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Theorem dvef 12845
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 7737 . . . . . . . 8 ℂ ∈ V
2 eff 11358 . . . . . . . 8 exp:ℂ⟶ℂ
3 fpmg 6561 . . . . . . . 8 ((ℂ ∈ V ∧ ℂ ∈ V ∧ exp:ℂ⟶ℂ) → exp ∈ (ℂ ↑pm ℂ))
41, 1, 2, 3mp3an 1315 . . . . . . 7 exp ∈ (ℂ ↑pm ℂ)
5 dvfcnpm 12817 . . . . . . 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 5270 . . . . . 6 ((ℂ D exp):dom (ℂ D exp)⟶ℂ → Fun (ℂ D exp))
86, 7ax-mp 5 . . . . 5 Fun (ℂ D exp)
9 subcl 7954 . . . . . . . . . . . 12 ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑧𝑥) ∈ ℂ)
109ancoms 266 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧𝑥) ∈ ℂ)
11 efadd 11370 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ (𝑧𝑥) ∈ ℂ) → (exp‘(𝑥 + (𝑧𝑥))) = ((exp‘𝑥) · (exp‘(𝑧𝑥))))
1210, 11syldan 280 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧𝑥))) = ((exp‘𝑥) · (exp‘(𝑧𝑥))))
13 pncan3 7963 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 + (𝑧𝑥)) = 𝑧)
1413fveq2d 5418 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧𝑥))) = (exp‘𝑧))
1512, 14eqtr3d 2172 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((exp‘𝑥) · (exp‘(𝑧𝑥))) = (exp‘𝑧))
1615mpteq2dva 4013 . . . . . . . 8 (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ ((exp‘𝑥) · (exp‘(𝑧𝑥)))) = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
171a1i 9 . . . . . . . . 9 (𝑥 ∈ ℂ → ℂ ∈ V)
18 efcl 11359 . . . . . . . . . 10 (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
1918adantr 274 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘𝑥) ∈ ℂ)
20 efcl 11359 . . . . . . . . . 10 ((𝑧𝑥) ∈ ℂ → (exp‘(𝑧𝑥)) ∈ ℂ)
2110, 20syl 14 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑧𝑥)) ∈ ℂ)
22 fconstmpt 4581 . . . . . . . . . 10 (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥))
2322a1i 9 . . . . . . . . 9 (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥)))
24 eqidd 2138 . . . . . . . . 9 (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥))))
2517, 19, 21, 23, 24offval2 5990 . . . . . . . 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 5467 . . . . . . . 8 (𝑥 ∈ ℂ → exp = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
2816, 25, 273eqtr4d 2180 . . . . . . 7 (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))) = exp)
2928oveq2d 5783 . . . . . 6 (𝑥 ∈ ℂ → (ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥))))) = (ℂ D exp))
30 fconstg 5314 . . . . . . . . . 10 ((exp‘𝑥) ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶{(exp‘𝑥)})
3118, 30syl 14 . . . . . . . . 9 (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶{(exp‘𝑥)})
3218snssd 3660 . . . . . . . . 9 (𝑥 ∈ ℂ → {(exp‘𝑥)} ⊆ ℂ)
3331, 32fssd 5280 . . . . . . . 8 (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶ℂ)
34 ssidd 3113 . . . . . . . 8 (𝑥 ∈ ℂ → ℂ ⊆ ℂ)
3521fmpttd 5568 . . . . . . . 8 (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥))):ℂ⟶ℂ)
36 c0ex 7753 . . . . . . . . . . . 12 0 ∈ V
3736snid 3551 . . . . . . . . . . 11 0 ∈ {0}
38 opelxpi 4566 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 0 ∈ {0}) → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
3937, 38mpan2 421 . . . . . . . . . 10 (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
40 dvconst 12819 . . . . . . . . . . 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 2217 . . . . . . . . 9 (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
43 df-br 3925 . . . . . . . . 9 (𝑥(ℂ D (ℂ × {(exp‘𝑥)}))0 ↔ ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
4442, 43sylibr 133 . . . . . . . 8 (𝑥 ∈ ℂ → 𝑥(ℂ D (ℂ × {(exp‘𝑥)}))0)
4526, 10cofmpt 5582 . . . . . . . . . 10 (𝑥 ∈ ℂ → (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥))))
4645oveq2d 5783 . . . . . . . . 9 (𝑥 ∈ ℂ → (ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧𝑥)))) = (ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))))
4710fmpttd 5568 . . . . . . . . . . 11 (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝑧𝑥)):ℂ⟶ℂ)
48 simpr 109 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → 𝑢 ∈ ℂ)
4948adantr 274 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → 𝑢 ∈ ℂ)
50 simpl 108 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → 𝑥 ∈ ℂ)
5150adantr 274 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → 𝑥 ∈ ℂ)
