dvexp - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > dvexp		GIF version

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 ↦ (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 5877	. . . . 5 ⊢ (𝑛 = 1 → (𝑥↑𝑛) = (𝑥↑1))
2	1	mpteq2dv 4091	. . . 4 ⊢ (𝑛 = 1 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑1)))
3	2	oveq2d 5885	. . 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 5876	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
6	5	oveq2d 5885	. . . . 5 ⊢ (𝑛 = 1 → (𝑥↑(𝑛 − 1)) = (𝑥↑(1 − 1)))
7	4, 6	oveq12d 5887	. . . 4 ⊢ (𝑛 = 1 → (𝑛 · (𝑥↑(𝑛 − 1))) = (1 · (𝑥↑(1 − 1))))
8	7	mpteq2dv 4091	. . 3 ⊢ (𝑛 = 1 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))))
9	3, 8	eqeq12d 2192	. 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 5877	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑥↑𝑛) = (𝑥↑𝑘))
11	10	mpteq2dv 4091	. . . 4 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
12	11	oveq2d 5885	. . 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 5876	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 − 1) = (𝑘 − 1))
15	14	oveq2d 5885	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑥↑(𝑛 − 1)) = (𝑥↑(𝑘 − 1)))
16	13, 15	oveq12d 5887	. . . 4 ⊢ (𝑛 = 𝑘 → (𝑛 · (𝑥↑(𝑛 − 1))) = (𝑘 · (𝑥↑(𝑘 − 1))))
17	16	mpteq2dv 4091	. . 3 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))))
18	12, 17	eqeq12d 2192	. 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 5877	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → (𝑥↑𝑛) = (𝑥↑(𝑘 + 1)))
20	19	mpteq2dv 4091	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
21	20	oveq2d 5885	. . 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 5876	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 − 1) = ((𝑘 + 1) − 1))
24	23	oveq2d 5885	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → (𝑥↑(𝑛 − 1)) = (𝑥↑((𝑘 + 1) − 1)))
25	22, 24	oveq12d 5887	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 · (𝑥↑(𝑛 − 1))) = ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))))
26	25	mpteq2dv 4091	. . 3 ⊢ (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))))
27	21, 26	eqeq12d 2192	. 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 5877	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑥↑𝑛) = (𝑥↑𝑁))
29	28	mpteq2dv 4091	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
30	29	oveq2d 5885	. . 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 5876	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1))
33	32	oveq2d 5885	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑥↑(𝑛 − 1)) = (𝑥↑(𝑁 − 1)))
34	31, 33	oveq12d 5887	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 · (𝑥↑(𝑛 − 1))) = (𝑁 · (𝑥↑(𝑁 − 1))))
35	34	mpteq2dv 4091	. . 3 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑛 · (𝑥↑(𝑛 − 1)))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
36	30, 35	eqeq12d 2192	. 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 10512	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (𝑥↑1) = 𝑥)
38	37	mpteq2ia 4086	. . . . 5 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑1)) = (𝑥 ∈ ℂ ↦ 𝑥)
39		mptresid 4957	. . . . 5 ⊢ (𝑥 ∈ ℂ ↦ 𝑥) = ( I ↾ ℂ)
40	38, 39	eqtri 2198	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑1)) = ( I ↾ ℂ)
41	40	oveq2i 5880	. . 3 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑1))) = (ℂ D ( I ↾ ℂ))
42		1m1e0 8977	. . . . . . . . . 10 ⊢ (1 − 1) = 0
43	42	oveq2i 5880	. . . . . . . . 9 ⊢ (𝑥↑(1 − 1)) = (𝑥↑0)
44		exp0 10510	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
45	43, 44	eqtrid 2222	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑥↑(1 − 1)) = 1)
46	45	oveq2d 5885	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (1 · (𝑥↑(1 − 1))) = (1 · 1))
47		1t1e1 9060	. . . . . . 7 ⊢ (1 · 1) = 1
48	46, 47	eqtrdi 2226	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (1 · (𝑥↑(1 − 1))) = 1)
49	48	mpteq2ia 4086	. . . . 5 ⊢ (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (𝑥 ∈ ℂ ↦ 1)
50		fconstmpt 4670	. . . . 5 ⊢ (ℂ × {1}) = (𝑥 ∈ ℂ ↦ 1)
51	49, 50	eqtr4i 2201	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (ℂ × {1})
52		dvid 13829	. . . 4 ⊢ (ℂ D ( I ↾ ℂ)) = (ℂ × {1})
53	51, 52	eqtr4i 2201	. . 3 ⊢ (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1)))) = (ℂ D ( I ↾ ℂ))
54	41, 53	eqtr4i 2201	. 2 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑1))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑(1 − 1))))
55		nncn 8916	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
56	55	adantr 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)
59	56, 57, 58	sylancl 413	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
60	59	oveq2d 5885	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑((𝑘 + 1) − 1)) = (𝑥↑𝑘))
61	60	oveq2d 5885	. . . . . . . 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 9172	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
65		expcl 10524	. . . . . . . . . 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 7973	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) · (𝑥↑𝑘)) = ((𝑘 · (𝑥↑𝑘)) + (1 · (𝑥↑𝑘))))
68	66	mulid2d 7966	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (1 · (𝑥↑𝑘)) = (𝑥↑𝑘))
69	68	oveq2d 5885	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥↑𝑘)) + (1 · (𝑥↑𝑘))) = ((𝑘 · (𝑥↑𝑘)) + (𝑥↑𝑘)))
