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Theorem dvexp 15400

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 ↦ (𝑥↑𝑁))) = (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.

Step	Hyp	Ref	Expression
1		oveq2 6015	. . . . 5 ⊢ (𝑛 = 1 → (𝑥↑𝑛) = (𝑥↑1))
2	1	mpteq2dv 4175	. . . 4 ⊢ (𝑛 = 1 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑1)))
3	2	oveq2d 6023	. . 3 ⊢ (𝑛 = 1 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑛))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑1))))
4		id 19	. . . . 5 ⊢ (𝑛 = 1 → 𝑛 = 1)
5		oveq1 6014	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
6	5	oveq2d 6023	. . . . 5 ⊢ (𝑛 = 1 → (𝑥↑(𝑛 − 1)) = (𝑥↑(1 − 1)))
7	4, 6	oveq12d 6025	. . . 4 ⊢ (𝑛 = 1 → (𝑛 · (𝑥↑(𝑛 − 1))) = (1 · (𝑥↑(1 − 1))))
8	7	mpteq2dv 4175	. . 3 ⊢ (𝑛 = 1 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))))
9	3, 8	eqeq12d 2244	. 2 ⊢ (𝑛 = 1 → ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑛))) = (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) ↔ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑1))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1))))))
10		oveq2 6015	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑥↑𝑛) = (𝑥↑𝑘))
11	10	mpteq2dv 4175	. . . 4 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
12	11	oveq2d 6023	. . 3 ⊢ (𝑛 = 𝑘 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑛))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))))
13		id 19	. . . . 5 ⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘)
14		oveq1 6014	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 − 1) = (𝑘 − 1))
15	14	oveq2d 6023	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑥↑(𝑛 − 1)) = (𝑥↑(𝑘 − 1)))
16	13, 15	oveq12d 6025	. . . 4 ⊢ (𝑛 = 𝑘 → (𝑛 · (𝑥↑(𝑛 − 1))) = (𝑘 · (𝑥↑(𝑘 − 1))))
17	16	mpteq2dv 4175	. . 3 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))))
18	12, 17	eqeq12d 2244	. 2 ⊢ (𝑛 = 𝑘 → ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑛))) = (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) ↔ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))))
19		oveq2 6015	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → (𝑥↑𝑛) = (𝑥↑(𝑘 + 1)))
20	19	mpteq2dv 4175	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
21	20	oveq2d 6023	. . 3 ⊢ (𝑛 = (𝑘 + 1) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑛))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))))
22		id 19	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → 𝑛 = (𝑘 + 1))
23		oveq1 6014	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 − 1) = ((𝑘 + 1) − 1))
24	23	oveq2d 6023	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → (𝑥↑(𝑛 − 1)) = (𝑥↑((𝑘 + 1) − 1)))
25	22, 24	oveq12d 6025	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 · (𝑥↑(𝑛 − 1))) = ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))))
26	25	mpteq2dv 4175	. . 3 ⊢ (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))))
27	21, 26	eqeq12d 2244	. 2 ⊢ (𝑛 = (𝑘 + 1) → ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑛))) = (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) ↔ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))))))
28		oveq2 6015	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑥↑𝑛) = (𝑥↑𝑁))
29	28	mpteq2dv 4175	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
30	29	oveq2d 6023	. . 3 ⊢ (𝑛 = 𝑁 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑛))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))))
31		id 19	. . . . 5 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
32		oveq1 6014	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1))
33	32	oveq2d 6023	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑥↑(𝑛 − 1)) = (𝑥↑(𝑁 − 1)))
34	31, 33	oveq12d 6025	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 · (𝑥↑(𝑛 − 1))) = (𝑁 · (𝑥↑(𝑁 − 1))))
35	34	mpteq2dv 4175	. . 3 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
36	30, 35	eqeq12d 2244	. 2 ⊢ (𝑛 = 𝑁 → ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑛))) = (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) ↔ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1))))))
37		exp1 10779	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (𝑥↑1) = 𝑥)
38	37	mpteq2ia 4170	. . . . 5 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑1)) = (𝑥 ∈ ℂ ↦ 𝑥)
39		mptresid 5059	. . . . 5 ⊢ ( I ↾ ℂ) = (𝑥 ∈ ℂ ↦ 𝑥)
40	38, 39	eqtr4i 2253	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑1)) = ( I ↾ ℂ)
41	40	oveq2i 6018	. . 3 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑1))) = (ℂ D ( I ↾ ℂ))
42		1m1e0 9190	. . . . . . . . . 10 ⊢ (1 − 1) = 0
43	42	oveq2i 6018	. . . . . . . . 9 ⊢ (𝑥↑(1 − 1)) = (𝑥↑0)
44		exp0 10777	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
45	43, 44	eqtrid 2274	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑥↑(1 − 1)) = 1)
46	45	oveq2d 6023	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (1 · (𝑥↑(1 − 1))) = (1 · 1))
47		1t1e1 9274	. . . . . . 7 ⊢ (1 · 1) = 1
48	46, 47	eqtrdi 2278	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (1 · (𝑥↑(1 − 1))) = 1)
