Theorem dvrecap 15127

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 5308	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))
2		funforn 5504	. . . . . . . . 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 5497	. . . . . . . 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 4046	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (𝑤 # 0 ↔ 𝑥 # 0))
8	7	elrab 2928	. . . . . . . . . . . . 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 8862	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / 𝑥) ∈ ℂ)
14	13	ralrimiva 2578	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ∀𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} (𝐴 / 𝑥) ∈ ℂ)
15		eqid 2204	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))
16	15	rnmptss 5740	. . . . . . . 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 5436	. . . . . . 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 5177	. . . . . . 7 ⊢ dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) = {𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ (𝐴 / 𝑥) ∈ V}
21		ssrab2 3277	. . . . . . . 8 ⊢ {𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ (𝐴 / 𝑥) ∈ V} ⊆ {𝑤 ∈ ℂ ∣ 𝑤 # 0}
22		ssrab2 3277	. . . . . . . 8 ⊢ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ⊆ ℂ
23	21, 22	sstri 3201	. . . . . . 7 ⊢ {𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ (𝐴 / 𝑥) ∈ V} ⊆ ℂ
24	20, 23	eqsstri 3224	. . . . . 6 ⊢ dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ⊆ ℂ
25		cnex 8048	. . . . . . 7 ⊢ ℂ ∈ V
26	25, 25	elpm2 6766	. . . . . 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 15104	. . . . 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 3213	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ℂ ⊆ ℂ)
31		divclap 8750	. . . . . . . . . 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 5734	. . . . . . 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 15099	. . . . . 6 ⊢ (𝐴 ∈ ℂ → dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) ⊆ {𝑤 ∈ ℂ ∣ 𝑤 # 0})
37		elrabi 2925	. . . . . . . 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 10814	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑦↑2) ∈ ℂ)
41		breq1 4046	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (𝑤 # 0 ↔ 𝑦 # 0))
42	41	elrab 2928	. . . . . . . . . . . 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 10749	. . . . . . . . . . 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 8862	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑦↑2)) ∈ ℂ)
49	48	negcld 8369	. . . . . . 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 2204	. . . . . . . . . . 11 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
52	51	cntoptop 14947	. . . . . . . . . 10 ⊢ (MetOpen‘(abs ∘ − )) ∈ Top
53		0cn 8063	. . . . . . . . . . 11 ⊢ 0 ∈ ℂ
54		cnopnap 15025	. . . . . . . . . . 11 ⊢ (0 ∈ ℂ → {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∈ (MetOpen‘(abs ∘ − )))
55	53, 54	ax-mp 5	. . . . . . . . . 10 ⊢ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∈ (MetOpen‘(abs ∘ − ))
56		isopn3i 14549	. . . . . . . . . 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 2298	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 ∈ ((int‘(MetOpen‘(abs ∘ − )))‘{𝑤 ∈ ℂ ∣ 𝑤 # 0}))
59	38	sqvald 10813	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑦↑2) = (𝑦 · 𝑦))
60	59	oveq2d 5959	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑦↑2)) = (𝐴 / (𝑦 · 𝑦)))
61	39, 38, 38, 44, 44	divdivap1d 8894	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝐴 / 𝑦) / 𝑦) = (𝐴 / (𝑦 · 𝑦)))
62	60, 61	eqtr4d 2240	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑦↑2)) = ((𝐴 / 𝑦) / 𝑦))
63	62	negeqd 8266	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) = -((𝐴 / 𝑦) / 𝑦))
64	39, 38, 44	divclapd 8862	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / 𝑦) ∈ ℂ)
65	64, 38, 44	divnegapd 8875	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -((𝐴 / 𝑦) / 𝑦) = (-(𝐴 / 𝑦) / 𝑦))
66	63, 65	eqtrd 2237	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) = (-(𝐴 / 𝑦) / 𝑦))
67	64	negcld 8369	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / 𝑦) ∈ ℂ)
68		eqid 2204	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) = (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧))
69	68	cdivcncfap 15018	. . . . . . . . . . . 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 5951	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (-(𝐴 / 𝑦) / 𝑧) = (-(𝐴 / 𝑦) / 𝑦))
72	70, 50, 71	cnmptlimc 15088	. . . . . . . . . 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 2281	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) ∈ ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) limℂ 𝑦))
74		cncff 14991	. . . . . . . . . . . 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 15076	. . . . . . . . . 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 2925	. . . . . . . . . . . . . . . . . . . . 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 4046	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑧 → (𝑤 # 0 ↔ 𝑧 # 0))
81	80	elrab 2928	. . . . . . . . . . . . . . . . . . . 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 8382	. . . . . . . . . . . . . . . . 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 8862	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝐴 / 𝑦) / 𝑧) ∈ ℂ)
90		mulneg12 8468	. . . . . . . . . . . . . . . . 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 8486	. . . . . . . . . . . . . . . . 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 8398	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → -(𝑧 − 𝑦) = (𝑦 − 𝑧))
