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Theorem dvexp 13842
Description: Derivative of a power function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
Assertion
Ref Expression
dvexp (𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑁))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
Distinct variable group:   𝑥,𝑁

Proof of Theorem dvexp
Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5877 . . . . 5 (𝑛 = 1 → (𝑥𝑛) = (𝑥↑1))
21mpteq2dv 4091 . . . 4 (𝑛 = 1 → (𝑥 ∈ ℂ ↦ (𝑥𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑1)))
32oveq2d 5885 . . 3 (𝑛 = 1 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑛))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑1))))
4 id 19 . . . . 5 (𝑛 = 1 → 𝑛 = 1)
5 oveq1 5876 . . . . . 6 (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
65oveq2d 5885 . . . . 5 (𝑛 = 1 → (𝑥↑(𝑛 − 1)) = (𝑥↑(1 − 1)))
74, 6oveq12d 5887 . . . 4 (𝑛 = 1 → (𝑛 · (𝑥↑(𝑛 − 1))) = (1 · (𝑥↑(1 − 1))))
87mpteq2dv 4091 . . 3 (𝑛 = 1 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))))
93, 8eqeq12d 2192 . 2 (𝑛 = 1 → ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑛))) = (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) ↔ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑1))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1))))))
10 oveq2 5877 . . . . 5 (𝑛 = 𝑘 → (𝑥𝑛) = (𝑥𝑘))
1110mpteq2dv 4091 . . . 4 (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥𝑘)))
1211oveq2d 5885 . . 3 (𝑛 = 𝑘 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑛))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))))
13 id 19 . . . . 5 (𝑛 = 𝑘𝑛 = 𝑘)
14 oveq1 5876 . . . . . 6 (𝑛 = 𝑘 → (𝑛 − 1) = (𝑘 − 1))
1514oveq2d 5885 . . . . 5 (𝑛 = 𝑘 → (𝑥↑(𝑛 − 1)) = (𝑥↑(𝑘 − 1)))
1613, 15oveq12d 5887 . . . 4 (𝑛 = 𝑘 → (𝑛 · (𝑥↑(𝑛 − 1))) = (𝑘 · (𝑥↑(𝑘 − 1))))
1716mpteq2dv 4091 . . 3 (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))))
1812, 17eqeq12d 2192 . 2 (𝑛 = 𝑘 → ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑛))) = (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) ↔ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))))
19 oveq2 5877 . . . . 5 (𝑛 = (𝑘 + 1) → (𝑥𝑛) = (𝑥↑(𝑘 + 1)))
2019mpteq2dv 4091 . . . 4 (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
2120oveq2d 5885 . . 3 (𝑛 = (𝑘 + 1) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑛))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))))
22 id 19 . . . . 5 (𝑛 = (𝑘 + 1) → 𝑛 = (𝑘 + 1))
23 oveq1 5876 . . . . . 6 (𝑛 = (𝑘 + 1) → (𝑛 − 1) = ((𝑘 + 1) − 1))
2423oveq2d 5885 . . . . 5 (𝑛 = (𝑘 + 1) → (𝑥↑(𝑛 − 1)) = (𝑥↑((𝑘 + 1) − 1)))
2522, 24oveq12d 5887 . . . 4 (𝑛 = (𝑘 + 1) → (𝑛 · (𝑥↑(𝑛 − 1))) = ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))))
2625mpteq2dv 4091 . . 3 (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))))
2721, 26eqeq12d 2192 . 2 (𝑛 = (𝑘 + 1) → ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑛))) = (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) ↔ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))))))
28 oveq2 5877 . . . . 5 (𝑛 = 𝑁 → (𝑥𝑛) = (𝑥𝑁))
2928mpteq2dv 4091 . . . 4 (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥𝑁)))
3029oveq2d 5885 . . 3 (𝑛 = 𝑁 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑛))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑁))))
31 id 19 . . . . 5 (𝑛 = 𝑁𝑛 = 𝑁)
32 oveq1 5876 . . . . . 6 (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1))
3332oveq2d 5885 . . . . 5 (𝑛 = 𝑁 → (𝑥↑(𝑛 − 1)) = (𝑥↑(𝑁 − 1)))
3431, 33oveq12d 5887 . . . 4 (𝑛 = 𝑁 → (𝑛 · (𝑥↑(𝑛 − 1))) = (𝑁 · (𝑥↑(𝑁 − 1))))
3534mpteq2dv 4091 . . 3 (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
