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Theorem dvexp 12883

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 5790	. . . . 5 ⊢ (𝑛 = 1 → (𝑥↑𝑛) = (𝑥↑1))
2	1	mpteq2dv 4027	. . . 4 ⊢ (𝑛 = 1 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑1)))
3	2	oveq2d 5798	. . 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 5789	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
6	5	oveq2d 5798	. . . . 5 ⊢ (𝑛 = 1 → (𝑥↑(𝑛 − 1)) = (𝑥↑(1 − 1)))
7	4, 6	oveq12d 5800	. . . 4 ⊢ (𝑛 = 1 → (𝑛 · (𝑥↑(𝑛 − 1))) = (1 · (𝑥↑(1 − 1))))
8	7	mpteq2dv 4027	. . 3 ⊢ (𝑛 = 1 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))))
9	3, 8	eqeq12d 2155	. 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 5790	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑥↑𝑛) = (𝑥↑𝑘))
11	10	mpteq2dv 4027	. . . 4 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
12	11	oveq2d 5798	. . 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 5789	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 − 1) = (𝑘 − 1))
15	14	oveq2d 5798	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑥↑(𝑛 − 1)) = (𝑥↑(𝑘 − 1)))
16	13, 15	oveq12d 5800	. . . 4 ⊢ (𝑛 = 𝑘 → (𝑛 · (𝑥↑(𝑛 − 1))) = (𝑘 · (𝑥↑(𝑘 − 1))))
17	16	mpteq2dv 4027	. . 3 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))))
18	12, 17	eqeq12d 2155	. 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 5790	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → (𝑥↑𝑛) = (𝑥↑(𝑘 + 1)))
20	19	mpteq2dv 4027	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
21	20	oveq2d 5798	. . 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 5789	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 − 1) = ((𝑘 + 1) − 1))
24	23	oveq2d 5798	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → (𝑥↑(𝑛 − 1)) = (𝑥↑((𝑘 + 1) − 1)))
25	22, 24	oveq12d 5800	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 · (𝑥↑(𝑛 − 1))) = ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))))
26	25	mpteq2dv 4027	. . 3 ⊢ (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))))
27	21, 26	eqeq12d 2155	. 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 5790	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑥↑𝑛) = (𝑥↑𝑁))
29	28	mpteq2dv 4027	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
30	29	oveq2d 5798	. . 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 5789	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1))
33	32	oveq2d 5798	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑥↑(𝑛 − 1)) = (𝑥↑(𝑁 − 1)))
34	31, 33	oveq12d 5800	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 · (𝑥↑(𝑛 − 1))) = (𝑁 · (𝑥↑(𝑁 − 1))))
35	34	mpteq2dv 4027	. . 3 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
36	30, 35	eqeq12d 2155	. 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 10330	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (𝑥↑1) = 𝑥)
38	37	mpteq2ia 4022	. . . . 5 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑1)) = (𝑥 ∈ ℂ ↦ 𝑥)
39		mptresid 4881	. . . . 5 ⊢ (𝑥 ∈ ℂ ↦ 𝑥) = ( I ↾ ℂ)
40	38, 39	eqtri 2161	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑1)) = ( I ↾ ℂ)
41	40	oveq2i 5793	. . 3 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑1))) = (ℂ D ( I ↾ ℂ))
42		1m1e0 8813	. . . . . . . . . 10 ⊢ (1 − 1) = 0
43	42	oveq2i 5793	. . . . . . . . 9 ⊢ (𝑥↑(1 − 1)) = (𝑥↑0)
44		exp0 10328	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
45	43, 44	syl5eq 2185	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑥↑(1 − 1)) = 1)
46	45	oveq2d 5798	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (1 · (𝑥↑(1 − 1))) = (1 · 1))
47		1t1e1 8896	. . . . . . 7 ⊢ (1 · 1) = 1
48	46, 47	eqtrdi 2189	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (1 · (𝑥↑(1 − 1))) = 1)
49	48	mpteq2ia 4022	. . . . 5 ⊢ (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (𝑥 ∈ ℂ ↦ 1)
50		fconstmpt 4594	. . . . 5 ⊢ (ℂ × {1}) = (𝑥 ∈ ℂ ↦ 1)
51	49, 50	eqtr4i 2164	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (ℂ × {1})
52		dvid 12870	. . . 4 ⊢ (ℂ D ( I ↾ ℂ)) = (ℂ × {1})
53	51, 52	eqtr4i 2164	. . 3 ⊢ (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (ℂ D ( I ↾ ℂ))