49	48	mpteq2ia 4170	. . . . 5 ⊢ (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (𝑥 ∈ ℂ ↦ 1)
50		fconstmpt 4766	. . . . 5 ⊢ (ℂ × {1}) = (𝑥 ∈ ℂ ↦ 1)
51	49, 50	eqtr4i 2253	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (ℂ × {1})
52		dvid 15384	. . . 4 ⊢ (ℂ D ( I ↾ ℂ)) = (ℂ × {1})
53	51, 52	eqtr4i 2253	. . 3 ⊢ (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (ℂ D ( I ↾ ℂ))
54	41, 53	eqtr4i 2253	. 2 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑1))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1))))
55		nncn 9129	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
56	55	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → 𝑘 ∈ ℂ)
57		ax-1cn 8103	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
58		pncan 8363	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
59	56, 57, 58	sylancl 413	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
60	59	oveq2d 6023	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑((𝑘 + 1) − 1)) = (𝑥↑𝑘))
61	60	oveq2d 6023	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))) = ((𝑘 + 1) · (𝑥↑𝑘)))
62	57	a1i 9	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
63		id 19	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
64		nnnn0 9387	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
65		expcl 10791	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑥↑𝑘) ∈ ℂ)
66	63, 64, 65	syl2anr 290	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑘) ∈ ℂ)
67	56, 62, 66	adddird 8183	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) · (𝑥↑𝑘)) = ((𝑘 · (𝑥↑𝑘)) + (1 · (𝑥↑𝑘))))
68	66	mulid2d 8176	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (1 · (𝑥↑𝑘)) = (𝑥↑𝑘))
69	68	oveq2d 6023	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥↑𝑘)) + (1 · (𝑥↑𝑘))) = ((𝑘 · (𝑥↑𝑘)) + (𝑥↑𝑘)))
70	61, 67, 69	3eqtrd 2266	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))) = ((𝑘 · (𝑥↑𝑘)) + (𝑥↑𝑘)))
71	70	mpteq2dva 4174	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑𝑘)) + (𝑥↑𝑘))))
72		cnex 8134	. . . . . . . 8 ⊢ ℂ ∈ V
73	72	a1i 9	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ℂ ∈ V)
74	56, 66	mulcld 8178	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑘 · (𝑥↑𝑘)) ∈ ℂ)
75		nnm1nn0 9421	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
76		expcl 10791	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (𝑘 − 1) ∈ ℕ₀) → (𝑥↑(𝑘 − 1)) ∈ ℂ)
77	63, 75, 76	syl2anr 290	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑(𝑘 − 1)) ∈ ℂ)
78	56, 77	mulcld 8178	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑘 · (𝑥↑(𝑘 − 1))) ∈ ℂ)
79		simpr 110	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
80		eqidd 2230	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))))
81	39	a1i 9	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ( I ↾ ℂ) = (𝑥 ∈ ℂ ↦ 𝑥))
82	73, 78, 79, 80, 81	offval2 6240	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥)))
83	56, 77, 79	mulassd 8181	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥) = (𝑘 · ((𝑥↑(𝑘 − 1)) · 𝑥)))
84		expm1t 10801	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝑥↑𝑘) = ((𝑥↑(𝑘 − 1)) · 𝑥))
85	84	ancoms 268	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑘) = ((𝑥↑(𝑘 − 1)) · 𝑥))
86	85	oveq2d 6023	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑘 · (𝑥↑𝑘)) = (𝑘 · ((𝑥↑(𝑘 − 1)) · 𝑥)))
87	83, 86	eqtr4d 2265	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥) = (𝑘 · (𝑥↑𝑘)))
88	87	mpteq2dva 4174	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥)) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑𝑘))))
89	82, 88	eqtrd 2262	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑𝑘))))
90	52, 50	eqtri 2250	. . . . . . . . . 10 ⊢ (ℂ D ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ 1)
91	90	a1i 9	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (ℂ D ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ 1))
92		eqidd 2230	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
93	73, 62, 66, 91, 92	offval2 6240	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑𝑘))))
94	68	mpteq2dva 4174	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (1 · (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
95	93, 94	eqtrd 2262	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
96	73, 74, 66, 89, 95	offval2 6240	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘_𝑓 · ( I ↾ ℂ)) ∘_𝑓 + ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))) = (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑𝑘)) + (𝑥↑𝑘))))
97	71, 96	eqtr4d 2265	. . . . 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 ↦ (𝑥↑𝑘)))))
98		oveq1 6014	. . . . . . 7 ⊢ ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) → ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) ∘_𝑓 · ( I ↾ ℂ)) = ((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘_𝑓 · ( I ↾ ℂ)))
99	98	oveq1d 6022	. . . . . 6 ⊢ ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) → (((ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) ∘_𝑓 · ( I ↾ ℂ)) ∘_𝑓 + ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))) = (((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘_𝑓 · ( I ↾ ℂ)) ∘_𝑓 + ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))))