3630, 35eqeq12d 2192 . 2 (𝑛 = 𝑁 → ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑛))) = (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) ↔ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑁))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1))))))
37 exp1 10512 . . . . . 6 (𝑥 ∈ ℂ → (𝑥↑1) = 𝑥)
3837mpteq2ia 4086 . . . . 5 (𝑥 ∈ ℂ ↦ (𝑥↑1)) = (𝑥 ∈ ℂ ↦ 𝑥)
39 mptresid 4957 . . . . 5 (𝑥 ∈ ℂ ↦ 𝑥) = ( I ↾ ℂ)
4038, 39eqtri 2198 . . . 4 (𝑥 ∈ ℂ ↦ (𝑥↑1)) = ( I ↾ ℂ)
4140oveq2i 5880 . . 3 (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑1))) = (ℂ D ( I ↾ ℂ))
42 1m1e0 8977 . . . . . . . . . 10 (1 − 1) = 0
4342oveq2i 5880 . . . . . . . . 9 (𝑥↑(1 − 1)) = (𝑥↑0)
44 exp0 10510 . . . . . . . . 9 (𝑥 ∈ ℂ → (𝑥↑0) = 1)
4543, 44eqtrid 2222 . . . . . . . 8 (𝑥 ∈ ℂ → (𝑥↑(1 − 1)) = 1)
4645oveq2d 5885 . . . . . . 7 (𝑥 ∈ ℂ → (1 · (𝑥↑(1 − 1))) = (1 · 1))
47 1t1e1 9060 . . . . . . 7 (1 · 1) = 1
4846, 47eqtrdi 2226 . . . . . 6 (𝑥 ∈ ℂ → (1 · (𝑥↑(1 − 1))) = 1)
4948mpteq2ia 4086 . . . . 5 (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (𝑥 ∈ ℂ ↦ 1)
50 fconstmpt 4670 . . . . 5 (ℂ × {1}) = (𝑥 ∈ ℂ ↦ 1)
5149, 50eqtr4i 2201 . . . 4 (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (ℂ × {1})
52 dvid 13829 . . . 4 (ℂ D ( I ↾ ℂ)) = (ℂ × {1})
5351, 52eqtr4i 2201 . . 3 (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (ℂ D ( I ↾ ℂ))
5441, 53eqtr4i 2201 . 2 (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑1))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1))))
55 nncn 8916 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
5655adantr 276 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → 𝑘 ∈ ℂ)
57 ax-1cn 7895 . . . . . . . . . . 11 1 ∈ ℂ
58 pncan 8153 . . . . . . . . . . 11 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
5956, 57, 58sylancl 413 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
6059oveq2d 5885 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑((𝑘 + 1) − 1)) = (𝑥𝑘))
6160oveq2d 5885 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))) = ((𝑘 + 1) · (𝑥𝑘)))
6257a1i 9 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
63 id 19 . . . . . . . . . 10 (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
64 nnnn0 9172 . . . . . . . . . 10 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
65 expcl 10524 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑥𝑘) ∈ ℂ)
6663, 64, 65syl2anr 290 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥𝑘) ∈ ℂ)
6756, 62, 66adddird 7973 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) · (𝑥𝑘)) = ((𝑘 · (𝑥𝑘)) + (1 · (𝑥𝑘))))
6866mulid2d 7966 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (1 · (𝑥𝑘)) = (𝑥𝑘))
6968oveq2d 5885 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥𝑘)) + (1 · (𝑥𝑘))) = ((𝑘 · (𝑥𝑘)) + (𝑥𝑘)))
7061, 67, 693eqtrd 2214 . . . . . . 7 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))) = ((𝑘 · (𝑥𝑘)) + (𝑥𝑘)))
7170mpteq2dva 4090 . . . . . 6 (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥𝑘)) + (𝑥𝑘))))
72 cnex 7926 . . . . . . . 8 ℂ ∈ V
7372a1i 9 . . . . . . 7 (𝑘 ∈ ℕ → ℂ ∈ V)
7456, 66mulcld 7968 . . . . . . 7 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑘 · (𝑥𝑘)) ∈ ℂ)
75 nnm1nn0 9206 . . . . . . . . . . 11 (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ0)
76 expcl 10524 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ (𝑘 − 1) ∈ ℕ0) → (𝑥↑(𝑘 − 1)) ∈ ℂ)
7763, 75, 76syl2anr 290 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑(𝑘 − 1)) ∈ ℂ)
7856, 77mulcld 7968 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑘 · (𝑥↑(𝑘 − 1))) ∈ ℂ)
79 simpr 110 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
