Theorem dvrecap 14862

Description: Derivative of the reciprocal function. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)

Assertion

Ref	Expression
dvrecap	⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2))))

Distinct variable group:   𝑥,𝑤,𝐴

Proof of Theorem dvrecap

Dummy variables 𝑦 𝑧 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funmpt 5292	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))
2		funforn 5483	. . . . . . . . 9 ⊢ (Fun (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ↔ (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))–onto→ran (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))
3	1, 2	mpbi 145	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))–onto→ran (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))
4		fof 5476	. . . . . . . 8 ⊢ ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))–onto→ran (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))⟶ran (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))⟶ran (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))
6		simpl 109	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝐴 ∈ ℂ)
7		breq1 4032	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (𝑤 # 0 ↔ 𝑥 # 0))
8	7	elrab 2916	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↔ (𝑥 ∈ ℂ ∧ 𝑥 # 0))
9	8	biimpi 120	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} → (𝑥 ∈ ℂ ∧ 𝑥 # 0))
10	9	adantl 277	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑥 ∈ ℂ ∧ 𝑥 # 0))
11	10	simpld 112	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑥 ∈ ℂ)
12	10	simprd 114	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑥 # 0)
13	6, 11, 12	divclapd 8809	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / 𝑥) ∈ ℂ)
14	13	ralrimiva 2567	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ∀𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} (𝐴 / 𝑥) ∈ ℂ)
15		eqid 2193	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))
16	15	rnmptss 5719	. . . . . . . 8 ⊢ (∀𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} (𝐴 / 𝑥) ∈ ℂ → ran (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ⊆ ℂ)
17	14, 16	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ran (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ⊆ ℂ)
18		fss 5415	. . . . . . 7 ⊢ (((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))⟶ran (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ∧ ran (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ⊆ ℂ) → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))⟶ℂ)
19	5, 17, 18	sylancr 414	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))⟶ℂ)
20	15	dmmpt 5161	. . . . . . 7 ⊢ dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) = {𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ (𝐴 / 𝑥) ∈ V}
21		ssrab2 3264	. . . . . . . 8 ⊢ {𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ (𝐴 / 𝑥) ∈ V} ⊆ {𝑤 ∈ ℂ ∣ 𝑤 # 0}
22		ssrab2 3264	. . . . . . . 8 ⊢ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ⊆ ℂ
23	21, 22	sstri 3188	. . . . . . 7 ⊢ {𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ (𝐴 / 𝑥) ∈ V} ⊆ ℂ
24	20, 23	eqsstri 3211	. . . . . 6 ⊢ dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ⊆ ℂ
25		cnex 7996	. . . . . . 7 ⊢ ℂ ∈ V
26	25, 25	elpm2 6734	. . . . . 6 ⊢ ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ∈ (ℂ ↑pm ℂ) ↔ ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))⟶ℂ ∧ dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ⊆ ℂ))
27	19, 24, 26	sylanblrc 416	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ∈ (ℂ ↑pm ℂ))
28		dvfcnpm 14844	. . . . 5 ⊢ ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ∈ (ℂ ↑pm ℂ) → (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))):dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))⟶ℂ)
29	27, 28	syl 14	. . . 4 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))):dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))⟶ℂ)
30		ssidd 3200	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ℂ ⊆ ℂ)
31		divclap 8697	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑥 # 0) → (𝐴 / 𝑥) ∈ ℂ)
32	31	3expb 1206	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 # 0)) → (𝐴 / 𝑥) ∈ ℂ)
