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Theorem dvrecap 12856
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.
StepHypRef Expression
1 funmpt 5161 . . . . . . . . 9 Fun (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))
2 funforn 5352 . . . . . . . . 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)))
31, 2mpbi 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 5345 . . . . . . . 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)))
53, 4ax-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 3932 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑤 # 0 ↔ 𝑥 # 0))
87elrab 2840 . . . . . . . . . . . . 13 (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↔ (𝑥 ∈ ℂ ∧ 𝑥 # 0))
98biimpi 119 . . . . . . . . . . . 12 (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} → (𝑥 ∈ ℂ ∧ 𝑥 # 0))
109adantl 275 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑥 ∈ ℂ ∧ 𝑥 # 0))
1110simpld 111 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑥 ∈ ℂ)
1210simprd 113 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑥 # 0)
136, 11, 12divclapd 8557 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / 𝑥) ∈ ℂ)
1413ralrimiva 2505 . . . . . . . 8 (𝐴 ∈ ℂ → ∀𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} (𝐴 / 𝑥) ∈ ℂ)
15 eqid 2139 . . . . . . . . 9 (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))
1615rnmptss 5581 . . . . . . . 8 (∀𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} (𝐴 / 𝑥) ∈ ℂ → ran (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ⊆ ℂ)
1714, 16syl 14 . . . . . . 7 (𝐴 ∈ ℂ → ran (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ⊆ ℂ)
18 fss 5284 . . . . . . 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)
195, 17, 18sylancr 410 . . . . . 6 (𝐴 ∈ ℂ → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))⟶ℂ)
2015dmmpt 5034 . . . . . . 7 dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) = {𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ (𝐴 / 𝑥) ∈ V}
21 ssrab2 3182 . . . . . . . 8 {𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ (𝐴 / 𝑥) ∈ V} ⊆ {𝑤 ∈ ℂ ∣ 𝑤 # 0}
22 ssrab2 3182 . . . . . . . 8 {𝑤 ∈ ℂ ∣ 𝑤 # 0} ⊆ ℂ
2321, 22sstri 3106 . . . . . . 7 {𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ (𝐴 / 𝑥) ∈ V} ⊆ ℂ
2420, 23eqsstri 3129 . . . . . 6 dom (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ⊆ ℂ
25 cnex 7751 . . . . . . 7 ℂ ∈ V
2625, 25elpm2 6574 . . . . . 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))
2719, 24, 26sylanblrc 412 . . . . 5 (𝐴 ∈ ℂ → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)) ∈ (ℂ ↑pm ℂ))
28 dvfcnpm 12838 . . . . 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)
2927, 28syl 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 3118 . . . . . . 7 (𝐴 ∈ ℂ → ℂ ⊆ ℂ)
31 divclap 8445 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑥 # 0) → (𝐴 / 𝑥) ∈ ℂ)
32313expb 1182 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 # 0)) → (𝐴 / 𝑥) ∈ ℂ)
338, 32sylan2b 285 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / 𝑥) ∈ ℂ)
3433fmpttd 5575 . . . . . . 7 (𝐴 ∈ ℂ → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):{𝑤 ∈ ℂ ∣ 𝑤 # 0}⟶ℂ)
3522a1i 9 . . . . . . 7 (𝐴 ∈ ℂ → {𝑤 ∈ ℂ ∣ 𝑤 # 0} ⊆ ℂ)
3630, 34, 35dvbss 12833 . . . . . 6 (𝐴 ∈ ℂ → dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) ⊆ {𝑤 ∈ ℂ ∣ 𝑤 # 0})
37 elrabi 2837 . . . . . . . 8 (𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} → 𝑦 ∈ ℂ)
3837adantl 275 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 ∈ ℂ)
39 simpl 108 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝐴 ∈ ℂ)