54	41, 53	eqtr4i 2164	. 2 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑1))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1))))
55		nncn 8752	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
56	55	adantr 274	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → 𝑘 ∈ ℂ)
57		ax-1cn 7737	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
58		pncan 7992	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
59	56, 57, 58	sylancl 410	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
60	59	oveq2d 5798	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑((𝑘 + 1) − 1)) = (𝑥↑𝑘))
61	60	oveq2d 5798	. . . . . . . 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 9008	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
65		expcl 10342	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑥↑𝑘) ∈ ℂ)
66	63, 64, 65	syl2anr 288	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑘) ∈ ℂ)
67	56, 62, 66	adddird 7815	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) · (𝑥↑𝑘)) = ((𝑘 · (𝑥↑𝑘)) + (1 · (𝑥↑𝑘))))
68	66	mulid2d 7808	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (1 · (𝑥↑𝑘)) = (𝑥↑𝑘))
69	68	oveq2d 5798	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥↑𝑘)) + (1 · (𝑥↑𝑘))) = ((𝑘 · (𝑥↑𝑘)) + (𝑥↑𝑘)))
70	61, 67, 69	3eqtrd 2177	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))) = ((𝑘 · (𝑥↑𝑘)) + (𝑥↑𝑘)))
71	70	mpteq2dva 4026	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑𝑘)) + (𝑥↑𝑘))))
72		cnex 7768	. . . . . . . 8 ⊢ ℂ ∈ V
73	72	a1i 9	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ℂ ∈ V)
74	56, 66	mulcld 7810	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑘 · (𝑥↑𝑘)) ∈ ℂ)
75		nnm1nn0 9042	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
76		expcl 10342	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (𝑘 − 1) ∈ ℕ₀) → (𝑥↑(𝑘 − 1)) ∈ ℂ)
77	63, 75, 76	syl2anr 288	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑(𝑘 − 1)) ∈ ℂ)
78	56, 77	mulcld 7810	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑘 · (𝑥↑(𝑘 − 1))) ∈ ℂ)
79		simpr 109	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
80		eqidd 2141	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))))
81	39	eqcomi 2144	. . . . . . . . . 10 ⊢ ( I ↾ ℂ) = (𝑥 ∈ ℂ ↦ 𝑥)
82	81	a1i 9	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ( I ↾ ℂ) = (𝑥 ∈ ℂ ↦ 𝑥))
83	73, 78, 79, 80, 82	offval2 6005	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥)))
84	56, 77, 79	mulassd 7813	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥) = (𝑘 · ((𝑥↑(𝑘 − 1)) · 𝑥)))
85		expm1t 10352	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝑥↑𝑘) = ((𝑥↑(𝑘 − 1)) · 𝑥))
86	85	ancoms 266	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑘) = ((𝑥↑(𝑘 − 1)) · 𝑥))
87	86	oveq2d 5798	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑘 · (𝑥↑𝑘)) = (𝑘 · ((𝑥↑(𝑘 − 1)) · 𝑥)))
88	84, 87	eqtr4d 2176	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥) = (𝑘 · (𝑥↑𝑘)))
89	88	mpteq2dva 4026	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥)) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑𝑘))))
90	83, 89	eqtrd 2173	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑𝑘))))
91	52, 50	eqtri 2161	. . . . . . . . . 10 ⊢ (ℂ D ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ 1)
92	91	a1i 9	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (ℂ D ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ 1))
93		eqidd 2141	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
94	73, 62, 66, 92, 93	offval2 6005	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑𝑘))))
95	68	mpteq2dva 4026	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (1 · (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
96	94, 95	eqtrd 2173	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
97	73, 74, 66, 90, 96	offval2 6005	. . . . . 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))))
98	71, 97	eqtr4d 2176	. . . . 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)))))
99		oveq1 5789	. . . . . . 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)))
