Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  perfdvf Structured version   Visualization version   GIF version

Theorem perfdvf 23473
 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 23437 . . . . . . . . . . . . . . . . . . . 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 7124 . . . . . . . . . . . . . . . . . . 19 dom D ⊆ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑pm 𝑠))
3 simpl 472 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ⟨𝑆, 𝐹⟩ ∈ dom D )
42, 3sseldi 3566 . . . . . . . . . . . . . . . . . 18 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ⟨𝑆, 𝐹⟩ ∈ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑pm 𝑠)))
5 oveq2 6557 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑆 → (ℂ ↑pm 𝑠) = (ℂ ↑pm 𝑆))
65opeliunxp2 5182 . . . . . . . . . . . . . . . . . 18 (⟨𝑆, 𝐹⟩ ∈ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑pm 𝑠)) ↔ (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)))
74, 6sylib 207 . . . . . . . . . . . . . . . . 17 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)))
87simprd 478 . . . . . . . . . . . . . . . 16 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → 𝐹 ∈ (ℂ ↑pm 𝑆))
9 cnex 9896 . . . . . . . . . . . . . . . . 17 ℂ ∈ V
107simpld 474 . . . . . . . . . . . . . . . . 17 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → 𝑆 ∈ 𝒫 ℂ)
11 elpm2g 7760 . . . . . . . . . . . . . . . . 17 ((ℂ ∈ V ∧ 𝑆 ∈ 𝒫 ℂ) → (𝐹 ∈ (ℂ ↑pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹𝑆)))
129, 10, 11sylancr 694 . . . . . . . . . . . . . . . 16 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝐹 ∈ (ℂ ↑pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹𝑆)))
138, 12mpbid 221 . . . . . . . . . . . . . . 15 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹𝑆))
1413simpld 474 . . . . . . . . . . . . . 14 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → 𝐹:dom 𝐹⟶ℂ)
1514adantr 480 . . . . . . . . . . . . 13 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) → 𝐹:dom 𝐹⟶ℂ)
162sseli 3564 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑆, 𝐹⟩ ∈ dom D → ⟨𝑆, 𝐹⟩ ∈ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑pm 𝑠)))
1716, 6sylib 207 . . . . . . . . . . . . . . . . . . 19 (⟨𝑆, 𝐹⟩ ∈ dom D → (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)))
1817simprd 478 . . . . . . . . . . . . . . . . . 18 (⟨𝑆, 𝐹⟩ ∈ dom D → 𝐹 ∈ (ℂ ↑pm 𝑆))
1917simpld 474 . . . . . . . . . . . . . . . . . . 19 (⟨𝑆, 𝐹⟩ ∈ dom D → 𝑆 ∈ 𝒫 ℂ)
209, 19, 11sylancr 694 . . . . . . . . . . . . . . . . . 18 (⟨𝑆, 𝐹⟩ ∈ dom D → (𝐹 ∈ (ℂ ↑pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹𝑆)))
2118, 20mpbid 221 . . . . . . . . . . . . . . . . 17 (⟨𝑆, 𝐹⟩ ∈ dom D → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹𝑆))
2221simprd 478 . . . . . . . . . . . . . . . 16 (⟨𝑆, 𝐹⟩ ∈ dom D → dom 𝐹𝑆)
2322adantr 480 . . . . . . . . . . . . . . 15 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → dom 𝐹𝑆)
2410elpwid 4118 . . . . . . . . . . . . . . 15 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → 𝑆 ⊆ ℂ)
2523, 24sstrd 3578 . . . . . . . . . . . . . 14 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → dom 𝐹 ⊆ ℂ)
2625adantr 480 . . . . . . . . . . . . 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 22396 . . . . . . . . . . . . . . . . 17 𝐾 ∈ (TopOn‘ℂ)
29 resttopon 20775 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾t 𝑆) ∈ (TopOn‘𝑆))
3028, 24, 29sylancr 694 . . . . . . . . . . . . . . . 16 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝐾t 𝑆) ∈ (TopOn‘𝑆))
31 topontop 20541 . . . . . . . . . . . . . . . 16 ((𝐾t 𝑆) ∈ (TopOn‘𝑆) → (𝐾t 𝑆) ∈ Top)
3230, 31syl 17 . . . . . . . . . . . . . . 15 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝐾t 𝑆) ∈ Top)
33 toponuni 20542 . . . . . . . . . . . . . . . . 17 ((𝐾t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = (𝐾t 𝑆))
