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Theorem dvexp 13315

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 5850	. . . . 5 ⊢ (𝑛 = 1 → (𝑥↑𝑛) = (𝑥↑1))
2	1	mpteq2dv 4073	. . . 4 ⊢ (𝑛 = 1 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑1)))
3	2	oveq2d 5858	. . 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 5849	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
6	5	oveq2d 5858	. . . . 5 ⊢ (𝑛 = 1 → (𝑥↑(𝑛 − 1)) = (𝑥↑(1 − 1)))
7	4, 6	oveq12d 5860	. . . 4 ⊢ (𝑛 = 1 → (𝑛 · (𝑥↑(𝑛 − 1))) = (1 · (𝑥↑(1 − 1))))
8	7	mpteq2dv 4073	. . 3 ⊢ (𝑛 = 1 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))))
9	3, 8	eqeq12d 2180	. 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 5850	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑥↑𝑛) = (𝑥↑𝑘))
11	10	mpteq2dv 4073	. . . 4 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
12	11	oveq2d 5858	. . 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 5849	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 − 1) = (𝑘 − 1))
15	14	oveq2d 5858	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑥↑(𝑛 − 1)) = (𝑥↑(𝑘 − 1)))
16	13, 15	oveq12d 5860	. . . 4 ⊢ (𝑛 = 𝑘 → (𝑛 · (𝑥↑(𝑛 − 1))) = (𝑘 · (𝑥↑(𝑘 − 1))))
17	16	mpteq2dv 4073	. . 3 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))))
18	12, 17	eqeq12d 2180	. 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 5850	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → (𝑥↑𝑛) = (𝑥↑(𝑘 + 1)))
20	19	mpteq2dv 4073	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
21	20	oveq2d 5858	. . 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 5849	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 − 1) = ((𝑘 + 1) − 1))
24	23	oveq2d 5858	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → (𝑥↑(𝑛 − 1)) = (𝑥↑((𝑘 + 1) − 1)))
25	22, 24	oveq12d 5860	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 · (𝑥↑(𝑛 − 1))) = ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))))
26	25	mpteq2dv 4073	. . 3 ⊢ (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))))
27	21, 26	eqeq12d 2180	. 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 5850	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑥↑𝑛) = (𝑥↑𝑁))
29	28	mpteq2dv 4073	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
30	29	oveq2d 5858	. . 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 5849	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1))
33	32	oveq2d 5858	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑥↑(𝑛 − 1)) = (𝑥↑(𝑁 − 1)))
34	31, 33	oveq12d 5860	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 · (𝑥↑(𝑛 − 1))) = (𝑁 · (𝑥↑(𝑁 − 1))))
35	34	mpteq2dv 4073	. . 3 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
36	30, 35	eqeq12d 2180	. 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 10461	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (𝑥↑1) = 𝑥)
38	37	mpteq2ia 4068	. . . . 5 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑1)) = (𝑥 ∈ ℂ ↦ 𝑥)
39		mptresid 4938	. . . . 5 ⊢ (𝑥 ∈ ℂ ↦ 𝑥) = ( I ↾ ℂ)
40	38, 39	eqtri 2186	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑1)) = ( I ↾ ℂ)
41	40	oveq2i 5853	. . 3 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑1))) = (ℂ D ( I ↾ ℂ))
42		1m1e0 8926	. . . . . . . . . 10 ⊢ (1 − 1) = 0
43	42	oveq2i 5853	. . . . . . . . 9 ⊢ (𝑥↑(1 − 1)) = (𝑥↑0)
44		exp0 10459	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
45	43, 44	syl5eq 2211	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑥↑(1 − 1)) = 1)
46	45	oveq2d 5858	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (1 · (𝑥↑(1 − 1))) = (1 · 1))
47		1t1e1 9009	. . . . . . 7 ⊢ (1 · 1) = 1
48	46, 47	eqtrdi 2215	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (1 · (𝑥↑(1 − 1))) = 1)
49	48	mpteq2ia 4068	. . . . 5 ⊢ (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (𝑥 ∈ ℂ ↦ 1)
50		fconstmpt 4651	. . . . 5 ⊢ (ℂ × {1}) = (𝑥 ∈ ℂ ↦ 1)
51	49, 50	eqtr4i 2189	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (ℂ × {1})
52		dvid 13302	. . . 4 ⊢ (ℂ D ( I ↾ ℂ)) = (ℂ × {1})
53	51, 52	eqtr4i 2189	. . 3 ⊢ (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (ℂ D ( I ↾ ℂ))
54	41, 53	eqtr4i 2189	. 2 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑1))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1))))
55		nncn 8865	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
56	55	adantr 274	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → 𝑘 ∈ ℂ)
57		ax-1cn 7846	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
58		pncan 8104	. . . . . . . . . . 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 5858	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑((𝑘 + 1) − 1)) = (𝑥↑𝑘))
61	60	oveq2d 5858	. . . . . . . 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 9121	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
65		expcl 10473	. . . . . . . . . 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 7924	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) · (𝑥↑𝑘)) = ((𝑘 · (𝑥↑𝑘)) + (1 · (𝑥↑𝑘))))
68	66	mulid2d 7917	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (1 · (𝑥↑𝑘)) = (𝑥↑𝑘))
