom 𝐹))))
6257, 58, 61sylancr 693 . . . . . . . . . . . . . . 15 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((𝐾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) ∈ Perf ↔ ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ ((limPt‘𝐾)‘((int‘(𝐾t 𝑆))‘dom 𝐹))))
6356, 62mpbid 220 . . . . . . . . . . . . . 14 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ ((limPt‘𝐾)‘((int‘(𝐾t 𝑆))‘dom 𝐹)))
6457a1i 11 . . . . . . . . . . . . . . 15 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → 𝐾 ∈ Top)
6559lpss3 20706 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Top ∧ dom 𝐹 ⊆ ℂ ∧ ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ dom 𝐹) → ((limPt‘𝐾)‘((int‘(𝐾t 𝑆))‘dom 𝐹)) ⊆ ((limPt‘𝐾)‘dom 𝐹))
6664, 25, 38, 65syl3anc 1317 . . . . . . . . . . . . . 14 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((limPt‘𝐾)‘((int‘(𝐾t 𝑆))‘dom 𝐹)) ⊆ ((limPt‘𝐾)‘dom 𝐹))
6763, 66sstrd 3577 . . . . . . . . . . . . 13 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ ((limPt‘𝐾)‘dom 𝐹))
6867sselda 3567 . . . . . . . . . . . 12 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) → 𝑥 ∈ ((limPt‘𝐾)‘dom 𝐹))
6959lpdifsn 20705 . . . . . . . . . . . . 13 ((𝐾 ∈ Top ∧ dom 𝐹 ⊆ ℂ) → (𝑥 ∈ ((limPt‘𝐾)‘dom 𝐹) ↔ 𝑥 ∈ ((limPt‘𝐾)‘(dom 𝐹 ∖ {𝑥}))))
7057, 26, 69sylancr 693 . . . . . . . . . . . 12 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) → (𝑥 ∈ ((limPt‘𝐾)‘dom 𝐹) ↔ 𝑥 ∈ ((limPt‘𝐾)‘(dom 𝐹 ∖ {𝑥}))))
7168, 70mpbid 220 . . . . . . . . . . 11 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) → 𝑥 ∈ ((limPt‘𝐾)‘(dom 𝐹 ∖ {𝑥})))
7242, 43, 71, 27limcmo 23397 . . . . . . . . . 10 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) → ∃*𝑦 𝑦 ∈ ((𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥))) lim 𝑥))
7372ex 448 . . . . . . . . 9 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹) → ∃*𝑦 𝑦 ∈ ((𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥))) lim 𝑥)))
74 moanimv 2518 . . . . . . . . 9 (∃*𝑦(𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹) ∧ 𝑦 ∈ ((𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥))) lim 𝑥)) ↔ (𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹) → ∃*𝑦 𝑦 ∈ ((𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥))) lim 𝑥)))
7573, 74sylibr 222 . . . . . . . 8 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ∃*𝑦(𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹) ∧ 𝑦 ∈ ((𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥))) lim 𝑥)))
76 eqid 2609 . . . . . . . . . 10 (𝐾t 𝑆) = (𝐾t 𝑆)
7776, 27, 41, 24, 14, 23eldv 23413 . . . . . . . . 9 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝑥(𝑆 D 𝐹)𝑦 ↔ (𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹) ∧ 𝑦 ∈ ((𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥))) lim 𝑥))))
7877mobidv 2478 . . . . . . . 8 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (∃*𝑦 𝑥(𝑆 D 𝐹)𝑦 ↔ ∃*𝑦(𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹) ∧ 𝑦 ∈ ((𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥))) lim 𝑥))))
7975, 78mpbird 245 . . . . . . 7 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ∃*𝑦 𝑥(𝑆 D 𝐹)𝑦)
8079alrimiv 1841 . . . . . 6 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ∀𝑥∃*𝑦 𝑥(𝑆 D 𝐹)𝑦)
81 reldv 23385 . . . . . . 7 Rel (𝑆 D 𝐹)
82 dffun6 5805 . . . . . . 7 (Fun (𝑆 D 𝐹) ↔ (Rel (𝑆 D 𝐹) ∧ ∀𝑥∃*𝑦 𝑥(𝑆 D 𝐹)𝑦))
8381, 82mpbiran 954 . . . . . 6 (Fun (𝑆 D 𝐹) ↔ ∀𝑥∃*𝑦 𝑥(𝑆 D 𝐹)𝑦)
8480, 83sylibr 222 . . . . 5 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → Fun (𝑆 D 𝐹))
85 funfn 5819 . . . . 5 (Fun (𝑆 D 𝐹) ↔ (𝑆 D 𝐹) Fn dom (𝑆 D 𝐹))
8684, 85sylib 206 . . . 4 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝑆 D 𝐹) Fn dom (𝑆 D 𝐹))