4038sqcld 10429 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑦↑2) ∈ ℂ)
41 breq1 3932 . . . . . . . . . . . . 13 (𝑤 = 𝑦 → (𝑤 # 0 ↔ 𝑦 # 0))
4241elrab 2840 . . . . . . . . . . . 12 (𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↔ (𝑦 ∈ ℂ ∧ 𝑦 # 0))
4342simprbi 273 . . . . . . . . . . 11 (𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} → 𝑦 # 0)
4443adantl 275 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 # 0)
45 sqap0 10366 . . . . . . . . . . 11 (𝑦 ∈ ℂ → ((𝑦↑2) # 0 ↔ 𝑦 # 0))
4638, 45syl 14 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝑦↑2) # 0 ↔ 𝑦 # 0))
4744, 46mpbird 166 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑦↑2) # 0)
4839, 40, 47divclapd 8557 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑦↑2)) ∈ ℂ)
4948negcld 8067 . . . . . . 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 2139 . . . . . . . . . . 11 (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
5251cntoptop 12712 . . . . . . . . . 10 (MetOpen‘(abs ∘ − )) ∈ Top
53 0cn 7765 . . . . . . . . . . 11 0 ∈ ℂ
54 cnopnap 12773 . . . . . . . . . . 11 (0 ∈ ℂ → {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∈ (MetOpen‘(abs ∘ − )))
5553, 54ax-mp 5 . . . . . . . . . 10 {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∈ (MetOpen‘(abs ∘ − ))
56 isopn3i 12314 . . . . . . . . . 10 (((MetOpen‘(abs ∘ − )) ∈ Top ∧ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∈ (MetOpen‘(abs ∘ − ))) → ((int‘(MetOpen‘(abs ∘ − )))‘{𝑤 ∈ ℂ ∣ 𝑤 # 0}) = {𝑤 ∈ ℂ ∣ 𝑤 # 0})
5752, 55, 56mp2an 422 . . . . . . . . 9 ((int‘(MetOpen‘(abs ∘ − )))‘{𝑤 ∈ ℂ ∣ 𝑤 # 0}) = {𝑤 ∈ ℂ ∣ 𝑤 # 0}
5850, 57eleqtrrdi 2233 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 ∈ ((int‘(MetOpen‘(abs ∘ − )))‘{𝑤 ∈ ℂ ∣ 𝑤 # 0}))
5938sqvald 10428 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑦↑2) = (𝑦 · 𝑦))
6059oveq2d 5790 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑦↑2)) = (𝐴 / (𝑦 · 𝑦)))
6139, 38, 38, 44, 44divdivap1d 8589 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝐴 / 𝑦) / 𝑦) = (𝐴 / (𝑦 · 𝑦)))
6260, 61eqtr4d 2175 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑦↑2)) = ((𝐴 / 𝑦) / 𝑦))
6362negeqd 7964 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) = -((𝐴 / 𝑦) / 𝑦))
6439, 38, 44divclapd 8557 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / 𝑦) ∈ ℂ)
6564, 38, 44divnegapd 8570 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -((𝐴 / 𝑦) / 𝑦) = (-(𝐴 / 𝑦) / 𝑦))
6663, 65eqtrd 2172 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) = (-(𝐴 / 𝑦) / 𝑦))
6764negcld 8067 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / 𝑦) ∈ ℂ)
68 eqid 2139 . . . . . . . . . . . . 13 (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) = (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧))
6968cdivcncfap 12766 . . . . . . . . . . . 12 (-(𝐴 / 𝑦) ∈ ℂ → (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ∈ ({𝑤 ∈ ℂ ∣ 𝑤 # 0}–cn→ℂ))
7067, 69syl 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 5782 . . . . . . . . . . 11 (𝑧 = 𝑦 → (-(𝐴 / 𝑦) / 𝑧) = (-(𝐴 / 𝑦) / 𝑦))
7270, 50, 71cnmptlimc 12822 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (-(𝐴 / 𝑦) / 𝑦) ∈ ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) lim 𝑦))
7366, 72eqeltrd 2216 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑦↑2)) ∈ ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) lim 𝑦))
74 cncff 12743 . . . . . . . . . . . 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)
7570, 74syl 14 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)):{𝑤 ∈ ℂ ∣ 𝑤 # 0}⟶ℂ)
7622a1i 9 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → {𝑤 ∈ ℂ ∣ 𝑤 # 0} ⊆ ℂ)