52 simpr 109 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → 𝑢 # 𝑥)
5349, 51, 52subap0d 8399 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → (𝑢𝑥) # 0)
54 eqid 2137 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℂ ↦ (𝑧𝑥)) = (𝑧 ∈ ℂ ↦ (𝑧𝑥))
55 oveq1 5774 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑢 → (𝑧𝑥) = (𝑢𝑥))
56 subcl 7954 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑢𝑥) ∈ ℂ)
5756ancoms 266 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑢𝑥) ∈ ℂ)
5854, 55, 48, 57fvmptd3 5507 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑢) = (𝑢𝑥))
59 oveq1 5774 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → (𝑧𝑥) = (𝑥𝑥))
60 id 19 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
6160, 60subcld 8066 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℂ → (𝑥𝑥) ∈ ℂ)
6261adantr 274 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑥𝑥) ∈ ℂ)
6354, 59, 50, 62fvmptd3 5507 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥) = (𝑥𝑥))
64 subid 7974 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℂ → (𝑥𝑥) = 0)
6564adantr 274 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑥𝑥) = 0)
6663, 65eqtrd 2170 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥) = 0)
6758, 66breq12d 3937 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥) ↔ (𝑢𝑥) # 0))
6867adantr 274 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → (((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥) ↔ (𝑢𝑥) # 0))
6953, 68mpbird 166 . . . . . . . . . . . . 13 (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥))
7069ex 114 . . . . . . . . . . . 12 ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑢 # 𝑥 → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥)))
7170ralrimiva 2503 . . . . . . . . . . 11 (𝑥 ∈ ℂ → ∀𝑢 ∈ ℂ (𝑢 # 𝑥 → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥)))
7254, 59, 60, 61fvmptd3 5507 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥) = (𝑥𝑥))
7372, 64eqtrd 2170 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥) = 0)
74 dveflem 12844 . . . . . . . . . . . 12 0(ℂ D exp)1
7573, 74eqbrtrdi 3962 . . . . . . . . . . 11 (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧𝑥))‘𝑥)(ℂ D exp)1)
76 1ex 7754 . . . . . . . . . . . . . . 15 1 ∈ V
7776snid 3551 . . . . . . . . . . . . . 14 1 ∈ {1}
78 opelxpi 4566 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℂ ∧ 1 ∈ {1}) → ⟨𝑥, 1⟩ ∈ (ℂ × {1}))
7977, 78mpan2 421 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ × {1}))
80 simpr 109 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
81 1cnd 7775 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
82 dvmptidcn 12836 . . . . . . . . . . . . . . . 16 (ℂ D (𝑧 ∈ ℂ ↦ 𝑧)) = (𝑧 ∈ ℂ ↦ 1)
8382a1i 9 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑧)) = (𝑧 ∈ ℂ ↦ 1))
84 simpl 108 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
85 0cnd 7752 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 0 ∈ ℂ)
8660dvmptccn 12837 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑥)) = (𝑧 ∈ ℂ ↦ 0))
8780, 81, 83, 84, 85, 86dvmptsubcn 12843 . . . . . . . . . . . . . 14 (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧𝑥))) = (𝑧 ∈ ℂ ↦ (1 − 0)))
88 1m0e1 8826 . . . . . . . . . . . . . . . 16 (1 − 0) = 1
8988mpteq2i 4010 . . . . . . . . . . . . . . 15 (𝑧 ∈ ℂ ↦ (1 − 0)) = (𝑧 ∈ ℂ ↦ 1)
90 fconstmpt 4581 . . . . . . . . . . . . . . 15 (ℂ × {1}) = (𝑧 ∈ ℂ ↦ 1)
9189, 90eqtr4i 2161 . . . . . . . . . . . . . 14 (𝑧 ∈ ℂ ↦ (1 − 0)) = (ℂ × {1})
9287, 91syl6eq 2186 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧𝑥))) = (ℂ × {1}))
9379, 92eleqtrrd 2217 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧𝑥))))
94 df-br 3925 . . . . . . . . . . . 12 (𝑥(ℂ D (𝑧 ∈ ℂ ↦ (𝑧𝑥)))1 ↔ ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧𝑥))))
9593, 94sylibr 133 . . . . . . . . . . 11 (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (𝑧𝑥)))1)
96 eqid 2137 . . . . . . . . . . 11 (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
9726, 34, 47, 34, 71, 34, 34, 75, 95, 96dvcoapbr 12829 . . . . . . . . . 10 (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧𝑥))))(1 · 1))
98 1t1e1 8865 . . . . . . . . . 10 (1 · 1) = 1
