dvexp3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvexp3		Structured version Visualization version GIF version

Theorem dvexp3 26137

Description: Derivative of an exponential of integer exponent. (Contributed by Mario Carneiro, 26-Feb-2015.)

**Assertion**
Ref	Expression
dvexp3	⊢ (𝑁 ∈ ℤ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))

Distinct variable group: 𝑥,𝑁

Proof of Theorem dvexp3 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elznn0nn 12600	. 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
2		cnelprrecn 11188	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
3	2	a1i 11	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → ℂ ∈ {ℝ, ℂ})
4		expcl 14111	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (𝑥↑𝑁) ∈ ℂ)
5	4	ancoms 463	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑁) ∈ ℂ)
6		c0ex 11195	. . . . . . 7 ⊢ 0 ∈ V
7		ovex 7443	. . . . . . 7 ⊢ (𝑁 · (𝑥↑(𝑁 − 1))) ∈ V
8	6, 7	ifex 4538	. . . . . 6 ⊢ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))) ∈ V
9	8	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ ℂ) → if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))) ∈ V)
10		dvexp2 26113	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
11		difssd 4091	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (ℂ ∖ {0}) ⊆ ℂ)
12		eqid 2763	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	cnfldtopon 24939	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
14	13	toponrestid 23078	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
15		cnn0opn 24944	. . . . . 6 ⊢ (ℂ ∖ {0}) ∈ (TopOpen‘ℂ_fld)
16	15	a1i 11	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (ℂ ∖ {0}) ∈ (TopOpen‘ℂ_fld))
17	3, 5, 9, 10, 11, 14, 12, 16	dvmptres 26122	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
18		ifid 4528	. . . . . 6 ⊢ if(𝑁 = 0, (𝑁 · (𝑥↑(𝑁 − 1))), (𝑁 · (𝑥↑(𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1)))
19		id 23	. . . . . . . . 9 ⊢ (𝑁 = 0 → 𝑁 = 0)
20		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝑁 − 1) = (0 − 1))
21	20	oveq2d 7426	. . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑥↑(𝑁 − 1)) = (𝑥↑(0 − 1)))
22	19, 21	oveq12d 7428	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁 · (𝑥↑(𝑁 − 1))) = (0 · (𝑥↑(0 − 1))))
23		eldifsn 4753	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
24		0z 12597	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
25		peano2zm 12632	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℤ → (0 − 1) ∈ ℤ)
26	24, 25	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0 − 1) ∈ ℤ
27		expclz 14116	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0 ∧ (0 − 1) ∈ ℤ) → (𝑥↑(0 − 1)) ∈ ℂ)
28	26, 27	mp3an3 1479	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (𝑥↑(0 − 1)) ∈ ℂ)
29	23, 28	sylbi 220	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → (𝑥↑(0 − 1)) ∈ ℂ)
30	29	adantl 486	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(0 − 1)) ∈ ℂ)
31	30	mul02d 11403	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) → (0 · (𝑥↑(0 − 1))) = 0)
32	22, 31	sylan9eqr 2820	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑁 = 0) → (𝑁 · (𝑥↑(𝑁 − 1))) = 0)
33	32	ifeq1da 4519	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) → if(𝑁 = 0, (𝑁 · (𝑥↑(𝑁 − 1))), (𝑁 · (𝑥↑(𝑁 − 1)))) = if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))))
34	18, 33	eqtr3id 2812	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑁 · (𝑥↑(𝑁 − 1))) = if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))))
35	34	mpteq2dva 5204	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
36	17, 35	eqtr4d 2801	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
37		eldifi 4085	. . . . . . . 8 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ∈ ℂ)
38	37	adantl 486	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
39		simpll 778	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑁 ∈ ℝ)
40	39	recnd 11232	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑁 ∈ ℂ)
41		nnnn0 12506	. . . . . . . 8 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℕ₀)
42	41	ad2antlr 739	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -𝑁 ∈ ℕ₀)
43		expneg2 14102	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ₀) → (𝑥↑𝑁) = (1 / (𝑥↑-𝑁)))
44	38, 40, 42, 43	syl3anc 1398	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑𝑁) = (1 / (𝑥↑-𝑁)))