7775, 76limcdifap 12810 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) lim 𝑦) = (((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ↾ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) lim 𝑦))
78 elrabi 2837 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} → 𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0})
7978adantl 275 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0})
80 breq1 3932 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑧 → (𝑤 # 0 ↔ 𝑧 # 0))
8180elrab 2840 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↔ (𝑧 ∈ ℂ ∧ 𝑧 # 0))
8279, 81sylib 121 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑧 ∈ ℂ ∧ 𝑧 # 0))
8382simpld 111 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑧 ∈ ℂ)
8437ad2antlr 480 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑦 ∈ ℂ)
8583, 84subcld 8080 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑧𝑦) ∈ ℂ)
8664adantr 274 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝐴 / 𝑦) ∈ ℂ)
8781simprbi 273 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} → 𝑧 # 0)
8879, 87syl 14 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑧 # 0)
8986, 83, 88divclapd 8557 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝐴 / 𝑦) / 𝑧) ∈ ℂ)
90 mulneg12 8166 . . . . . . . . . . . . . . . . 17 (((𝑧𝑦) ∈ ℂ ∧ ((𝐴 / 𝑦) / 𝑧) ∈ ℂ) → (-(𝑧𝑦) · ((𝐴 / 𝑦) / 𝑧)) = ((𝑧𝑦) · -((𝐴 / 𝑦) / 𝑧)))
9185, 89, 90syl2anc 408 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (-(𝑧𝑦) · ((𝐴 / 𝑦) / 𝑧)) = ((𝑧𝑦) · -((𝐴 / 𝑦) / 𝑧)))
9284, 83, 89subdird 8184 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝑦𝑧) · ((𝐴 / 𝑦) / 𝑧)) = ((𝑦 · ((𝐴 / 𝑦) / 𝑧)) − (𝑧 · ((𝐴 / 𝑦) / 𝑧))))
9383, 84negsubdi2d 8096 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → -(𝑧𝑦) = (𝑦𝑧))
9493oveq1d 5789 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (-(𝑧𝑦) · ((𝐴 / 𝑦) / 𝑧)) = ((𝑦𝑧) · ((𝐴 / 𝑦) / 𝑧)))
95 oveq2 5782 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑧 → (𝐴 / 𝑥) = (𝐴 / 𝑧))
96 simpll 518 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝐴 ∈ ℂ)
9796, 83, 88divclapd 8557 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝐴 / 𝑧) ∈ ℂ)
9815, 95, 79, 97fvmptd3 5514 . . . . . . . . . . . . . . . . . . 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))
9943ad2antlr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑦 # 0)
10096, 84, 99divcanap2d 8559 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑦 · (𝐴 / 𝑦)) = 𝐴)
101100oveq1d 5789 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝑦 · (𝐴 / 𝑦)) / 𝑧) = (𝐴 / 𝑧))
10284, 86, 83, 88divassapd 8593 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝑦 · (𝐴 / 𝑦)) / 𝑧) = (𝑦 · ((𝐴 / 𝑦) / 𝑧)))
10398, 101, 1023eqtr2d 2178 . . . . . . . . . . . . . . . . . 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 5782 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑦 → (𝐴 / 𝑥) = (𝐴 / 𝑦))
10550adantr 274 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0})
10615, 104, 105, 86fvmptd3 5514 . . . . . . . . . . . . . . . . . . 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))
10786, 83, 88divcanap2d 8559 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑧 · ((𝐴 / 𝑦) / 𝑧)) = (𝐴 / 𝑦))
108106, 107eqtr4d 2175 . . . . . . . . . . . . . . . . . 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)))
109103, 108oveq12d 5792 . . . . . . . . . . . . . . . . 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))))
11092, 94, 1093eqtr4d 2182 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (-(𝑧𝑦) · ((𝐴 / 𝑦) / 𝑧)) = (((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)))