33	8, 32	sylan2b 287	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / 𝑥) ∈ ℂ)
34	33	fmpttd 5713	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):{𝑤 ∈ ℂ ∣ 𝑤 # 0}⟶ℂ)
35	22	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → {𝑤 ∈ ℂ ∣ 𝑤 # 0} ⊆ ℂ)
36	30, 34, 35	dvbss 14839	. . . . . 6 ⊢ (𝐴 ∈ ℂ → dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) ⊆ {𝑤 ∈ ℂ ∣ 𝑤 # 0})
37		elrabi 2913	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} → 𝑦 ∈ ℂ)
38	37	adantl 277	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 ∈ ℂ)
39		simpl 109	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝐴 ∈ ℂ)
40	38	sqcld 10742	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑦↑2) ∈ ℂ)
41		breq1 4032	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (𝑤 # 0 ↔ 𝑦 # 0))
42	41	elrab 2916	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↔ (𝑦 ∈ ℂ ∧ 𝑦 # 0))
43	42	simprbi 275	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} → 𝑦 # 0)
44	43	adantl 277	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 # 0)
45		sqap0 10677	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) # 0 ↔ 𝑦 # 0))
46	38, 45	syl 14	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝑦↑2) # 0 ↔ 𝑦 # 0))
47	44, 46	mpbird 167	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑦↑2) # 0)
48	39, 40, 47	divclapd 8809	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑦↑2)) ∈ ℂ)
49	48	negcld 8317	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) ∈ ℂ)
50		simpr 110	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0})
51		eqid 2193	. . . . . . . . . . 11 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
52	51	cntoptop 14701	. . . . . . . . . 10 ⊢ (MetOpen‘(abs ∘ − )) ∈ Top
53		0cn 8011	. . . . . . . . . . 11 ⊢ 0 ∈ ℂ
54		cnopnap 14765	. . . . . . . . . . 11 ⊢ (0 ∈ ℂ → {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∈ (MetOpen‘(abs ∘ − )))
55	53, 54	ax-mp 5	. . . . . . . . . 10 ⊢ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∈ (MetOpen‘(abs ∘ − ))
56		isopn3i 14303	. . . . . . . . . 10 ⊢ (((MetOpen‘(abs ∘ − )) ∈ Top ∧ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∈ (MetOpen‘(abs ∘ − ))) → ((int‘(MetOpen‘(abs ∘ − )))‘{𝑤 ∈ ℂ ∣ 𝑤 # 0}) = {𝑤 ∈ ℂ ∣ 𝑤 # 0})
57	52, 55, 56	mp2an 426	. . . . . . . . 9 ⊢ ((int‘(MetOpen‘(abs ∘ − )))‘{𝑤 ∈ ℂ ∣ 𝑤 # 0}) = {𝑤 ∈ ℂ ∣ 𝑤 # 0}
58	50, 57	eleqtrrdi 2287	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 ∈ ((int‘(MetOpen‘(abs ∘ − )))‘{𝑤 ∈ ℂ ∣ 𝑤 # 0}))
59	38	sqvald 10741	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑦↑2) = (𝑦 · 𝑦))
60	59	oveq2d 5934	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑦↑2)) = (𝐴 / (𝑦 · 𝑦)))
61	39, 38, 38, 44, 44	divdivap1d 8841	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝐴 / 𝑦) / 𝑦) = (𝐴 / (𝑦 · 𝑦)))
62	60, 61	eqtr4d 2229	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑦↑2)) = ((𝐴 / 𝑦) / 𝑦))
63	62	negeqd 8214	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) = -((𝐴 / 𝑦) / 𝑦))
64	39, 38, 44	divclapd 8809	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / 𝑦) ∈ ℂ)
65	64, 38, 44	divnegapd 8822	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -((𝐴 / 𝑦) / 𝑦) = (-(𝐴 / 𝑦) / 𝑦))
66	63, 65	eqtrd 2226	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) = (-(𝐴 / 𝑦) / 𝑦))
67	64	negcld 8317	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / 𝑦) ∈ ℂ)
68		eqid 2193	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) = (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧))
69	68	cdivcncfap 14758	. . . . . . . . . . . 12 ⊢ (-(𝐴 / 𝑦) ∈ ℂ → (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ∈ ({𝑤 ∈ ℂ ∣ 𝑤 # 0}–cn→ℂ))
70	67, 69	syl 14	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ∈ ({𝑤 ∈ ℂ ∣ 𝑤 # 0}–cn→ℂ))
71		oveq2 5926	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (-(𝐴 / 𝑦) / 𝑧) = (-(𝐴 / 𝑦) / 𝑦))
72	70, 50, 71	cnmptlimc 14828	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (-(𝐴 / 𝑦) / 𝑦) ∈ ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) limℂ 𝑦))
73	66, 72	eqeltrd 2270	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) ∈ ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) limℂ 𝑦))
