Theorem dvrecap 13430

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 5234	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))
2		funforn 5425	. . . . . . . . 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 144	. . . . . . . 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 5418	. . . . . . . 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 108	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝐴 ∈ ℂ)
7		breq1 3990	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (𝑤 # 0 ↔ 𝑥 # 0))
8	7	elrab 2886	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↔ (𝑥 ∈ ℂ ∧ 𝑥 # 0))
9	8	biimpi 119	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} → (𝑥 ∈ ℂ ∧ 𝑥 # 0))
10	9	adantl 275	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑥 ∈ ℂ ∧ 𝑥 # 0))
11	10	simpld 111	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑥 ∈ ℂ)
12	10	simprd 113	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑥 # 0)
13	6, 11, 12	divclapd 8694	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / 𝑥) ∈ ℂ)
14	13	ralrimiva 2543	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ∀𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} (𝐴 / 𝑥) ∈ ℂ)
15		eqid 2170	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))
16	15	rnmptss 5654	. . . . . . . 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 5357	. . . . . . 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 412	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))⟶ℂ)
20	15	dmmpt 5104	. . . . . . 7 ⊢ dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) = {𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ (𝐴 / 𝑥) ∈ V}
21		ssrab2 3232	. . . . . . . 8 ⊢ {𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ (𝐴 / 𝑥) ∈ V} ⊆ {𝑤 ∈ ℂ ∣ 𝑤 # 0}
22		ssrab2 3232	. . . . . . . 8 ⊢ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ⊆ ℂ
23	21, 22	sstri 3156	. . . . . . 7 ⊢ {𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ (𝐴 / 𝑥) ∈ V} ⊆ ℂ
24	20, 23	eqsstri 3179	. . . . . 6 ⊢ dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ⊆ ℂ
25		cnex 7885	. . . . . . 7 ⊢ ℂ ∈ V
26	25, 25	elpm2 6654	. . . . . 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 414	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ∈ (ℂ ↑pm ℂ))
28		dvfcnpm 13412	. . . . 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 3168	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ℂ ⊆ ℂ)
31		divclap 8582	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑥 # 0) → (𝐴 / 𝑥) ∈ ℂ)
32	31	3expb 1199	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 # 0)) → (𝐴 / 𝑥) ∈ ℂ)
33	8, 32	sylan2b 285	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / 𝑥) ∈ ℂ)
34	33	fmpttd 5648	. . . . . . 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 13407	. . . . . 6 ⊢ (𝐴 ∈ ℂ → dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) ⊆ {𝑤 ∈ ℂ ∣ 𝑤 # 0})
37		elrabi 2883	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} → 𝑦 ∈ ℂ)
38	37	adantl 275	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 ∈ ℂ)
39		simpl 108	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝐴 ∈ ℂ)
40	38	sqcld 10594	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑦↑2) ∈ ℂ)
41		breq1 3990	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (𝑤 # 0 ↔ 𝑦 # 0))
42	41	elrab 2886	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↔ (𝑦 ∈ ℂ ∧ 𝑦 # 0))
43	42	simprbi 273	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} → 𝑦 # 0)
44	43	adantl 275	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 # 0)
45		sqap0 10529	. . . . . . . . . . 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 166	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑦↑2) # 0)
48	39, 40, 47	divclapd 8694	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑦↑2)) ∈ ℂ)
49	48	negcld 8204	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) ∈ ℂ)
50		simpr 109	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0})
51		eqid 2170	. . . . . . . . . . 11 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
52	51	cntoptop 13286	. . . . . . . . . 10 ⊢ (MetOpen‘(abs ∘ − )) ∈ Top
53		0cn 7899	. . . . . . . . . . 11 ⊢ 0 ∈ ℂ
54		cnopnap 13347	. . . . . . . . . . 11 ⊢ (0 ∈ ℂ → {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∈ (MetOpen‘(abs ∘ − )))
55	53, 54	ax-mp 5	. . . . . . . . . 10 ⊢ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∈ (MetOpen‘(abs ∘ − ))
56		isopn3i 12888	. . . . . . . . . 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 424	. . . . . . . . 9 ⊢ ((int‘(MetOpen‘(abs ∘ − )))‘{𝑤 ∈ ℂ ∣ 𝑤 # 0}) = {𝑤 ∈ ℂ ∣ 𝑤 # 0}
58	50, 57	eleqtrrdi 2264	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 ∈ ((int‘(MetOpen‘(abs ∘ − )))‘{𝑤 ∈ ℂ ∣ 𝑤 # 0}))
59	38	sqvald 10593	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑦↑2) = (𝑦 · 𝑦))
60	59	oveq2d 5866	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑦↑2)) = (𝐴 / (𝑦 · 𝑦)))
61	39, 38, 38, 44, 44	divdivap1d 8726	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝐴 / 𝑦) / 𝑦) = (𝐴 / (𝑦 · 𝑦)))