94	93	oveq1d 5958	. . . . . . . . . . . . . . . . 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 5951	. . . . . . . . . . . . . . . . . . . 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 8862	. . . . . . . . . . . . . . . . . . . 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 5672	. . . . . . . . . . . . . . . . . . 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 8864	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑦 · (𝐴 / 𝑦)) = 𝐴)
101	100	oveq1d 5958	. . . . . . . . . . . . . . . . . . 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 8898	. . . . . . . . . . . . . . . . . . 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 2243	. . . . . . . . . . . . . . . . . 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 5951	. . . . . . . . . . . . . . . . . . . 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 5672	. . . . . . . . . . . . . . . . . . 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 8864	. . . . . . . . . . . . . . . . . . 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 2240	. . . . . . . . . . . . . . . . . 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 5961	. . . . . . . . . . . . . . . . 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 2247	. . . . . . . . . . . . . . . 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 8875	. . . . . . . . . . . . . . . . 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 5959	. . . . . . . . . . . . . . . 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 2245	. . . . . . . . . . . . . . 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 5958	. . . . . . . . . . . . . 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 8369	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → -(𝐴 / 𝑦) ∈ ℂ)
116	115, 83, 88	divclapd 8862	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (-(𝐴 / 𝑦) / 𝑧) ∈ ℂ)
117		breq1 4046	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑧 → (𝑘 # 𝑦 ↔ 𝑧 # 𝑦))
118	117	elrab 2928	. . . . . . . . . . . . . . . . . 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 8716	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑧 − 𝑦) # 0)
122	116, 85, 121	divcanap3d 8867	. . . . . . . . . . . . . 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 2237	. . . . . . . . . . . . 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 4133	. . . . . . . . . . . 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 3277	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ⊆ {𝑤 ∈ ℂ ∣ 𝑤 # 0}
126		resmpt 5006	. . . . . . . . . . . . 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 2255	. . . . . . . . . . 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 5958	. . . . . . . . . 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 2240	. . . . . . . . 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 2283	. . . . . . . 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 14946	. . . . . . . . . 10 ⊢ (MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ)
133	132	toponrestid 14435	. . . . . . . . 9 ⊢ (MetOpen‘(abs ∘ − )) = ((MetOpen‘(abs ∘ − )) ↾t ℂ)
134		eqid 2204	. . . . . . . . 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 3213	. . . . . . . . 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 15096	. . . . . . . 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 4883	. . . . . . 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 3211	. . . . 5 ⊢ (𝐴 ∈ ℂ → dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) = {𝑤 ∈ ℂ ∣ 𝑤 # 0})
142	141	feq2d 5412	. . . 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 5425	. 2 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) Fn {𝑤 ∈ ℂ ∣ 𝑤 # 0})
145	11	sqcld 10814	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑥↑2) ∈ ℂ)
146		sqap0 10749	. . . . . . . 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 8862	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑥↑2)) ∈ ℂ)
150	149	negcld 8369	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑥↑2)) ∈ ℂ)
151	150	ralrimiva 2578	. . 3 ⊢ (𝐴 ∈ ℂ → ∀𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}-(𝐴 / (𝑥↑2)) ∈ ℂ)
152		eqid 2204	. . . 4 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2)))
153	152	fnmpt 5401	. . 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 5428	. . . . 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 5616	. . . 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 5950	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2))
160	159	oveq2d 5959	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 / (𝑥↑2)) = (𝐴 / (𝑦↑2)))
161	160	negeqd 8266	. . . 4 ⊢ (𝑥 = 𝑦 → -(𝐴 / (𝑥↑2)) = -(𝐴 / (𝑦↑2)))
162	152, 161, 50, 49	fvmptd3 5672	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2)))‘𝑦) = -(𝐴 / (𝑦↑2)))
163	158, 162	eqtr4d 2240	. 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 5679	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 1372   ∈ wcel 2175  ∀wral 2483  {crab 2487  Vcvv 2771   ⊆ wss 3165   class class class wbr 4043   ↦ cmpt 4104  dom cdm 4674  ran crn 4675   ↾ cres 4676   ∘ ccom 4678  Fun wfun 5264   Fn wfn 5265  ⟶wf 5266  –onto→wfo 5268  ‘cfv 5270  (class class class)co 5943   ↑_pm cpm 6735  ℂcc 7922  0cc0 7924   · cmul 7929   − cmin 8242  -cneg 8243   # cap 8653   / cdiv 8744  2c2 9086  ↑cexp 10681  abscabs 11250  MetOpencmopn 14245  Topctop 14411  intcnt 14507  –cn→ccncf 14984   lim_ℂ climc 15068   D cdv 15069

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-isom 5279  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-map 6736  df-pm 6737  df-sup 7085  df-inf 7086  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-xneg 9893  df-xadd 9894  df-seqfrec 10591  df-exp 10682  df-cj 11095  df-re 11096  df-im 11097  df-rsqrt 11251  df-abs 11252  df-rest 13015  df-topgen 13034  df-psmet 14247  df-xmet 14248  df-met 14249  df-bl 14250  df-mopn 14251  df-top 14412  df-topon 14425  df-bases 14457  df-ntr 14510  df-cn 14602  df-cnp 14603  df-cncf 14985  df-limced 15070  df-dvap 15071

This theorem is referenced by: (None)