45	44	mpteq2dva 5204	. . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / (𝑥↑-𝑁))))
46	45	oveq2d 7426	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / (𝑥↑-𝑁)))))
47	2	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → ℂ ∈ {ℝ, ℂ})
48		eldifsni 4758	. . . . . . . 8 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ≠ 0)
49	48	adantl 486	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0)
50		nnz 12607	. . . . . . . 8 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
51	50	ad2antlr 739	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -𝑁 ∈ ℤ)
52	38, 49, 51	expclzd 14183	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑-𝑁) ∈ ℂ)
53	38, 49, 51	expne0d 14184	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑-𝑁) ≠ 0)
54		eldifsn 4753	. . . . . 6 ⊢ ((𝑥↑-𝑁) ∈ (ℂ ∖ {0}) ↔ ((𝑥↑-𝑁) ∈ ℂ ∧ (𝑥↑-𝑁) ≠ 0))
55	52, 53, 54	sylanbrc 594	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑-𝑁) ∈ (ℂ ∖ {0}))
56		ovex 7443	. . . . . 6 ⊢ (-𝑁 · (𝑥↑(-𝑁 − 1))) ∈ V
57	56	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · (𝑥↑(-𝑁 − 1))) ∈ V)
58		eldifsn 4753	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
59	58	bilani 509	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
60		reccl 11874	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (1 / 𝑦) ∈ ℂ)
61	59, 60	syl 18	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (1 / 𝑦) ∈ ℂ)
62		negex 11450	. . . . . 6 ⊢ -(1 / (𝑦↑2)) ∈ V
63	62	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(1 / (𝑦↑2)) ∈ V)
64		simpr 489	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
65	41	ad2antlr 739	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → -𝑁 ∈ ℕ₀)
66	64, 65	expcld 14178	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → (𝑥↑-𝑁) ∈ ℂ)
67	56	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → (-𝑁 · (𝑥↑(-𝑁 − 1))) ∈ V)
68		dvexp 26112	. . . . . . 7 ⊢ (-𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑-𝑁))) = (𝑥 ∈ ℂ ↦ (-𝑁 · (𝑥↑(-𝑁 − 1)))))
69	68	adantl 486	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑-𝑁))) = (𝑥 ∈ ℂ ↦ (-𝑁 · (𝑥↑(-𝑁 − 1)))))
70		difssd 4091	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ ∖ {0}) ⊆ ℂ)
71	15	a1i 11	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ ∖ {0}) ∈ (TopOpen‘ℂ_fld))
72	47, 66, 67, 69, 70, 14, 12, 71	dvmptres 26122	. . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑-𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (-𝑁 · (𝑥↑(-𝑁 − 1)))))
73		ax-1cn 11153	. . . . . 6 ⊢ 1 ∈ ℂ
74		dvrec 26114	. . . . . 6 ⊢ (1 ∈ ℂ → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
75	73, 74	mp1i 14	. . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
76		oveq2 7418	. . . . 5 ⊢ (𝑦 = (𝑥↑-𝑁) → (1 / 𝑦) = (1 / (𝑥↑-𝑁)))
77		oveq1 7417	. . . . . . 7 ⊢ (𝑦 = (𝑥↑-𝑁) → (𝑦↑2) = ((𝑥↑-𝑁)↑2))
78	77	oveq2d 7426	. . . . . 6 ⊢ (𝑦 = (𝑥↑-𝑁) → (1 / (𝑦↑2)) = (1 / ((𝑥↑-𝑁)↑2)))
79	78	negeqd 11446	. . . . 5 ⊢ (𝑦 = (𝑥↑-𝑁) → -(1 / (𝑦↑2)) = -(1 / ((𝑥↑-𝑁)↑2)))
80	47, 47, 55, 57, 61, 63, 72, 75, 76, 79	dvmptco 26131	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / (𝑥↑-𝑁)))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (-(1 / ((𝑥↑-𝑁)↑2)) · (-𝑁 · (𝑥↑(-𝑁 − 1))))))
81		2z 12621	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
82	81	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 2 ∈ ℤ)
83		expmulz 14140	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (-𝑁 ∈ ℤ ∧ 2 ∈ ℤ)) → (𝑥↑(-𝑁 · 2)) = ((𝑥↑-𝑁)↑2))
84	38, 49, 51, 82, 83	syl22anc 851	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 · 2)) = ((𝑥↑-𝑁)↑2))
85	84	eqcomd 2769	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((𝑥↑-𝑁)↑2) = (𝑥↑(-𝑁 · 2)))
86	85	oveq2d 7426	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (1 / ((𝑥↑-𝑁)↑2)) = (1 / (𝑥↑(-𝑁 · 2))))
87	86	negeqd 11446	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -(1 / ((𝑥↑-𝑁)↑2)) = -(1 / (𝑥↑(-𝑁 · 2))))
88		peano2zm 12632	. . . . . . . . . 10 ⊢ (-𝑁 ∈ ℤ → (-𝑁 − 1) ∈ ℤ)
89	51, 88	syl 18	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − 1) ∈ ℤ)
90	38, 49, 89	expclzd 14183	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 − 1)) ∈ ℂ)
91	40, 90	mulneg1d 11662	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · (𝑥↑(-𝑁 − 1))) = -(𝑁 · (𝑥↑(-𝑁 − 1))))
92	87, 91	oveq12d 7428	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-(1 / ((𝑥↑-𝑁)↑2)) · (-𝑁 · (𝑥↑(-𝑁 − 1)))) = (-(1 / (𝑥↑(-𝑁 · 2))) · -(𝑁 · (𝑥↑(-𝑁 − 1)))))
93		zmulcl 12638	. . . . . . . . . 10 ⊢ ((-𝑁 ∈ ℤ ∧ 2 ∈ ℤ) → (-𝑁 · 2) ∈ ℤ)