100	99	eqcomd 2235	. . . . 5 ⊢ ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) → (((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘_𝑓 · ( I ↾ ℂ)) ∘_𝑓 + ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))) = (((ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) ∘_𝑓 · ( I ↾ ℂ)) ∘_𝑓 + ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))))
101	97, 100	sylan9eq 2282	. . . 4 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))) = (((ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) ∘_𝑓 · ( I ↾ ℂ)) ∘_𝑓 + ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))))
102		cnelprrecn 8146	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
103	102	a1i 9	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → ℂ ∈ {ℝ, ℂ})
104		ssidd 3245	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → ℂ ⊆ ℂ)
105	66	fmpttd 5792	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)):ℂ⟶ℂ)
106	105	adantr 276	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)):ℂ⟶ℂ)
107		f1oi 5613	. . . . . 6 ⊢ ( I ↾ ℂ):ℂ–1-1-onto→ℂ
108		f1of 5574	. . . . . 6 ⊢ (( I ↾ ℂ):ℂ–1-1-onto→ℂ → ( I ↾ ℂ):ℂ⟶ℂ)
109	107, 108	mp1i 10	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → ( I ↾ ℂ):ℂ⟶ℂ)
110		simpr 110	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))))
111	110	dmeqd 4925	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → dom (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = dom (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))))
112	78	fmpttd 5792	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))):ℂ⟶ℂ)
113	112	adantr 276	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))):ℂ⟶ℂ)
114	113	fdmd 5480	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → dom (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) = ℂ)
115	111, 114	eqtrd 2262	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → dom (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = ℂ)
116		1ex 8152	. . . . . . . . 9 ⊢ 1 ∈ V
117	116	fconst 5523	. . . . . . . 8 ⊢ (ℂ × {1}):ℂ⟶{1}
118	52	feq1i 5466	. . . . . . . 8 ⊢ ((ℂ D ( I ↾ ℂ)):ℂ⟶{1} ↔ (ℂ × {1}):ℂ⟶{1})
119	117, 118	mpbir 146	. . . . . . 7 ⊢ (ℂ D ( I ↾ ℂ)):ℂ⟶{1}
120	119	fdmi 5481	. . . . . 6 ⊢ dom (ℂ D ( I ↾ ℂ)) = ℂ
121	120	a1i 9	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → dom (ℂ D ( I ↾ ℂ)) = ℂ)
122	103, 104, 106, 109, 115, 121	dvimulf 15395	. . . 4 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (ℂ D ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∘_𝑓 · ( I ↾ ℂ))) = (((ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) ∘_𝑓 · ( I ↾ ℂ)) ∘_𝑓 + ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))))
123	73, 66, 79, 92, 81	offval2 6240	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)))
124		expp1 10780	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑥↑(𝑘 + 1)) = ((𝑥↑𝑘) · 𝑥))
125	63, 64, 124	syl2anr 290	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑(𝑘 + 1)) = ((𝑥↑𝑘) · 𝑥))
126	125	mpteq2dva 4174	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)))
127	123, 126	eqtr4d 2265	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
128	127	oveq2d 6023	. . . . 5 ⊢ (𝑘 ∈ ℕ → (ℂ D ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∘_𝑓 · ( I ↾ ℂ))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))))
129	128	adantr 276	. . . 4 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (ℂ D ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∘_𝑓 · ( I ↾ ℂ))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))))
130	101, 122, 129	3eqtr2rd 2269	. . 3 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))))
131	130	ex 115	. 2 ⊢ (𝑘 ∈ ℕ → ((ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))))))
132	9, 18, 27, 36, 54, 131	nnind 9137	1 ⊢ (𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1395 ∈ wcel 2200 Vcvv 2799 {csn 3666 {cpr 3667 ↦ cmpt 4145 I cid 4379 × cxp 4717 dom cdm 4719 ↾ cres 4721 ⟶wf 5314 –1-1-onto→wf1o 5317 (class class class)co 6007 ∘_𝑓 cof 6222 ℂcc 8008 ℝcr 8009 0cc0 8010 1c1 8011 + caddc 8013 · cmul 8015 − cmin 8328 ℕcn 9121 ℕ₀cn0 9380 ↑cexp 10772 D cdv 15344

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128 ax-arch 8129 ax-caucvg 8130 ax-addf 8132 ax-mulf 8133

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-of 6224 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-map 6805 df-pm 6806 df-sup 7162 df-inf 7163 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-inn 9122 df-2 9180 df-3 9181 df-4 9182 df-n0 9381 df-z 9458 df-uz 9734 df-q 9827 df-rp 9862 df-xneg 9980 df-xadd 9981 df-seqfrec 10682 df-exp 10773 df-cj 11368 df-re 11369 df-im 11370 df-rsqrt 11524 df-abs 11525 df-rest 13289 df-topgen 13308 df-psmet 14522 df-xmet 14523 df-met 14524 df-bl 14525 df-mopn 14526 df-top 14687 df-topon 14700 df-bases 14732 df-ntr 14785 df-cn 14877 df-cnp 14878 df-tx 14942 df-cncf 15260 df-limced 15345 df-dvap 15346

This theorem is referenced by: dvexp2 15401

W3C validator