87 vex 3175 . . . . . . 7 𝑦 ∈ V
8887elrn 5274 . . . . . 6 (𝑦 ∈ ran (𝑆 D 𝐹) ↔ ∃𝑥 𝑥(𝑆 D 𝐹)𝑦)
8924, 14, 23dvcl 23414 . . . . . . . 8 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥(𝑆 D 𝐹)𝑦) → 𝑦 ∈ ℂ)
9089ex 448 . . . . . . 7 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝑥(𝑆 D 𝐹)𝑦𝑦 ∈ ℂ))
9190exlimdv 1847 . . . . . 6 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (∃𝑥 𝑥(𝑆 D 𝐹)𝑦𝑦 ∈ ℂ))
9288, 91syl5bi 230 . . . . 5 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝑦 ∈ ran (𝑆 D 𝐹) → 𝑦 ∈ ℂ))
9392ssrdv 3573 . . . 4 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ran (𝑆 D 𝐹) ⊆ ℂ)
94 df-f 5794 . . . 4 ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ ((𝑆 D 𝐹) Fn dom (𝑆 D 𝐹) ∧ ran (𝑆 D 𝐹) ⊆ ℂ))
9586, 93, 94sylanbrc 694 . . 3 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
9695ex 448 . 2 (⟨𝑆, 𝐹⟩ ∈ dom D → ((𝐾t 𝑆) ∈ Perf → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ))
97 f0 5984 . . . 4 ∅:∅⟶ℂ
98 df-ov 6530 . . . . . 6 (𝑆 D 𝐹) = ( D ‘⟨𝑆, 𝐹⟩)
99 ndmfv 6113 . . . . . 6 (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → ( D ‘⟨𝑆, 𝐹⟩) = ∅)
10098, 99syl5eq 2655 . . . . 5 (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → (𝑆 D 𝐹) = ∅)
101100dmeqd 5235 . . . . . 6 (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) = dom ∅)
102 dm0 5247 . . . . . 6 dom ∅ = ∅
103101, 102syl6eq 2659 . . . . 5 (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) = ∅)
104100, 103feq12d 5932 . . . 4 (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ ∅:∅⟶ℂ))
10597, 104mpbiri 246 . . 3 (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
106105a1d 25 . 2 (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → ((𝐾t 𝑆) ∈ Perf → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ))
10796, 106pm2.61i 174 1 ((𝐾t 𝑆) ∈ Perf → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wex 1694  wcel 1976  ∃*wmo 2458  Vcvv 3172  cdif 3536  wss 3539  c0 3873  𝒫 cpw 4107  {csn 4124  cop 4130   cuni 4366   ciun 4449   class class class wbr 4577  cmpt 4637   × cxp 5026  dom cdm 5028  ran crn 5029  Rel wrel 5033  Fun wfun 5784   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  pm cpm 7723  cc 9791  cmin 10118   / cdiv 10536  t crest 15853  TopOpenctopn 15854  fldccnfld 19516  Topctop 20465  TopOnctopon 20466  intcnt 20579  limPtclp 20696  Perfcperf 20697   lim climc 23377   D cdv 23378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-map 7724  df-pm 7725  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-fi 8178  df-sup 8209  df-inf 8210  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-3 10930  df-4 10931  df-5 10932  df-6 10933  df-7 10934  df-8 10935  df-9 10936  df-n0 11143  df-z 11214  df-dec 11329  df-uz 11523  df-q 11624  df-rp 11668  df-xneg 11781  df-xadd 11782  df-xmul 11783  df-icc 12012  df-fz 12156  df-seq 12622  df-exp 12681  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-struct 15646  df-ndx 15647  df-slot 15648  df-base 15649  df-plusg 15730  df-mulr 15731  df-starv 15732  df-tset 15736  df-ple 15737  df-ds 15740  df-unif 15741  df-rest 15855  df-topn 15856  df-topgen 15876  df-psmet 19508  df-xmet 19509  df-met 19510  df-bl 19511  df-mopn 19512  df-fbas 19513  df-fg 19514  df-cnfld 19517  df-top 20469  df-bases 20470  df-topon 20471  df-topsp 20472  df-cld 20581  df-ntr 20582  df-cls 20583  df-nei 20660  df-lp 20698  df-perf 20699  df-cnp 20790  df-haus 20877  df-fil 21408  df-fm 21500  df-flim 21501  df-flf 21502  df-xms 21883  df-ms 21884  df-limc 23381  df-dv 23382
This theorem is referenced by:  dvfg  23421
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