94	51, 81, 93	sylancl 597	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) ∈ ℤ)
95	38, 49, 94	expclzd 14183	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 · 2)) ∈ ℂ)
96	38, 49, 94	expne0d 14184	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 · 2)) ≠ 0)
97	95, 96	reccld 11979	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (1 / (𝑥↑(-𝑁 · 2))) ∈ ℂ)
98	40, 90	mulcld 11224	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑁 · (𝑥↑(-𝑁 − 1))) ∈ ℂ)
99	97, 98	mul2negd 11664	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-(1 / (𝑥↑(-𝑁 · 2))) · -(𝑁 · (𝑥↑(-𝑁 − 1)))) = ((1 / (𝑥↑(-𝑁 · 2))) · (𝑁 · (𝑥↑(-𝑁 − 1)))))
100	97, 40, 90	mul12d 11414	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((1 / (𝑥↑(-𝑁 · 2))) · (𝑁 · (𝑥↑(-𝑁 − 1)))) = (𝑁 · ((1 / (𝑥↑(-𝑁 · 2))) · (𝑥↑(-𝑁 − 1)))))
101	38, 49, 94, 89	expsubd 14189	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑((-𝑁 − 1) − (-𝑁 · 2))) = ((𝑥↑(-𝑁 − 1)) / (𝑥↑(-𝑁 · 2))))
102		nncn 12236	. . . . . . . . . . . . 13 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℂ)
103	102	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -𝑁 ∈ ℂ)
104	73	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 1 ∈ ℂ)
105	94	zcnd 12696	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) ∈ ℂ)
106	103, 104, 105	sub32d 11596	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((-𝑁 − 1) − (-𝑁 · 2)) = ((-𝑁 − (-𝑁 · 2)) − 1))
107	103	times2d 12483	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) = (-𝑁 + -𝑁))
108	103, 40	negsubd 11570	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 + -𝑁) = (-𝑁 − 𝑁))
109	107, 108	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) = (-𝑁 − 𝑁))
110	109	oveq2d 7426	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − (-𝑁 · 2)) = (-𝑁 − (-𝑁 − 𝑁)))
111	103, 40	nncand 11569	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − (-𝑁 − 𝑁)) = 𝑁)
112	110, 111	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − (-𝑁 · 2)) = 𝑁)
113	112	oveq1d 7425	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((-𝑁 − (-𝑁 · 2)) − 1) = (𝑁 − 1))
114	106, 113	eqtrd 2798	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((-𝑁 − 1) − (-𝑁 · 2)) = (𝑁 − 1))
115	114	oveq2d 7426	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑((-𝑁 − 1) − (-𝑁 · 2))) = (𝑥↑(𝑁 − 1)))
116	90, 95, 96	divrec2d 11990	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((𝑥↑(-𝑁 − 1)) / (𝑥↑(-𝑁 · 2))) = ((1 / (𝑥↑(-𝑁 · 2))) · (𝑥↑(-𝑁 − 1))))
117	101, 115, 116	3eqtr3rd 2807	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((1 / (𝑥↑(-𝑁 · 2))) · (𝑥↑(-𝑁 − 1))) = (𝑥↑(𝑁 − 1)))
118	117	oveq2d 7426	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑁 · ((1 / (𝑥↑(-𝑁 · 2))) · (𝑥↑(-𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1))))
119	100, 118	eqtrd 2798	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((1 / (𝑥↑(-𝑁 · 2))) · (𝑁 · (𝑥↑(-𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1))))
120	92, 99, 119	3eqtrd 2802	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-(1 / ((𝑥↑-𝑁)↑2)) · (-𝑁 · (𝑥↑(-𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1))))
121	120	mpteq2dva 5204	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ (ℂ ∖ {0}) ↦ (-(1 / ((𝑥↑-𝑁)↑2)) · (-𝑁 · (𝑥↑(-𝑁 − 1))))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
122	46, 80, 121	3eqtrd 2802	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
123	36, 122	jaoi 870	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
124	1, 123	sylbi 220	1 ⊢ (𝑁 ∈ ℤ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 Vcvv 3455 ∖ cdif 3902 ifcif 4487 {csn 4589 {cpr 4591 ↦ cmpt 5192 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 0cc0 11095 1c1 11096 + caddc 11098 · cmul 11100 − cmin 11436 -cneg 11437 / cdiv 11866 ℕcn 12228 2c2 12290 ℕ₀cn0 12499 ℤcz 12586 ↑cexp 14093 TopOpenctopn 17469 ℂ_fldccnfld 21522 D cdv 26022

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-icc 13374 df-fz 13531 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-fbas 21519 df-fg 21520 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-nei 23255 df-lp 23293 df-perf 23294 df-cn 23384 df-cnp 23385 df-t1 23471 df-haus 23472 df-tx 23719 df-hmeo 23912 df-fil 24003 df-fm 24095 df-flim 24096 df-flf 24097 df-xms 24477 df-ms 24478 df-tms 24479 df-cncf 25037 df-limc 26025 df-dv 26026

This theorem is referenced by: (None)

W3C validator