3430, 33syl 17 . . . . . . . . . . . . . . . 16 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → 𝑆 = (𝐾t 𝑆))
3523, 34sseqtrd 3604 . . . . . . . . . . . . . . 15 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → dom 𝐹 (𝐾t 𝑆))
36 eqid 2610 . . . . . . . . . . . . . . . 16 (𝐾t 𝑆) = (𝐾t 𝑆)
3736ntrss2 20671 . . . . . . . . . . . . . . 15 (((𝐾t 𝑆) ∈ Top ∧ dom 𝐹 (𝐾t 𝑆)) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ dom 𝐹)
3832, 35, 37syl2anc 691 . . . . . . . . . . . . . 14 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ dom 𝐹)
3938sselda 3568 . . . . . . . . . . . . 13 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) → 𝑥 ∈ dom 𝐹)
4015, 26, 39dvlem 23466 . . . . . . . . . . . 12 ((((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) ∧ 𝑧 ∈ (dom 𝐹 ∖ {𝑥})) → (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥)) ∈ ℂ)
41 eqid 2610 . . . . . . . . . . . 12 (𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥))) = (𝑧 ∈ (dom 𝐹 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥)))
4240, 41fmptd 6292 . . . . . . . . . . 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 3710 . . . . . . . . . . 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 20674 . . . . . . . . . . . . . . . . . . 19 (((𝐾t 𝑆) ∈ Top ∧ dom 𝐹 (𝐾t 𝑆)) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ (𝐾t 𝑆))
4632, 35, 45syl2anc 691 . . . . . . . . . . . . . . . . . 18 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ (𝐾t 𝑆))
4746, 34sseqtr4d 3605 . . . . . . . . . . . . . . . . 17 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ 𝑆)
48 restabs 20779 . . . . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . . . . 16 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((𝐾t 𝑆) ↾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) = (𝐾t ((int‘(𝐾t 𝑆))‘dom 𝐹)))
50 simpr 476 . . . . . . . . . . . . . . . . 17 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝐾t 𝑆) ∈ Perf)
5136ntropn 20663 . . . . . . . . . . . . . . . . . 18 (((𝐾t 𝑆) ∈ Top ∧ dom 𝐹 (𝐾t 𝑆)) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ∈ (𝐾t 𝑆))
5232, 35, 51syl2anc 691 . . . . . . . . . . . . . . . . 17 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ∈ (𝐾t 𝑆))
53 eqid 2610 . . . . . . . . . . . . . . . . . 18 ((𝐾t 𝑆) ↾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) = ((𝐾t 𝑆) ↾t ((int‘(𝐾t 𝑆))‘dom 𝐹))
5436, 53perfopn 20799 . . . . . . . . . . . . . . . . 17 (((𝐾t 𝑆) ∈ Perf ∧ ((int‘(𝐾t 𝑆))‘dom 𝐹) ∈ (𝐾t 𝑆)) → ((𝐾t 𝑆) ↾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) ∈ Perf)
5550, 52, 54syl2anc 691 . . . . . . . . . . . . . . . 16 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((𝐾t 𝑆) ↾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) ∈ Perf)
5649, 55eqeltrrd 2689 . . . . . . . . . . . . . . 15 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝐾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) ∈ Perf)
5727cnfldtop 22397 . . . . . . . . . . . . . . . 16 𝐾 ∈ Top
5847, 24sstrd 3578 . . . . . . . . . . . . . . . 16 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ ℂ)
5928toponunii 20547 . . . . . . . . . . . . . . . . 17 ℂ = 𝐾
60 eqid 2610 . . . . . . . . . . . . . . . . 17 (𝐾t ((int‘(𝐾t 𝑆))‘dom 𝐹)) = (𝐾t ((int‘(𝐾t 𝑆))‘dom 𝐹))
6159, 60restperf 20798 . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . . 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 20758 . . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . . 14 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((limPt‘𝐾)‘((int‘(𝐾t 𝑆))‘dom 𝐹)) ⊆ ((limPt‘𝐾)‘dom 𝐹))
6763, 66sstrd 3578 . . . . . . . . . . . . 13 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ((int‘(𝐾t 𝑆))‘dom 𝐹) ⊆ ((limPt‘𝐾)‘dom 𝐹))
6867sselda 3568 . . . . . . . . . . . 12 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) → 𝑥 ∈ ((limPt‘𝐾)‘dom 𝐹))
6959lpdifsn 20757 . . . . . . . . . . . . 13 ((𝐾 ∈ Top ∧ dom 𝐹 ⊆ ℂ) → (𝑥 ∈ ((limPt‘𝐾)‘dom 𝐹) ↔ 𝑥 ∈ ((limPt‘𝐾)‘(dom 𝐹 ∖ {𝑥}))))