62	60, 61	eqtr4d 2206	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑦↑2)) = ((𝐴 / 𝑦) / 𝑦))
63	62	negeqd 8101	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) = -((𝐴 / 𝑦) / 𝑦))
64	39, 38, 44	divclapd 8694	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / 𝑦) ∈ ℂ)
65	64, 38, 44	divnegapd 8707	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -((𝐴 / 𝑦) / 𝑦) = (-(𝐴 / 𝑦) / 𝑦))
66	63, 65	eqtrd 2203	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) = (-(𝐴 / 𝑦) / 𝑦))
67	64	negcld 8204	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / 𝑦) ∈ ℂ)
68		eqid 2170	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) = (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧))
69	68	cdivcncfap 13340	. . . . . . . . . . . 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 5858	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (-(𝐴 / 𝑦) / 𝑧) = (-(𝐴 / 𝑦) / 𝑦))
72	70, 50, 71	cnmptlimc 13396	. . . . . . . . . 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 2247	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) ∈ ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) limℂ 𝑦))
74		cncff 13317	. . . . . . . . . . . 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 13384	. . . . . . . . . 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 2883	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} → 𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0})
79	78	adantl 275	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0})
80		breq1 3990	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑧 → (𝑤 # 0 ↔ 𝑧 # 0))
81	80	elrab 2886	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↔ (𝑧 ∈ ℂ ∧ 𝑧 # 0))
82	79, 81	sylib 121	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑧 ∈ ℂ ∧ 𝑧 # 0))
83	82	simpld 111	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑧 ∈ ℂ)
84	37	ad2antlr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑦 ∈ ℂ)
85	83, 84	subcld 8217	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑧 − 𝑦) ∈ ℂ)
86	64	adantr 274	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝐴 / 𝑦) ∈ ℂ)
87	81	simprbi 273	. . . . . . . . . . . . . . . . . . 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 8694	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝐴 / 𝑦) / 𝑧) ∈ ℂ)
90		mulneg12 8303	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 − 𝑦) ∈ ℂ ∧ ((𝐴 / 𝑦) / 𝑧) ∈ ℂ) → (-(𝑧 − 𝑦) · ((𝐴 / 𝑦) / 𝑧)) = ((𝑧 − 𝑦) · -((𝐴 / 𝑦) / 𝑧)))
91	85, 89, 90	syl2anc 409	. . . . . . . . . . . . . . . 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 8321	. . . . . . . . . . . . . . . . 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 8233	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → -(𝑧 − 𝑦) = (𝑦 − 𝑧))
94	93	oveq1d 5865	. . . . . . . . . . . . . . . . 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 5858	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → (𝐴 / 𝑥) = (𝐴 / 𝑧))
96		simpll 524	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝐴 ∈ ℂ)
97	96, 83, 88	divclapd 8694	. . . . . . . . . . . . . . . . . . . 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 5587	. . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑦 # 0)
100	96, 84, 99	divcanap2d 8696	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑦 · (𝐴 / 𝑦)) = 𝐴)
101	100	oveq1d 5865	. . . . . . . . . . . . . . . . . . 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 8730	. . . . . . . . . . . . . . . . . . 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 2209	. . . . . . . . . . . . . . . . . 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 5858	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (𝐴 / 𝑥) = (𝐴 / 𝑦))
105	50	adantr 274	. . . . . . . . . . . . . . . . . . . 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 5587	. . . . . . . . . . . . . . . . . . 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 8696	. . . . . . . . . . . . . . . . . . 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 2206	. . . . . . . . . . . . . . . . . 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 5868	. . . . . . . . . . . . . . . . 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 2213	. . . . . . . . . . . . . . . 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 8707	. . . . . . . . . . . . . . . . 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 5866	. . . . . . . . . . . . . . . 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 2211	. . . . . . . . . . . . . . 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 5865	. . . . . . . . . . . . . 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 8204	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → -(𝐴 / 𝑦) ∈ ℂ)
116	115, 83, 88	divclapd 8694	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (-(𝐴 / 𝑦) / 𝑧) ∈ ℂ)
117		breq1 3990	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑧 → (𝑘 # 𝑦 ↔ 𝑧 # 𝑦))
118	117	elrab 2886	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↔ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∧ 𝑧 # 𝑦))
