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Theorem dvef 13855
Description: Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)
Assertion
Ref Expression
dvef (ℂ D exp) = exp

Proof of Theorem dvef
Dummy variables 𝑥 𝑧 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnex 7926 . . . . . . . 8 ℂ ∈ V
2 eff 11655 . . . . . . . 8 exp:ℂ⟶ℂ
3 fpmg 6668 . . . . . . . 8 ((ℂ ∈ V ∧ ℂ ∈ V ∧ exp:ℂ⟶ℂ) → exp ∈ (ℂ ↑pm ℂ))
41, 1, 2, 3mp3an 1337 . . . . . . 7 exp ∈ (ℂ ↑pm ℂ)
5 dvfcnpm 13826 . . . . . . 7 (exp ∈ (ℂ ↑pm ℂ) → (ℂ D exp):dom (ℂ D exp)⟶ℂ)
64, 5ax-mp 5 . . . . . 6 (ℂ D exp):dom (ℂ D exp)⟶ℂ
7 ffun 5364 . . . . . 6 ((ℂ D exp):dom (ℂ D exp)⟶ℂ → Fun (ℂ D exp))
86, 7ax-mp 5 . . . . 5 Fun (ℂ D exp)
9 subcl 8146 . . . . . . . . . . . 12 ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑧𝑥) ∈ ℂ)
109ancoms 268 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧𝑥) ∈ ℂ)
11 efadd 11667 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ (𝑧𝑥) ∈ ℂ) → (exp‘(𝑥 + (𝑧𝑥))) = ((exp‘𝑥) · (exp‘(𝑧𝑥))))
1210, 11syldan 282 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧𝑥))) = ((exp‘𝑥) · (exp‘(𝑧𝑥))))
13 pncan3 8155 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 + (𝑧𝑥)) = 𝑧)
1413fveq2d 5515 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧𝑥))) = (exp‘𝑧))
1512, 14eqtr3d 2212 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((exp‘𝑥) · (exp‘(𝑧𝑥))) = (exp‘𝑧))
1615mpteq2dva 4090 . . . . . . . 8 (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ ((exp‘𝑥) · (exp‘(𝑧𝑥)))) = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
171a1i 9 . . . . . . . . 9 (𝑥 ∈ ℂ → ℂ ∈ V)
18 efcl 11656 . . . . . . . . . 10 (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
1918adantr 276 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘𝑥) ∈ ℂ)
20 efcl 11656 . . . . . . . . . 10 ((𝑧𝑥) ∈ ℂ → (exp‘(𝑧𝑥)) ∈ ℂ)
2110, 20syl 14 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑧𝑥)) ∈ ℂ)
22 fconstmpt 4670 . . . . . . . . . 10 (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥))
2322a1i 9 . . . . . . . . 9 (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥)))
24 eqidd 2178 . . . . . . . . 9 (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥))))
2517, 19, 21, 23, 24offval2 6092 . . . . . . . 8 (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))) = (𝑧 ∈ ℂ ↦ ((exp‘𝑥) · (exp‘(𝑧𝑥)))))
262a1i 9 . . . . . . . . 9 (𝑥 ∈ ℂ → exp:ℂ⟶ℂ)
2726feqmptd 5565 . . . . . . . 8 (𝑥 ∈ ℂ → exp = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
2816, 25, 273eqtr4d 2220 . . . . . . 7 (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))) = exp)
2928oveq2d 5885 . . . . . 6 (𝑥 ∈ ℂ → (ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥))))) = (ℂ D exp))
30 fconstg 5408 . . . . . . . . . 10 ((exp‘𝑥) ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶{(exp‘𝑥)})
3118, 30syl 14 . . . . . . . . 9 (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶{(exp‘𝑥)})
3218snssd 3736 . . . . . . . . 9 (𝑥 ∈ ℂ → {(exp‘𝑥)} ⊆ ℂ)
3331, 32fssd 5374 . . . . . . . 8 (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶ℂ)
34 ssidd 3176 . . . . . . . 8 (𝑥 ∈ ℂ → ℂ ⊆ ℂ)
3521fmpttd 5667 . . . . . . . 8 (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥))):ℂ⟶ℂ)
36 c0ex 7942 . . . . . . . . . . . 12 0 ∈ V