11186, 83, 88divnegapd 8570 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → -((𝐴 / 𝑦) / 𝑧) = (-(𝐴 / 𝑦) / 𝑧))
112111oveq2d 5790 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → ((𝑧𝑦) · -((𝐴 / 𝑦) / 𝑧)) = ((𝑧𝑦) · (-(𝐴 / 𝑦) / 𝑧)))
11391, 110, 1123eqtr3d 2180 . . . . . . . . . . . . . . 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)) = ((𝑧𝑦) · (-(𝐴 / 𝑦) / 𝑧)))
114113oveq1d 5789 . . . . . . . . . . . . . 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)) / (𝑧𝑦)) = (((𝑧𝑦) · (-(𝐴 / 𝑦) / 𝑧)) / (𝑧𝑦)))
11586negcld 8067 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → -(𝐴 / 𝑦) ∈ ℂ)
116115, 83, 88divclapd 8557 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (-(𝐴 / 𝑦) / 𝑧) ∈ ℂ)
117 breq1 3932 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑧 → (𝑘 # 𝑦𝑧 # 𝑦))
118117elrab 2840 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↔ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∧ 𝑧 # 𝑦))
119118simprbi 273 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} → 𝑧 # 𝑦)
120119adantl 275 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → 𝑧 # 𝑦)
12183, 84, 120subap0d 8413 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (𝑧𝑦) # 0)
122116, 85, 121divcanap3d 8562 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) ∧ 𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) → (((𝑧𝑦) · (-(𝐴 / 𝑦) / 𝑧)) / (𝑧𝑦)) = (-(𝐴 / 𝑦) / 𝑧))
123114, 122eqtrd 2172 . . . . . . . . . . . . 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)) / (𝑧𝑦)) = (-(𝐴 / 𝑦) / 𝑧))
124123mpteq2dva 4018 . . . . . . . . . . . 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)) / (𝑧𝑦))) = (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↦ (-(𝐴 / 𝑦) / 𝑧)))
125 ssrab2 3182 . . . . . . . . . . . . 13 {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ⊆ {𝑤 ∈ ℂ ∣ 𝑤 # 0}
126 resmpt 4867 . . . . . . . . . . . . 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)))
127125, 126ax-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))
128124, 127syl6eqr 2190 . . . . . . . . . . 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)) / (𝑧𝑦))) = ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ↾ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}))
129128oveq1d 5789 . . . . . . . . . 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)) / (𝑧𝑦))) lim 𝑦) = (((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ↾ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦}) lim 𝑦))
13077, 129eqtr4d 2175 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) lim 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)) / (𝑧𝑦))) lim 𝑦))
13173, 130eleqtrd 2218 . . . . . . . 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)) / (𝑧𝑦))) lim 𝑦))
13251cntoptopon 12711 . . . . . . . . . 10 (MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ)
133132toponrestid 12198 . . . . . . . . 9 (MetOpen‘(abs ∘ − )) = ((MetOpen‘(abs ∘ − )) ↾t ℂ)
134 eqid 2139 . . . . . . . . 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)) / (𝑧𝑦))) = (𝑧 ∈ {𝑘 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ∣ 𝑘 # 𝑦} ↦ ((((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧𝑦)))
135 ssidd 3118 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ℂ ⊆ ℂ)
13634adantr 274 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)):{𝑤 ∈ ℂ ∣ 𝑤 # 0}⟶ℂ)
137133, 51, 134, 135, 136, 76eldvap 12830 . . . . . . . 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)) / (𝑧𝑦))) lim 𝑦))))
13858, 131, 137mpbir2and 928 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦(ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))-(𝐴 / (𝑦↑2)))
139 breldmg 4745 . . . . . . 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))))