74		cncff 14732	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ∈ ({𝑤 ∈ ℂ ∣ 𝑤 # 0}–cn→ℂ) → (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)):{𝑤 ∈ ℂ ∣ 𝑤 # 0}⟶ℂ)
75	70, 74	syl 14	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)):{𝑤 ∈ ℂ ∣ 𝑤 # 0}⟶ℂ)
76	22	a1i 9	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → {𝑤 ∈ ℂ ∣ 𝑤 # 0} ⊆ ℂ)
77	75, 76	limcdifap 14816	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) limℂ 𝑦) = (((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ↾ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) limℂ 𝑦))
78		elrabi 2913	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} → 𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0})
79	78	adantl 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0})
80		breq1 4032	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑧 → (𝑤 # 0 ↔ 𝑧 # 0))
81	80	elrab 2916	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↔ (𝑧 ∈ ℂ ∧ 𝑧 # 0))
82	79, 81	sylib 122	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑧 ∈ ℂ ∧ 𝑧 # 0))
83	82	simpld 112	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑧 ∈ ℂ)
84	37	ad2antlr 489	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑦 ∈ ℂ)
85	83, 84	subcld 8330	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑧 − 𝑦) ∈ ℂ)
86	64	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝐴 / 𝑦) ∈ ℂ)
87	81	simprbi 275	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} → 𝑧 # 0)
88	79, 87	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑧 # 0)
89	86, 83, 88	divclapd 8809	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝐴 / 𝑦) / 𝑧) ∈ ℂ)
90		mulneg12 8416	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 − 𝑦) ∈ ℂ ∧ ((𝐴 / 𝑦) / 𝑧) ∈ ℂ) → (-(𝑧 − 𝑦) · ((𝐴 / 𝑦) / 𝑧)) = ((𝑧 − 𝑦) · -((𝐴 / 𝑦) / 𝑧)))
91	85, 89, 90	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (-(𝑧 − 𝑦) · ((𝐴 / 𝑦) / 𝑧)) = ((𝑧 − 𝑦) · -((𝐴 / 𝑦) / 𝑧)))
92	84, 83, 89	subdird 8434	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝑦 − 𝑧) · ((𝐴 / 𝑦) / 𝑧)) = ((𝑦 · ((𝐴 / 𝑦) / 𝑧)) − (𝑧 · ((𝐴 / 𝑦) / 𝑧))))
93	83, 84	negsubdi2d 8346	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → -(𝑧 − 𝑦) = (𝑦 − 𝑧))
94	93	oveq1d 5933	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (-(𝑧 − 𝑦) · ((𝐴 / 𝑦) / 𝑧)) = ((𝑦 − 𝑧) · ((𝐴 / 𝑦) / 𝑧)))
95		oveq2 5926	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → (𝐴 / 𝑥) = (𝐴 / 𝑧))
96		simpll 527	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝐴 ∈ ℂ)
97	96, 83, 88	divclapd 8809	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝐴 / 𝑧) ∈ ℂ)
98	15, 95, 79, 97	fvmptd3 5651	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) = (𝐴 / 𝑧))
99	43	ad2antlr 489	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑦 # 0)
100	96, 84, 99	divcanap2d 8811	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑦 · (𝐴 / 𝑦)) = 𝐴)
101	100	oveq1d 5933	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝑦 · (𝐴 / 𝑦)) / 𝑧) = (𝐴 / 𝑧))
102	84, 86, 83, 88	divassapd 8845	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝑦 · (𝐴 / 𝑦)) / 𝑧) = (𝑦 · ((𝐴 / 𝑦) / 𝑧)))
103	98, 101, 102	3eqtr2d 2232	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) = (𝑦 · ((𝐴 / 𝑦) / 𝑧)))
104		oveq2 5926	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (𝐴 / 𝑥) = (𝐴 / 𝑦))
105	50	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0})
106	15, 104, 105, 86	fvmptd3 5651	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦) = (𝐴 / 𝑦))
107	86, 83, 88	divcanap2d 8811	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑧 · ((𝐴 / 𝑦) / 𝑧)) = (𝐴 / 𝑦))
108	106, 107	eqtr4d 2229	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦) = (𝑧 · ((𝐴 / 𝑦) / 𝑧)))
109	103, 108	oveq12d 5936	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)) = ((𝑦 · ((𝐴 / 𝑦) / 𝑧)) − (𝑧 · ((𝐴 / 𝑦) / 𝑧))))
110	92, 94, 109	3eqtr4d 2236	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (-(𝑧 − 𝑦) · ((𝐴 / 𝑦) / 𝑧)) = (((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)))
111	86, 83, 88	divnegapd 8822	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → -((𝐴 / 𝑦) / 𝑧) = (-(𝐴 / 𝑦) / 𝑧))