3736snid 3622 . . . . . . . . . . 11 0 ∈ {0}
38 opelxpi 4655 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 0 ∈ {0}) → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
3937, 38mpan2 425 . . . . . . . . . 10 (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
40 dvconst 13828 . . . . . . . . . . 11 ((exp‘𝑥) ∈ ℂ → (ℂ D (ℂ × {(exp‘𝑥)})) = (ℂ × {0}))
4118, 40syl 14 . . . . . . . . . 10 (𝑥 ∈ ℂ → (ℂ D (ℂ × {(exp‘𝑥)})) = (ℂ × {0}))
4239, 41eleqtrrd 2257 . . . . . . . . 9 (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
43 df-br 4001 . . . . . . . . 9 (𝑥(ℂ D (ℂ × {(exp‘𝑥)}))0 ↔ ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
4442, 43sylibr 134 . . . . . . . 8 (𝑥 ∈ ℂ → 𝑥(ℂ D (ℂ × {(exp‘𝑥)}))0)
4526, 10cofmpt 5681 . . . . . . . . . 10 (𝑥 ∈ ℂ → (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥))))
4645oveq2d 5885 . . . . . . . . 9 (𝑥 ∈ ℂ → (ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧𝑥)))) = (ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))))
4710fmpttd 5667 . . . . . . . . . . 11 (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝑧𝑥)):ℂ⟶ℂ)
48 simpr 110 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → 𝑢 ∈ ℂ)
4948adantr 276 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → 𝑢 ∈ ℂ)
50 simpl 109 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → 𝑥 ∈ ℂ)
5150adantr 276 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → 𝑥 ∈ ℂ)
52 simpr 110 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → 𝑢 # 𝑥)
5349, 51, 52subap0d 8591 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → (𝑢𝑥) # 0)
54 eqid 2177 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℂ ↦ (𝑧𝑥)) = (𝑧 ∈ ℂ ↦ (𝑧𝑥))
55 oveq1 5876 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑢 → (𝑧𝑥) = (𝑢𝑥))
56 subcl 8146 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑢𝑥) ∈ ℂ)
5756ancoms 268 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑢𝑥) ∈ ℂ)
5854, 55, 48, 57fvmptd3 5605 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑢) = (𝑢𝑥))
59 oveq1 5876 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → (𝑧𝑥) = (𝑥𝑥))
60 id 19 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
6160, 60subcld 8258 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℂ → (𝑥𝑥) ∈ ℂ)
6261adantr 276 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑥𝑥) ∈ ℂ)
6354, 59, 50, 62fvmptd3 5605 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥) = (𝑥𝑥))
64 subid 8166 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℂ → (𝑥𝑥) = 0)
6564adantr 276 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑥𝑥) = 0)
6663, 65eqtrd 2210 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥) = 0)
6758, 66breq12d 4013 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥) ↔ (𝑢𝑥) # 0))
6867adantr 276 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → (((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥) ↔ (𝑢𝑥) # 0))
6953, 68mpbird 167 . . . . . . . . . . . . 13 (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥))
7069ex 115 . . . . . . . . . . . 12 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑢 # 𝑥 → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥)))
7170ralrimiva 2550 . . . . . . . . . . 11 (𝑥 ∈ ℂ → ∀𝑢 ∈ ℂ (𝑢 # 𝑥 → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥)))