70	61, 67, 69	3eqtrd 2214	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1))) = ((𝑘 · (𝑥↑𝑘)) + (𝑥↑𝑘)))
71	70	mpteq2dva 4090	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ ((𝑘 + 1) · (𝑥↑((𝑘 + 1) − 1)))) = (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑𝑘)) + (𝑥↑𝑘))))
72		cnex 7926	. . . . . . . 8 ⊢ ℂ ∈ V
73	72	a1i 9	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ℂ ∈ V)
74	56, 66	mulcld 7968	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑘 · (𝑥↑𝑘)) ∈ ℂ)
75		nnm1nn0 9206	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
76		expcl 10524	. . . . . . . . . . 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 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)))))
81	39	eqcomi 2181	. . . . . . . . . 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 6092	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥)))
84	56, 77, 79	mulassd 7971	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥) = (𝑘 · ((𝑥↑(𝑘 − 1)) · 𝑥)))
85		expm1t 10534	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝑥↑𝑘) = ((𝑥↑(𝑘 − 1)) · 𝑥))
86	85	ancoms 268	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑘) = ((𝑥↑(𝑘 − 1)) · 𝑥))
87	86	oveq2d 5885	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑘 · (𝑥↑𝑘)) = (𝑘 · ((𝑥↑(𝑘 − 1)) · 𝑥)))
88	84, 87	eqtr4d 2213	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥) = (𝑘 · (𝑥↑𝑘)))
89	88	mpteq2dva 4090	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ ((𝑘 · (𝑥↑(𝑘 − 1))) · 𝑥)) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑𝑘))))
90	83, 89	eqtrd 2210	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑𝑘))))
91	52, 50	eqtri 2198	. . . . . . . . . 10 ⊢ (ℂ D ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ 1)
92	91	a1i 9	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (ℂ D ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ 1))
93		eqidd 2178	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
94	73, 62, 66, 92, 93	offval2 6092	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (1 · (𝑥↑𝑘))))
95	68	mpteq2dva 4090	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (1 · (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
96	94, 95	eqtrd 2210	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((ℂ D ( I ↾ ℂ)) ∘_𝑓 · (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
97	73, 74, 66, 90, 96	offval2 6092	. . . . . 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 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 ↦ (𝑥↑𝑘)))))
99		oveq1 5876	. . . . . . 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 5884	. . . . . 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 2183	. . . . 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 2230	. . . 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 7938	. . . . . 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 3176	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → ℂ ⊆ ℂ)
106	66	fmpttd 5667	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)):ℂ⟶ℂ)
107	106	adantr 276	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1))))) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)):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)
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 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)))))
112	111	dmeqd 4825	. . . . . 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 5667	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑘 · (𝑥↑(𝑘 − 1)))):ℂ⟶ℂ)
114	113	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)
115	114	fdmd 5368	. . . . . 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 2210	. . . . 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 7943	. . . . . . . . 9 ⊢ 1 ∈ V
118	117	fconst 5407	. . . . . . . 8 ⊢ (ℂ × {1}):ℂ⟶{1}
119	52	feq1i 5354	. . . . . . . 8 ⊢ ((ℂ D ( I ↾ ℂ)):ℂ⟶{1} ↔ (ℂ × {1}):ℂ⟶{1})
120	118, 119	mpbir 146	. . . . . . 7 ⊢ (ℂ D ( I ↾ ℂ)):ℂ⟶{1}
121	120	fdmi 5369	. . . . . 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 13837	. . . 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 6092	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)))
125		expp1 10513	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑥↑(𝑘 + 1)) = ((𝑥↑𝑘) · 𝑥))
126	63, 64, 125	syl2anr 290	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ ℂ) → (𝑥↑(𝑘 + 1)) = ((𝑥↑𝑘) · 𝑥))
127	126	mpteq2dva 4090	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)))
128	124, 127	eqtr4d 2213	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∘_𝑓 · ( I ↾ ℂ)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
129	128	oveq2d 5885	. . . . 5 ⊢ (𝑘 ∈ ℕ → (ℂ D ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∘_𝑓 · ( I ↾ ℂ))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))))
130	129	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)))))
131	102, 123, 130	3eqtr2rd 2217	. . 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 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))))))
133	9, 18, 27, 36, 54, 132	nnind 8924	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 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-onto→wf1o 5211 (class class class)co 5869 ∘_𝑓 cof 6075 ℂcc 7800 ℝcr 7801 0cc0 7802 1c1 7803 + caddc 7805 · cmul 7807 − cmin 8118 ℕcn 8908 ℕ₀cn0 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

W3C validator