112	111	oveq2d 5934	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝑧 − 𝑦) · -((𝐴 / 𝑦) / 𝑧)) = ((𝑧 − 𝑦) · (-(𝐴 / 𝑦) / 𝑧)))
113	91, 110, 112	3eqtr3d 2234	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)) = ((𝑧 − 𝑦) · (-(𝐴 / 𝑦) / 𝑧)))
114	113	oveq1d 5933	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦)) = (((𝑧 − 𝑦) · (-(𝐴 / 𝑦) / 𝑧)) / (𝑧 − 𝑦)))
115	86	negcld 8317	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → -(𝐴 / 𝑦) ∈ ℂ)
116	115, 83, 88	divclapd 8809	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (-(𝐴 / 𝑦) / 𝑧) ∈ ℂ)
117		breq1 4032	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑧 → (𝑘 # 𝑦 ↔ 𝑧 # 𝑦))
118	117	elrab 2916	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↔ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∧ 𝑧 # 𝑦))
119	118	simprbi 275	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} → 𝑧 # 𝑦)
120	119	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑧 # 𝑦)
121	83, 84, 120	subap0d 8663	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑧 − 𝑦) # 0)
122	116, 85, 121	divcanap3d 8814	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (((𝑧 − 𝑦) · (-(𝐴 / 𝑦) / 𝑧)) / (𝑧 − 𝑦)) = (-(𝐴 / 𝑦) / 𝑧))
123	114, 122	eqtrd 2226	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦)) = (-(𝐴 / 𝑦) / 𝑧))
124	123	mpteq2dva 4119	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↦ ((((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) = (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↦ (-(𝐴 / 𝑦) / 𝑧)))
125		ssrab2 3264	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ⊆ {𝑤 ∈ ℂ ∣ 𝑤 # 0}
126		resmpt 4990	. . . . . . . . . . . . 13 ⊢ ({𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ⊆ {𝑤 ∈ ℂ ∣ 𝑤 # 0} → ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ↾ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) = (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↦ (-(𝐴 / 𝑦) / 𝑧)))
127	125, 126	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ↾ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) = (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↦ (-(𝐴 / 𝑦) / 𝑧))
128	124, 127	eqtr4di 2244	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↦ ((((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) = ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ↾ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}))
129	128	oveq1d 5933	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↦ ((((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) limℂ 𝑦) = (((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ↾ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) limℂ 𝑦))
130	77, 129	eqtr4d 2229	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) limℂ 𝑦) = ((𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↦ ((((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) limℂ 𝑦))
131	73, 130	eleqtrd 2272	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) ∈ ((𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↦ ((((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) limℂ 𝑦))
132	51	cntoptopon 14700	. . . . . . . . . 10 ⊢ (MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ)
133	132	toponrestid 14189	. . . . . . . . 9 ⊢ (MetOpen‘(abs ∘ − )) = ((MetOpen‘(abs ∘ − )) ↾t ℂ)
134		eqid 2193	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↦ ((((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) = (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↦ ((((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦)))
135		ssidd 3200	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ℂ ⊆ ℂ)
136	34	adantr 276	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):{𝑤 ∈ ℂ ∣ 𝑤 # 0}⟶ℂ)
137	133, 51, 134, 135, 136, 76	eldvap 14836	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑦(ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))-(𝐴 / (𝑦↑2)) ↔ (𝑦 ∈ ((int‘(MetOpen‘(abs ∘ − )))‘{𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ -(𝐴 / (𝑦↑2)) ∈ ((𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↦ ((((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) limℂ 𝑦))))
138	58, 131, 137	mpbir2and 946	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦(ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))-(𝐴 / (𝑦↑2)))