7254, 59, 60, 61fvmptd3 5605 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥) = (𝑥𝑥))
7372, 64eqtrd 2210 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥) = 0)
74 dveflem 13854 . . . . . . . . . . . 12 0(ℂ D exp)1
7573, 74eqbrtrdi 4039 . . . . . . . . . . 11 (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥)(ℂ D exp)1)
76 1ex 7943 . . . . . . . . . . . . . . 15 1 ∈ V
7776snid 3622 . . . . . . . . . . . . . 14 1 ∈ {1}
78 opelxpi 4655 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℂ ∧ 1 ∈ {1}) → ⟨𝑥, 1⟩ ∈ (ℂ × {1}))
7977, 78mpan2 425 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ × {1}))
80 simpr 110 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
81 1cnd 7964 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
82 dvmptidcn 13845 . . . . . . . . . . . . . . . 16 (ℂ D (𝑧 ∈ ℂ ↦ 𝑧)) = (𝑧 ∈ ℂ ↦ 1)
8382a1i 9 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑧)) = (𝑧 ∈ ℂ ↦ 1))
84 simpl 109 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
85 0cnd 7941 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 0 ∈ ℂ)
8660dvmptccn 13846 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑥)) = (𝑧 ∈ ℂ ↦ 0))
8780, 81, 83, 84, 85, 86dvmptsubcn 13852 . . . . . . . . . . . . . 14 (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧𝑥))) = (𝑧 ∈ ℂ ↦ (1 − 0)))
88 1m0e1 9021 . . . . . . . . . . . . . . . 16 (1 − 0) = 1
8988mpteq2i 4087 . . . . . . . . . . . . . . 15 (𝑧 ∈ ℂ ↦ (1 − 0)) = (𝑧 ∈ ℂ ↦ 1)
90 fconstmpt 4670 . . . . . . . . . . . . . . 15 (ℂ × {1}) = (𝑧 ∈ ℂ ↦ 1)
9189, 90eqtr4i 2201 . . . . . . . . . . . . . 14 (𝑧 ∈ ℂ ↦ (1 − 0)) = (ℂ × {1})
9287, 91eqtrdi 2226 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧𝑥))) = (ℂ × {1}))
9379, 92eleqtrrd 2257 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧𝑥))))
94 df-br 4001 . . . . . . . . . . . 12 (𝑥(ℂ D (𝑧 ∈ ℂ ↦ (𝑧𝑥)))1 ↔ ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧𝑥))))
9593, 94sylibr 134 . . . . . . . . . . 11 (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (𝑧𝑥)))1)
96 eqid 2177 . . . . . . . . . . 11 (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
9726, 34, 47, 34, 71, 34, 34, 75, 95, 96dvcoapbr 13838 . . . . . . . . . 10 (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧𝑥))))(1 · 1))
98 1t1e1 9060 . . . . . . . . . 10 (1 · 1) = 1
9997, 98breqtrdi 4041 . . . . . . . . 9 (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧𝑥))))1)
10046, 99breqdi 4015 . . . . . . . 8 (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥))))1)
10133, 34, 35, 34, 44, 100, 96dvmulxxbr 13833 . . . . . . 7 (𝑥 ∈ ℂ → 𝑥(ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))))((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))))
10235, 60ffvelcdmd 5648 . . . . . . . . . 10 (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))‘𝑥) ∈ ℂ)
103102mul02d 8339 . . . . . . . . 9 (𝑥 ∈ ℂ → (0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))‘𝑥)) = 0)
104 fvconst2g 5726 . . . . . . . . . . . 12 (((exp‘𝑥) ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((ℂ × {(exp‘𝑥)})‘𝑥) = (exp‘𝑥))
10518, 104mpancom 422 . . . . . . . . . . 11 (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)})‘𝑥) = (exp‘𝑥))
106105oveq2d 5885 . . . . . . . . . 10 (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (1 · (exp‘𝑥)))