80 eqidd 2178 . . . . . . . . 9 (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))))
8139eqcomi 2181 . . . . . . . . . 10 ( I ↾ ℂ) = (𝑥 ∈ ℂ ↦ 𝑥)
8281a1i 9 . . . . . . . . 9 (𝑘 ∈ ℕ → ( I ↾ ℂ) = (𝑥 ∈ ℂ ↦ 𝑥))
8373, 78, 79, 80, 82offval2 6092 . . . . . . . 8 (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥)))
8456, 77, 79mulassd 7971 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥) = (𝑘 · ((𝑥↑(𝑘 − 1)) · 𝑥)))
85 expm1t 10534 . . . . . . . . . . . 12 ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝑥𝑘) = ((𝑥↑(𝑘 − 1)) · 𝑥))
8685ancoms 268 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥𝑘) = ((𝑥↑(𝑘 − 1)) · 𝑥))
8786oveq2d 5885 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑘 · (𝑥𝑘)) = (𝑘 · ((𝑥↑(𝑘 − 1)) · 𝑥)))
8884, 87eqtr4d 2213 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥) = (𝑘 · (𝑥𝑘)))
8988mpteq2dva 4090 . . . . . . . 8 (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥)) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥𝑘))))
9083, 89eqtrd 2210 . . . . . . 7 (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥𝑘))))
9152, 50eqtri 2198 . . . . . . . . . 10 (ℂ D ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ 1)
9291a1i 9 . . . . . . . . 9 (𝑘 ∈ ℕ → (ℂ D ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ 1))
93 eqidd 2178 . . . . . . . . 9 (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑥𝑘)) = (𝑥 ∈ ℂ ↦ (𝑥𝑘)))
9473, 62, 66, 92, 93offval2 6092 . . . . . . . 8 (𝑘 ∈ ℕ → ((ℂ D ( I ↾ ℂ)) ∘𝑓 · (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥𝑘))))
9568mpteq2dva 4090 . . . . . . . 8 (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (1 · (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑥𝑘)))
9694, 95eqtrd 2210 . . . . . . 7 (𝑘 ∈ ℕ → ((ℂ D ( I ↾ ℂ)) ∘𝑓 · (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑥𝑘)))
9773, 74, 66, 90, 96offval2 6092 . . . . . 6 (𝑘 ∈ ℕ → (((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘𝑓 · ( I ↾ ℂ)) ∘𝑓 + ((ℂ D ( I ↾ ℂ)) ∘𝑓 · (𝑥 ∈ ℂ ↦ (𝑥𝑘)))) = (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥𝑘)) + (𝑥𝑘))))
9871, 97eqtr4d 2213 . . . . 5 (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))) = (((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘𝑓 · ( I ↾ ℂ)) ∘𝑓 + ((ℂ D ( I ↾ ℂ)) ∘𝑓 · (𝑥 ∈ ℂ ↦ (𝑥𝑘)))))
99 oveq1 5876 . . . . . . 7 ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) → ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) ∘𝑓 · ( I ↾ ℂ)) = ((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘𝑓 · ( I ↾ ℂ)))
10099oveq1d 5884 . . . . . 6 ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) → (((ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) ∘𝑓 · ( I ↾ ℂ)) ∘𝑓 + ((ℂ D ( I ↾ ℂ)) ∘𝑓 · (𝑥 ∈ ℂ ↦ (𝑥𝑘)))) = (((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘𝑓 · ( I ↾ ℂ)) ∘𝑓 + ((ℂ D ( I ↾ ℂ)) ∘𝑓 · (𝑥 ∈ ℂ ↦ (𝑥𝑘)))))
101100eqcomd 2183 . . . . 5 ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) → (((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘𝑓 · ( I ↾ ℂ)) ∘𝑓 + ((ℂ D ( I ↾ ℂ)) ∘𝑓 · (𝑥 ∈ ℂ ↦ (𝑥𝑘)))) = (((ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) ∘𝑓 · ( I ↾ ℂ)) ∘𝑓 + ((ℂ D ( I ↾ ℂ)) ∘𝑓 · (𝑥 ∈ ℂ ↦ (𝑥𝑘)))))
10298, 101sylan9eq 2230 . . . 4 ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))) = (((ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) ∘𝑓 · ( I ↾ ℂ)) ∘𝑓 + ((ℂ D ( I ↾ ℂ)) ∘𝑓 · (𝑥 ∈ ℂ ↦ (𝑥𝑘)))))
103 cnelprrecn 7938 . . . . . 6 ℂ ∈ {ℝ, ℂ}
104103a1i 9 . . . . 5 ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → ℂ ∈ {ℝ, ℂ})
105 ssidd 3176 . . . . 5 ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → ℂ ⊆ ℂ)
10666fmpttd 5667 . . . . . 6 (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑥𝑘)):ℂ⟶ℂ)