139		breldmg 4868	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ -(𝐴 / (𝑦↑2)) ∈ ℂ ∧ 𝑦(ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))-(𝐴 / (𝑦↑2))) → 𝑦 ∈ dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))))
140	38, 49, 138, 139	syl3anc 1249	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 ∈ dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))))
141	36, 140	eqelssd 3198	. . . . 5 ⊢ (𝐴 ∈ ℂ → dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) = {𝑤 ∈ ℂ ∣ 𝑤 # 0})
142	141	feq2d 5391	. . . 4 ⊢ (𝐴 ∈ ℂ → ((ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))):dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))⟶ℂ ↔ (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))):{𝑤 ∈ ℂ ∣ 𝑤 # 0}⟶ℂ))
143	29, 142	mpbid 147	. . 3 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))):{𝑤 ∈ ℂ ∣ 𝑤 # 0}⟶ℂ)
144	143	ffnd 5404	. 2 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) Fn {𝑤 ∈ ℂ ∣ 𝑤 # 0})
145	11	sqcld 10742	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑥↑2) ∈ ℂ)
146		sqap0 10677	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) # 0 ↔ 𝑥 # 0))
147	11, 146	syl 14	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝑥↑2) # 0 ↔ 𝑥 # 0))
148	12, 147	mpbird 167	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑥↑2) # 0)
149	6, 145, 148	divclapd 8809	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑥↑2)) ∈ ℂ)
150	149	negcld 8317	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑥↑2)) ∈ ℂ)
151	150	ralrimiva 2567	. . 3 ⊢ (𝐴 ∈ ℂ → ∀𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}-(𝐴 / (𝑥↑2)) ∈ ℂ)
152		eqid 2193	. . . 4 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2)))
153	152	fnmpt 5380	. . 3 ⊢ (∀𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}-(𝐴 / (𝑥↑2)) ∈ ℂ → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2))) Fn {𝑤 ∈ ℂ ∣ 𝑤 # 0})
154	151, 153	syl 14	. 2 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2))) Fn {𝑤 ∈ ℂ ∣ 𝑤 # 0})
155	29	ffund 5407	. . . . 5 ⊢ (𝐴 ∈ ℂ → Fun (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))))
156	155	adantr 276	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → Fun (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))))
157		funbrfv 5595	. . . 4 ⊢ (Fun (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) → (𝑦(ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))-(𝐴 / (𝑦↑2)) → ((ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))‘𝑦) = -(𝐴 / (𝑦↑2))))
158	156, 138, 157	sylc 62	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))‘𝑦) = -(𝐴 / (𝑦↑2)))
159		oveq1 5925	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2))
160	159	oveq2d 5934	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 / (𝑥↑2)) = (𝐴 / (𝑦↑2)))
161	160	negeqd 8214	. . . 4 ⊢ (𝑥 = 𝑦 → -(𝐴 / (𝑥↑2)) = -(𝐴 / (𝑦↑2)))
162	152, 161, 50, 49	fvmptd3 5651	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2)))‘𝑦) = -(𝐴 / (𝑦↑2)))
163	158, 162	eqtr4d 2229	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))‘𝑦) = ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2)))‘𝑦))
164	144, 154, 163	eqfnfvd 5658	1 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∀wral 2472  {crab 2476  Vcvv 2760   ⊆ wss 3153   class class class wbr 4029   ↦ cmpt 4090  dom cdm 4659  ran crn 4660   ↾ cres 4661   ∘ ccom 4663  Fun wfun 5248   Fn wfn 5249  ⟶wf 5250  –onto→wfo 5252  ‘cfv 5254  (class class class)co 5918   ↑_pm cpm 6703  ℂcc 7870  0cc0 7872   · cmul 7877   − cmin 8190  -cneg 8191   # cap 8600   / cdiv 8691  2c2 9033  ↑cexp 10609  abscabs 11141  MetOpencmopn 14037  Topctop 14165  intcnt 14261  –cn→ccncf 14725   lim_ℂ climc 14808   D cdv 14809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-map 6704  df-pm 6705  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-xneg 9838  df-xadd 9839  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-rest 12852  df-topgen 12871  df-psmet 14039  df-xmet 14040  df-met 14041  df-bl 14042  df-mopn 14043  df-top 14166  df-topon 14179  df-bases 14211  df-ntr 14264  df-cn 14356  df-cnp 14357  df-cncf 14726  df-limced 14810  df-dvap 14811

This theorem is referenced by: (None)