7057, 26, 69sylancr 694 . . . . . . . . . . . 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 221 . . . . . . . . . . 11 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥 ∈ ((int‘(𝐾t 𝑆))‘dom 𝐹)) → 𝑥 ∈ ((limPt‘𝐾)‘(dom 𝐹 ∖ {𝑥})))
7242, 43, 71, 27limcmo 23452 . . . . . . . . . 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 449 . . . . . . . . 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 2519 . . . . . . . . 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 223 . . . . . . . 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 2610 . . . . . . . . . 10 (𝐾t 𝑆) = (𝐾t 𝑆)
7776, 27, 41, 24, 14, 23eldv 23468 . . . . . . . . 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 2479 . . . . . . . 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 246 . . . . . . 7 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ∃*𝑦 𝑥(𝑆 D 𝐹)𝑦)
8079alrimiv 1842 . . . . . 6 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ∀𝑥∃*𝑦 𝑥(𝑆 D 𝐹)𝑦)
81 reldv 23440 . . . . . . 7 Rel (𝑆 D 𝐹)
82 dffun6 5819 . . . . . . 7 (Fun (𝑆 D 𝐹) ↔ (Rel (𝑆 D 𝐹) ∧ ∀𝑥∃*𝑦 𝑥(𝑆 D 𝐹)𝑦))
8381, 82mpbiran 955 . . . . . 6 (Fun (𝑆 D 𝐹) ↔ ∀𝑥∃*𝑦 𝑥(𝑆 D 𝐹)𝑦)
8480, 83sylibr 223 . . . . 5 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → Fun (𝑆 D 𝐹))
85 funfn 5833 . . . . 5 (Fun (𝑆 D 𝐹) ↔ (𝑆 D 𝐹) Fn dom (𝑆 D 𝐹))
8684, 85sylib 207 . . . 4 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝑆 D 𝐹) Fn dom (𝑆 D 𝐹))
87 vex 3176 . . . . . . 7 𝑦 ∈ V
8887elrn 5287 . . . . . 6 (𝑦 ∈ ran (𝑆 D 𝐹) ↔ ∃𝑥 𝑥(𝑆 D 𝐹)𝑦)
8924, 14, 23dvcl 23469 . . . . . . . 8 (((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) ∧ 𝑥(𝑆 D 𝐹)𝑦) → 𝑦 ∈ ℂ)
9089ex 449 . . . . . . 7 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝑥(𝑆 D 𝐹)𝑦𝑦 ∈ ℂ))
9190exlimdv 1848 . . . . . 6 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (∃𝑥 𝑥(𝑆 D 𝐹)𝑦𝑦 ∈ ℂ))
9288, 91syl5bi 231 . . . . 5 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝑦 ∈ ran (𝑆 D 𝐹) → 𝑦 ∈ ℂ))
9392ssrdv 3574 . . . 4 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → ran (𝑆 D 𝐹) ⊆ ℂ)
94 df-f 5808 . . . 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 695 . . 3 ((⟨𝑆, 𝐹⟩ ∈ dom D ∧ (𝐾t 𝑆) ∈ Perf) → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
9695ex 449 . 2 (⟨𝑆, 𝐹⟩ ∈ dom D → ((𝐾t 𝑆) ∈ Perf → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ))
97 f0 5999 . . . 4 ∅:∅⟶ℂ
98 df-ov 6552 . . . . . 6 (𝑆 D 𝐹) = ( D ‘⟨𝑆, 𝐹⟩)
99 ndmfv 6128 . . . . . 6 (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → ( D ‘⟨𝑆, 𝐹⟩) = ∅)
10098, 99syl5eq 2656 . . . . 5 (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → (𝑆 D 𝐹) = ∅)
101100dmeqd 5248 . . . . . 6 (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) = dom ∅)
102 dm0 5260 . . . . . 6 dom ∅ = ∅
103101, 102syl6eq 2660 . . . . 5 (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) = ∅)
104100, 103feq12d 5946 . . . 4 (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ ∅:∅⟶ℂ))
10597, 104mpbiri 247 . . 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 175 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 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃*wmo 2459  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039  Rel wrel 5043  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑pm cpm 7745  ℂcc 9813   − cmin 10145   / cdiv 10563   ↾t crest 15904  TopOpenctopn 15905  ℂfldccnfld 19567  Topctop 20517  TopOnctopon 20518  intcnt 20631  limPtclp 20748  Perfcperf 20749   limℂ climc 23432   D cdv 23433 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-icc 12053  df-fz 12198  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-rest 15906  df-topn 15907  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cnp 20842  df-haus 20929  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-limc 23436  df-dv 23437 This theorem is referenced by:  dvfg  23476
 Copyright terms: Public domain W3C validator