107106adantr 276 . . . . 5 ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (𝑥 ∈ ℂ ↦ (𝑥𝑘)):ℂ⟶ℂ)
108 f1oi 5495 . . . . . 6 ( I ↾ ℂ):ℂ–1-1-onto→ℂ
109 f1of 5457 . . . . . 6 (( I ↾ ℂ):ℂ–1-1-onto→ℂ → ( I ↾ ℂ):ℂ⟶ℂ)
110108, 109mp1i 10 . . . . 5 ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → ( I ↾ ℂ):ℂ⟶ℂ)
111 simpr 110 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))))
112111dmeqd 4825 . . . . . 6 ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → dom (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = dom (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))))
11378fmpttd 5667 . . . . . . . 8 (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))):ℂ⟶ℂ)
114113adantr 276 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))):ℂ⟶ℂ)
115114fdmd 5368 . . . . . 6 ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → dom (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) = ℂ)
116112, 115eqtrd 2210 . . . . 5 ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → dom (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = ℂ)
117 1ex 7943 . . . . . . . . 9 1 ∈ V
118117fconst 5407 . . . . . . . 8 (ℂ × {1}):ℂ⟶{1}
11952feq1i 5354 . . . . . . . 8 ((ℂ D ( I ↾ ℂ)):ℂ⟶{1} ↔ (ℂ × {1}):ℂ⟶{1})
120118, 119mpbir 146 . . . . . . 7 (ℂ D ( I ↾ ℂ)):ℂ⟶{1}
121120fdmi 5369 . . . . . 6 dom (ℂ D ( I ↾ ℂ)) = ℂ
122121a1i 9 . . . . 5 ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → dom (ℂ D ( I ↾ ℂ)) = ℂ)
123104, 105, 107, 110, 116, 122dvimulf 13837 . . . 4 ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (ℂ D ((𝑥 ∈ ℂ ↦ (𝑥𝑘)) ∘𝑓 · ( I ↾ ℂ))) = (((ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) ∘𝑓 · ( I ↾ ℂ)) ∘𝑓 + ((ℂ D ( I ↾ ℂ)) ∘𝑓 · (𝑥 ∈ ℂ ↦ (𝑥𝑘)))))
12473, 66, 79, 93, 82offval2 6092 . . . . . . 7 (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑥𝑘)) ∘𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ ((𝑥𝑘) · 𝑥)))
125 expp1 10513 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑥↑(𝑘 + 1)) = ((𝑥𝑘) · 𝑥))
12663, 64, 125syl2anr 290 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑(𝑘 + 1)) = ((𝑥𝑘) · 𝑥))
127126mpteq2dva 4090 . . . . . . 7 (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑥 ∈ ℂ ↦ ((𝑥𝑘) · 𝑥)))
128124, 127eqtr4d 2213 . . . . . 6 (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑥𝑘)) ∘𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
129128oveq2d 5885 . . . . 5 (𝑘 ∈ ℕ → (ℂ D ((𝑥 ∈ ℂ ↦ (𝑥𝑘)) ∘𝑓 · ( I ↾ ℂ))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))))
130129adantr 276 . . . 4 ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (ℂ D ((𝑥 ∈ ℂ ↦ (𝑥𝑘)) ∘𝑓 · ( I ↾ ℂ))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))))
131102, 123, 1303eqtr2rd 2217 . . 3 ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))))
132131ex 115 . 2 (𝑘 ∈ ℕ → ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))))))
1339, 18, 27, 36, 54, 132nnind 8924 1 (𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑁))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  {csn 3591  {cpr 3592  cmpt 4061   I cid 4285   × cxp 4621  dom cdm 4623  cres 4625  wf 5208  1-1-ontowf1o 5211  (class class class)co 5869  𝑓 cof 6075  cc 7800  cr 7801  0cc0 7802  1c1 7803   + caddc 7805   · cmul 7807  cmin 8118  cn 8908  0cn0 9165  cexp 10505   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-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-frec 6386  df-map 6644  df-pm 6645  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-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  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:  dvexp2  13843
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