10718mulid2d 7966 . . . . . . . . . 10 (𝑥 ∈ ℂ → (1 · (exp‘𝑥)) = (exp‘𝑥))
108106, 107eqtrd 2210 . . . . . . . . 9 (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (exp‘𝑥))
109103, 108oveq12d 5887 . . . . . . . 8 (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (0 + (exp‘𝑥)))
11018addid2d 8097 . . . . . . . 8 (𝑥 ∈ ℂ → (0 + (exp‘𝑥)) = (exp‘𝑥))
111109, 110eqtrd 2210 . . . . . . 7 (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (exp‘𝑥))
112101, 111breqtrd 4026 . . . . . 6 (𝑥 ∈ ℂ → 𝑥(ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))))(exp‘𝑥))
11329, 112breqdi 4015 . . . . 5 (𝑥 ∈ ℂ → 𝑥(ℂ D exp)(exp‘𝑥))
114 funbrfv 5550 . . . . 5 (Fun (ℂ D exp) → (𝑥(ℂ D exp)(exp‘𝑥) → ((ℂ D exp)‘𝑥) = (exp‘𝑥)))
1158, 113, 114mpsyl 65 . . . 4 (𝑥 ∈ ℂ → ((ℂ D exp)‘𝑥) = (exp‘𝑥))
116115mpteq2ia 4086 . . 3 (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)) = (𝑥 ∈ ℂ ↦ (exp‘𝑥))
117 ssid 3175 . . . . . . . . 9 ℂ ⊆ ℂ
118 dvbsssg 13822 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ exp ∈ (ℂ ↑pm ℂ)) → dom (ℂ D exp) ⊆ ℂ)
119117, 4, 118mp2an 426 . . . . . . . 8 dom (ℂ D exp) ⊆ ℂ
120 breldmg 4829 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ (exp‘𝑥) ∈ ℂ ∧ 𝑥(ℂ D exp)(exp‘𝑥)) → 𝑥 ∈ dom (ℂ D exp))
12118, 113, 120mpd3an23 1339 . . . . . . . . 9 (𝑥 ∈ ℂ → 𝑥 ∈ dom (ℂ D exp))
122121ssriv 3159 . . . . . . . 8 ℂ ⊆ dom (ℂ D exp)
123119, 122eqssi 3171 . . . . . . 7 dom (ℂ D exp) = ℂ
124123feq2i 5355 . . . . . 6 ((ℂ D exp):dom (ℂ D exp)⟶ℂ ↔ (ℂ D exp):ℂ⟶ℂ)
1256, 124mpbi 145 . . . . 5 (ℂ D exp):ℂ⟶ℂ
126125a1i 9 . . . 4 (⊤ → (ℂ D exp):ℂ⟶ℂ)
127126feqmptd 5565 . . 3 (⊤ → (ℂ D exp) = (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)))
1282a1i 9 . . . 4 (⊤ → exp:ℂ⟶ℂ)
129128feqmptd 5565 . . 3 (⊤ → exp = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
130116, 127, 1293eqtr4a 2236 . 2 (⊤ → (ℂ D exp) = exp)
131130mptru 1362 1 (ℂ D exp) = exp
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wtru 1354  wcel 2148  Vcvv 2737  wss 3129  {csn 3591  cop 3594   class class class wbr 4000  cmpt 4061   × cxp 4621  dom cdm 4623  ccom 4627  Fun wfun 5206  wf 5208  cfv 5212  (class class class)co 5869  𝑓 cof 6075  pm cpm 6643  cc 7800  0cc0 7802  1c1 7803   + caddc 7805   · cmul 7807  cmin 8118   # cap 8528  abscabs 10990  expce 11634  MetOpencmopn 13152   D cdv 13791
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922  ax-addf 7924  ax-mulf 7925
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-disj 3978  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-of 6077  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-map 6644  df-pm 6645  df-en 6735  df-dom 6736  df-fin 6737  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-xneg 9759  df-xadd 9760  df-ico 9881  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-fac 10690  df-bc 10712  df-ihash 10740  df-shft 10808  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-sumdc 11346  df-ef 11640  df-rest 12638  df-topgen 12657  df-psmet 13154  df-xmet 13155  df-met 13156  df-bl 13157  df-mopn 13158  df-top 13163  df-topon 13176  df-bases 13208  df-ntr 13263  df-cn 13355  df-cnp 13356  df-tx 13420  df-cncf 13725  df-limced 13792  df-dvap 13793
This theorem is referenced by:  efcn  13856
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