9997, 98breqtrdi 3964 . . . . . . . . 9 (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧𝑥))))1)
10046, 99breqdi 3939 . . . . . . . 8 (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥))))1)
10133, 34, 35, 34, 44, 100, 96dvmulxxbr 12824 . . . . . . 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, 60ffvelrnd 5549 . . . . . . . . . 10 (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))‘𝑥) ∈ ℂ)
103102mul02d 8147 . . . . . . . . 9 (𝑥 ∈ ℂ → (0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))‘𝑥)) = 0)
104 fvconst2g 5627 . . . . . . . . . . . 12 (((exp‘𝑥) ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((ℂ × {(exp‘𝑥)})‘𝑥) = (exp‘𝑥))
10518, 104mpancom 418 . . . . . . . . . . 11 (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)})‘𝑥) = (exp‘𝑥))
106105oveq2d 5783 . . . . . . . . . 10 (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (1 · (exp‘𝑥)))
10718mulid2d 7777 . . . . . . . . . 10 (𝑥 ∈ ℂ → (1 · (exp‘𝑥)) = (exp‘𝑥))
108106, 107eqtrd 2170 . . . . . . . . 9 (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (exp‘𝑥))
109103, 108oveq12d 5785 . . . . . . . 8 (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (0 + (exp‘𝑥)))
11018addid2d 7905 . . . . . . . 8 (𝑥 ∈ ℂ → (0 + (exp‘𝑥)) = (exp‘𝑥))
111109, 110eqtrd 2170 . . . . . . 7 (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (exp‘𝑥))
112101, 111breqtrd 3949 . . . . . 6 (𝑥 ∈ ℂ → 𝑥(ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧𝑥)))))(exp‘𝑥))
11329, 112breqdi 3939 . . . . 5 (𝑥 ∈ ℂ → 𝑥(ℂ D exp)(exp‘𝑥))
114 funbrfv 5453 . . . . 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 4009 . . 3 (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)) = (𝑥 ∈ ℂ ↦ (exp‘𝑥))
117 ssid 3112 . . . . . . . . 9 ℂ ⊆ ℂ
118 dvbsssg 12813 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ exp ∈ (ℂ ↑pm ℂ)) → dom (ℂ D exp) ⊆ ℂ)
119117, 4, 118mp2an 422 . . . . . . . 8 dom (ℂ D exp) ⊆ ℂ
120 breldmg 4740 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ (exp‘𝑥) ∈ ℂ ∧ 𝑥(ℂ D exp)(exp‘𝑥)) → 𝑥 ∈ dom (ℂ D exp))
12118, 113, 120mpd3an23 1317 . . . . . . . . 9 (𝑥 ∈ ℂ → 𝑥 ∈ dom (ℂ D exp))
122121ssriv 3096 . . . . . . . 8 ℂ ⊆ dom (ℂ D exp)
123119, 122eqssi 3108 . . . . . . 7 dom (ℂ D exp) = ℂ
124123feq2i 5261 . . . . . 6 ((ℂ D exp):dom (ℂ D exp)⟶ℂ ↔ (ℂ D exp):ℂ⟶ℂ)
1256, 124mpbi 144 . . . . 5 (ℂ D exp):ℂ⟶ℂ
126125a1i 9 . . . 4 (⊤ → (ℂ D exp):ℂ⟶ℂ)
127126feqmptd 5467 . . 3 (⊤ → (ℂ D exp) = (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)))
1282a1i 9 . . . 4 (⊤ → exp:ℂ⟶ℂ)
129128feqmptd 5467 . . 3 (⊤ → exp = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
130116, 127, 1293eqtr4a 2196 . 2 (⊤ → (ℂ D exp) = exp)
131130mptru 1340 1 (ℂ D exp) = exp
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wtru 1332  wcel 1480  Vcvv 2681  wss 3066  {csn 3522  cop 3525   class class class wbr 3924  cmpt 3984   × cxp 4532  dom cdm 4534  ccom 4538  Fun wfun 5112  wf 5114  cfv 5118  (class class class)co 5767  𝑓 cof 5973  pm cpm 6536  cc 7611  0cc0 7613  1c1 7614   + caddc 7616   · cmul 7618  cmin 7926   # cap 8336  abscabs 10762  expce 11337  MetOpencmopn 12143   D cdv 12782
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 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733  ax-addf 7735  ax-mulf 7736
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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-disj 3902  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-of 5975  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-frec 6281  df-1o 6306  df-oadd 6310  df-er 6422  df-map 6537  df-pm 6538  df-en 6628  df-dom 6629  df-fin 6630  df-sup 6864  df-inf 6865  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-xneg 9552  df-xadd 9553  df-ico 9670  df-fz 9784  df-fzo 9913  df-seqfrec 10212  df-exp 10286  df-fac 10465  df-bc 10487  df-ihash 10515  df-shft 10580  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-clim 11041  df-sumdc 11116  df-ef 11343  df-rest 12111  df-topgen 12130  df-psmet 12145  df-xmet 12146  df-met 12147  df-bl 12148  df-mopn 12149  df-top 12154  df-topon 12167  df-bases 12199  df-ntr 12254  df-cn 12346  df-cnp 12347  df-tx 12411  df-cncf 12716  df-limced 12783  df-dvap 12784
This theorem is referenced by:  efcn  12846
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