100	99	oveq1d 5797	. . . . . 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)))))
101	100	eqcomd 2146	. . . . 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)))))
102	98, 101	sylan9eq 2193	. . . 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)))))
103		cnelprrecn 7780	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
104	103	a1i 9	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → ℂ ∈ {ℝ, ℂ})
105		ssidd 3123	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → ℂ ⊆ ℂ)
106	66	fmpttd 5583	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)):ℂ⟶ℂ)
107	106	adantr 274	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)):ℂ⟶ℂ)
108		f1oi 5413	. . . . . 6 ⊢ ( I ↾ ℂ):ℂ–1-1-onto→ℂ
109		f1of 5375	. . . . . 6 ⊢ (( I ↾ ℂ):ℂ–1-1-onto→ℂ → ( I ↾ ℂ):ℂ⟶ℂ)
110	108, 109	mp1i 10	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → ( I ↾ ℂ):ℂ⟶ℂ)
111		simpr 109	. . . . . . 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)))))
112	111	dmeqd 4749	. . . . . 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)))))
113	78	fmpttd 5583	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))):ℂ⟶ℂ)
114	113	adantr 274	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))):ℂ⟶ℂ)
115	114	fdmd 5287	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → dom (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) = ℂ)
116	112, 115	eqtrd 2173	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → dom (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = ℂ)
117		1ex 7785	. . . . . . . . 9 ⊢ 1 ∈ V
118	117	fconst 5326	. . . . . . . 8 ⊢ (ℂ × {1}):ℂ⟶{1}
119	52	feq1i 5273	. . . . . . . 8 ⊢ ((ℂ D ( I ↾ ℂ)):ℂ⟶{1} ↔ (ℂ × {1}):ℂ⟶{1})
120	118, 119	mpbir 145	. . . . . . 7 ⊢ (ℂ D ( I ↾ ℂ)):ℂ⟶{1}
121	120	fdmi 5288	. . . . . 6 ⊢ dom (ℂ D ( I ↾ ℂ)) = ℂ
122	121	a1i 9	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → dom (ℂ D ( I ↾ ℂ)) = ℂ)
123	104, 105, 107, 110, 116, 122	dvimulf 12878	. . . 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)))))
124	73, 66, 79, 93, 82	offval2 6005	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)))
125		expp1 10331	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑥↑(𝑘 + 1)) = ((𝑥↑𝑘) · 𝑥))
126	63, 64, 125	syl2anr 288	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑(𝑘 + 1)) = ((𝑥↑𝑘) · 𝑥))
127	126	mpteq2dva 4026	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)))
128	124, 127	eqtr4d 2176	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
129	128	oveq2d 5798	. . . . 5 ⊢ (𝑘 ∈ ℕ → (ℂ D ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∘_𝑓 · ( I ↾ ℂ))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))))
130	129	adantr 274	. . . 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)))))
131	102, 123, 130	3eqtr2rd 2180	. . 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)))))
132	131	ex 114	. 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))))))
133	9, 18, 27, 36, 54, 132	nnind 8760	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 103 = wceq 1332 ∈ wcel 1481 Vcvv 2689 {csn 3532 {cpr 3533 ↦ cmpt 3997 I cid 4218 × cxp 4545 dom cdm 4547 ↾ cres 4549 ⟶wf 5127 –1-1-onto→wf1o 5130 (class class class)co 5782 ∘_𝑓 cof 5988 ℂcc 7642 ℝcr 7643 0cc0 7644 1c1 7645 + caddc 7647 · cmul 7649 − cmin 7957 ℕcn 8744 ℕ₀cn0 9001 ↑cexp 10323 D cdv 12832

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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 ax-addf 7766 ax-mulf 7767

This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-of 5990 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-map 6552 df-pm 6553 df-sup 6879 df-inf 6880 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-xneg 9589 df-xadd 9590 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-rest 12161 df-topgen 12180 df-psmet 12195 df-xmet 12196 df-met 12197 df-bl 12198 df-mopn 12199 df-top 12204 df-topon 12217 df-bases 12249 df-ntr 12304 df-cn 12396 df-cnp 12397 df-tx 12461 df-cncf 12766 df-limced 12833 df-dvap 12834

This theorem is referenced by: dvexp2 12884

W3C validator