119	118	simprbi 273	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} → 𝑧 # 𝑦)
120	119	adantl 275	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑧 # 𝑦)
121	83, 84, 120	subap0d 8550	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑧 − 𝑦) # 0)
122	116, 85, 121	divcanap3d 8699	. . . . . . . . . . . . . 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 2203	. . . . . . . . . . . . 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 4077	. . . . . . . . . . . 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 3232	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ⊆ {𝑤 ∈ ℂ ∣ 𝑤 # 0}
126		resmpt 4937	. . . . . . . . . . . . 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 2221	. . . . . . . . . . 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 5865	. . . . . . . . . 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 2206	. . . . . . . . 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 2249	. . . . . . . 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 13285	. . . . . . . . . 10 ⊢ (MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ)
133	132	toponrestid 12772	. . . . . . . . 9 ⊢ (MetOpen‘(abs ∘ − )) = ((MetOpen‘(abs ∘ − )) ↾t ℂ)
134		eqid 2170	. . . . . . . . 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 3168	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ℂ ⊆ ℂ)
136	34	adantr 274	. . . . . . . . 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 13404	. . . . . . . 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 939	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦(ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))-(𝐴 / (𝑦↑2)))
139		breldmg 4815	. . . . . . 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 1233	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 ∈ dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))))
141	36, 140	eqelssd 3166	. . . . 5 ⊢ (𝐴 ∈ ℂ → dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) = {𝑤 ∈ ℂ ∣ 𝑤 # 0})
142	141	feq2d 5333	. . . 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 146	. . 3 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))):{𝑤 ∈ ℂ ∣ 𝑤 # 0}⟶ℂ)
144	143	ffnd 5346	. 2 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) Fn {𝑤 ∈ ℂ ∣ 𝑤 # 0})
145	11	sqcld 10594	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑥↑2) ∈ ℂ)
146		sqap0 10529	. . . . . . . 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 166	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑥↑2) # 0)
149	6, 145, 148	divclapd 8694	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑥↑2)) ∈ ℂ)
150	149	negcld 8204	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑥↑2)) ∈ ℂ)
151	150	ralrimiva 2543	. . 3 ⊢ (𝐴 ∈ ℂ → ∀𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}-(𝐴 / (𝑥↑2)) ∈ ℂ)
152		eqid 2170	. . . 4 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2)))
153	152	fnmpt 5322	. . 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 5349	. . . . 5 ⊢ (𝐴 ∈ ℂ → Fun (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))))
156	155	adantr 274	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → Fun (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))))
157		funbrfv 5533	. . . 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 5857	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2))
160	159	oveq2d 5866	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 / (𝑥↑2)) = (𝐴 / (𝑦↑2)))
161	160	negeqd 8101	. . . 4 ⊢ (𝑥 = 𝑦 → -(𝐴 / (𝑥↑2)) = -(𝐴 / (𝑦↑2)))
162	152, 161, 50, 49	fvmptd3 5587	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2)))‘𝑦) = -(𝐴 / (𝑦↑2)))
163	158, 162	eqtr4d 2206	. 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 5594	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 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ∀wral 2448  {crab 2452  Vcvv 2730   ⊆ wss 3121   class class class wbr 3987   ↦ cmpt 4048  dom cdm 4609  ran crn 4610   ↾ cres 4611   ∘ ccom 4613  Fun wfun 5190   Fn wfn 5191  ⟶wf 5192  –onto→wfo 5194  ‘cfv 5196  (class class class)co 5850   ↑_pm cpm 6623  ℂcc 7759  0cc0 7761   · cmul 7766   − cmin 8077  -cneg 8078   # cap 8487   / cdiv 8576  2c2 8916  ↑cexp 10462  abscabs 10948  MetOpencmopn 12738  Topctop 12748  intcnt 12846  –cn→ccncf 13310   lim_ℂ climc 13376   D cdv 13377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880  ax-caucvg 7881

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-map 6624  df-pm 6625  df-sup 6957  df-inf 6958  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-xneg 9716  df-xadd 9717  df-seqfrec 10389  df-exp 10463  df-cj 10793  df-re 10794  df-im 10795  df-rsqrt 10949  df-abs 10950  df-rest 12568  df-topgen 12587  df-psmet 12740  df-xmet 12741  df-met 12742  df-bl 12743  df-mopn 12744  df-top 12749  df-topon 12762  df-bases 12794  df-ntr 12849  df-cn 12941  df-cnp 12942  df-cncf 13311  df-limced 13378  df-dvap 13379

This theorem is referenced by: (None)