69	68	oveq2d 5858	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥↑𝑘)) + (1 · (𝑥↑𝑘))) = ((𝑘 · (𝑥↑𝑘)) + (𝑥↑𝑘)))
70	61, 67, 69	3eqtrd 2202	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))) = ((𝑘 · (𝑥↑𝑘)) + (𝑥↑𝑘)))
71	70	mpteq2dva 4072	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑𝑘)) + (𝑥↑𝑘))))
72		cnex 7877	. . . . . . . 8 ⊢ ℂ ∈ V
73	72	a1i 9	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ℂ ∈ V)
74	56, 66	mulcld 7919	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑘 · (𝑥↑𝑘)) ∈ ℂ)
75		nnm1nn0 9155	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
76		expcl 10473	. . . . . . . . . . 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 7919	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑘 · (𝑥↑(𝑘 − 1))) ∈ ℂ)
79		simpr 109	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
80		eqidd 2166	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))))
81	39	eqcomi 2169	. . . . . . . . . 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 6065	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥)))
84	56, 77, 79	mulassd 7922	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥) = (𝑘 · ((𝑥↑(𝑘 − 1)) · 𝑥)))
85		expm1t 10483	. . . . . . . . . . . 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 5858	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑘 · (𝑥↑𝑘)) = (𝑘 · ((𝑥↑(𝑘 − 1)) · 𝑥)))
88	84, 87	eqtr4d 2201	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥) = (𝑘 · (𝑥↑𝑘)))
89	88	mpteq2dva 4072	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥)) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑𝑘))))
90	83, 89	eqtrd 2198	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑𝑘))))
91	52, 50	eqtri 2186	. . . . . . . . . 10 ⊢ (ℂ D ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ 1)
92	91	a1i 9	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (ℂ D ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ 1))
93		eqidd 2166	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
94	73, 62, 66, 92, 93	offval2 6065	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑𝑘))))
95	68	mpteq2dva 4072	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (1 · (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
96	94, 95	eqtrd 2198	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
97	73, 74, 66, 90, 96	offval2 6065	. . . . . 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 2201	. . . . 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 5849	. . . . . . 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 5857	. . . . . 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 2171	. . . . 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 2219	. . . 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 7889	. . . . . 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 3163	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → ℂ ⊆ ℂ)
106	66	fmpttd 5640	. . . . . 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 5470	. . . . . 6 ⊢ ( I ↾ ℂ):ℂ–1-1-onto→ℂ
109		f1of 5432	. . . . . 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 4806	. . . . . 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 5640	. . . . . . . 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 5344	. . . . . 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 2198	. . . . 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 7894	. . . . . . . . 9 ⊢ 1 ∈ V
118	117	fconst 5383	. . . . . . . 8 ⊢ (ℂ × {1}):ℂ⟶{1}
119	52	feq1i 5330	. . . . . . . 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 5345	. . . . . 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 13310	. . . 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 6065	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)))
125		expp1 10462	. . . . . . . . 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 4072	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)))
128	124, 127	eqtr4d 2201	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
129	128	oveq2d 5858	. . . . 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 2205	. . 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 8873	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 1343 ∈ wcel 2136 Vcvv 2726 {csn 3576 {cpr 3577 ↦ cmpt 4043 I cid 4266 × cxp 4602 dom cdm 4604 ↾ cres 4606 ⟶wf 5184 –1-1-onto→wf1o 5187 (class class class)co 5842 ∘_𝑓 cof 6048 ℂcc 7751 ℝcr 7752 0cc0 7753 1c1 7754 + caddc 7756 · cmul 7758 − cmin 8069 ℕcn 8857 ℕ₀cn0 9114 ↑cexp 10454 D cdv 13264

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873 ax-addf 7875 ax-mulf 7876

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-of 6050 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-map 6616 df-pm 6617 df-sup 6949 df-inf 6950 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-xneg 9708 df-xadd 9709 df-seqfrec 10381 df-exp 10455 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-rest 12558 df-topgen 12577 df-psmet 12627 df-xmet 12628 df-met 12629 df-bl 12630 df-mopn 12631 df-top 12636 df-topon 12649 df-bases 12681 df-ntr 12736 df-cn 12828 df-cnp 12829 df-tx 12893 df-cncf 13198 df-limced 13265 df-dvap 13266

This theorem is referenced by: dvexp2 13316

W3C validator