14038, 49, 138, 139syl3anc 1216 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → 𝑦 ∈ dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))))
14136, 140eqelssd 3116 . . . . 5 (𝐴 ∈ ℂ → dom (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) = {𝑤 ∈ ℂ ∣ 𝑤 # 0})
142141feq2d 5260 . . . 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))
14329, 142mpbid 146 . . 3 (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))):{𝑤 ∈ ℂ ∣ 𝑤 # 0}⟶ℂ)
144143ffnd 5273 . 2 (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) Fn {𝑤 ∈ ℂ ∣ 𝑤 # 0})
14511sqcld 10429 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑥↑2) ∈ ℂ)
146 sqap0 10366 . . . . . . . 8 (𝑥 ∈ ℂ → ((𝑥↑2) # 0 ↔ 𝑥 # 0))
14711, 146syl 14 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝑥↑2) # 0 ↔ 𝑥 # 0))
14812, 147mpbird 166 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝑥↑2) # 0)
1496, 145, 148divclapd 8557 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → (𝐴 / (𝑥↑2)) ∈ ℂ)
150149negcld 8067 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → -(𝐴 / (𝑥↑2)) ∈ ℂ)
151150ralrimiva 2505 . . 3 (𝐴 ∈ ℂ → ∀𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}-(𝐴 / (𝑥↑2)) ∈ ℂ)
152 eqid 2139 . . . 4 (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2)))
153152fnmpt 5249 . . 3 (∀𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}-(𝐴 / (𝑥↑2)) ∈ ℂ → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2))) Fn {𝑤 ∈ ℂ ∣ 𝑤 # 0})
154151, 153syl 14 . 2 (𝐴 ∈ ℂ → (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2))) Fn {𝑤 ∈ ℂ ∣ 𝑤 # 0})
15529ffund 5276 . . . . 5 (𝐴 ∈ ℂ → Fun (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))))
156155adantr 274 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → Fun (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))))
157 funbrfv 5460 . . . 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))))
158156, 138, 157sylc 62 . . 3 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥)))‘𝑦) = -(𝐴 / (𝑦↑2)))
159 oveq1 5781 . . . . . 6 (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2))
160159oveq2d 5790 . . . . 5 (𝑥 = 𝑦 → (𝐴 / (𝑥↑2)) = (𝐴 / (𝑦↑2)))
161160negeqd 7964 . . . 4 (𝑥 = 𝑦 → -(𝐴 / (𝑥↑2)) = -(𝐴 / (𝑦↑2)))
162152, 161, 50, 49fvmptd3 5514 . . 3 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0}) → ((𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2)))‘𝑦) = -(𝐴 / (𝑦↑2)))
163158, 162eqtr4d 2175 . 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))
164144, 154, 163eqfnfvd 5521 1 (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wral 2416  {crab 2420  Vcvv 2686  wss 3071   class class class wbr 3929  cmpt 3989  dom cdm 4539  ran crn 4540  cres 4541  ccom 4543  Fun wfun 5117   Fn wfn 5118  wf 5119  ontowfo 5121  cfv 5123  (class class class)co 5774  pm cpm 6543  cc 7625  0cc0 7627   · cmul 7632  cmin 7940  -cneg 7941   # cap 8350   / cdiv 8439  2c2 8778  cexp 10299  abscabs 10776  MetOpencmopn 12164  Topctop 12174  intcnt 12272  cnccncf 12736   lim climc 12802   D cdv 12803
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-map 6544  df-pm 6545  df-sup 6871  df-inf 6872  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-xneg 9566  df-xadd 9567  df-seqfrec 10226  df-exp 10300  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-rest 12132  df-topgen 12151  df-psmet 12166  df-xmet 12167  df-met 12168  df-bl 12169  df-mopn 12170  df-top 12175  df-topon 12188  df-bases 12220  df-ntr 12275  df-cn 12367  df-cnp 12368  df-cncf 12737  df-limced 12804  df-dvap 12805
This theorem is referenced by: (None)
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