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 25858

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 12519	. 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
2		cnelprrecn 11137	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
3	2	a1i 11	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → ℂ ∈ {ℝ, ℂ})
4		expcl 14020	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (𝑥↑𝑁) ∈ ℂ)
5	4	ancoms 458	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑁) ∈ ℂ)
6		c0ex 11144	. . . . . . 7 ⊢ 0 ∈ V
7		ovex 7402	. . . . . . 7 ⊢ (𝑁 · (𝑥↑(𝑁 − 1))) ∈ V
8	6, 7	ifex 4535	. . . . . 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 25834	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
11		difssd 4096	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (ℂ ∖ {0}) ⊆ ℂ)
12		eqid 2729	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	cnfldtopon 24646	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
14	13	toponrestid 22784	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
15		cnn0opn 24651	. . . . . 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 25843	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
18		ifid 4525	. . . . . 6 ⊢ if(𝑁 = 0, (𝑁 · (𝑥↑(𝑁 − 1))), (𝑁 · (𝑥↑(𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1)))
19		id 22	. . . . . . . . 9 ⊢ (𝑁 = 0 → 𝑁 = 0)
20		oveq1 7376	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝑁 − 1) = (0 − 1))
21	20	oveq2d 7385	. . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑥↑(𝑁 − 1)) = (𝑥↑(0 − 1)))
22	19, 21	oveq12d 7387	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁 · (𝑥↑(𝑁 − 1))) = (0 · (𝑥↑(0 − 1))))
23		eldifsn 4746	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
24		0z 12516	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
25		peano2zm 12552	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℤ → (0 − 1) ∈ ℤ)
26	24, 25	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0 − 1) ∈ ℤ
27		expclz 14025	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0 ∧ (0 − 1) ∈ ℤ) → (𝑥↑(0 − 1)) ∈ ℂ)
28	26, 27	mp3an3 1452	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (𝑥↑(0 − 1)) ∈ ℂ)
29	23, 28	sylbi 217	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → (𝑥↑(0 − 1)) ∈ ℂ)
30	29	adantl 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(0 − 1)) ∈ ℂ)
31	30	mul02d 11348	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) → (0 · (𝑥↑(0 − 1))) = 0)
32	22, 31	sylan9eqr 2786	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑁 = 0) → (𝑁 · (𝑥↑(𝑁 − 1))) = 0)
33	32	ifeq1da 4516	. . . . . 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 2778	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑁 · (𝑥↑(𝑁 − 1))) = if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))))
35	34	mpteq2dva 5195	. . . 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 2767	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
37		eldifi 4090	. . . . . . . 8 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ∈ ℂ)
38	37	adantl 481	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
39		simpll 766	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑁 ∈ ℝ)
40	39	recnd 11178	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑁 ∈ ℂ)
41		nnnn0 12425	. . . . . . . 8 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℕ₀)
42	41	ad2antlr 727	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -𝑁 ∈ ℕ₀)
43		expneg2 14011	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ₀) → (𝑥↑𝑁) = (1 / (𝑥↑-𝑁)))
44	38, 40, 42, 43	syl3anc 1373	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑𝑁) = (1 / (𝑥↑-𝑁)))
45	44	mpteq2dva 5195	. . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / (𝑥↑-𝑁))))
46	45	oveq2d 7385	. . . 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 4750	. . . . . . . 8 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ≠ 0)
49	48	adantl 481	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0)
50		nnz 12526	. . . . . . . 8 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
51	50	ad2antlr 727	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -𝑁 ∈ ℤ)
52	38, 49, 51	expclzd 14092	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑-𝑁) ∈ ℂ)
53	38, 49, 51	expne0d 14093	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑-𝑁) ≠ 0)
54		eldifsn 4746	. . . . . 6 ⊢ ((𝑥↑-𝑁) ∈ (ℂ ∖ {0}) ↔ ((𝑥↑-𝑁) ∈ ℂ ∧ (𝑥↑-𝑁) ≠ 0))
55	52, 53, 54	sylanbrc 583	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑-𝑁) ∈ (ℂ ∖ {0}))
56		ovex 7402	. . . . . 6 ⊢ (-𝑁 · (𝑥↑(-𝑁 − 1))) ∈ V
57	56	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · (𝑥↑(-𝑁 − 1))) ∈ V)
58		simpr 484	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ (ℂ ∖ {0}))
59		eldifsn 4746	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
60	58, 59	sylib 218	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
61		reccl 11820	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (1 / 𝑦) ∈ ℂ)
62	60, 61	syl 17	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (1 / 𝑦) ∈ ℂ)
63		negex 11395	. . . . . 6 ⊢ -(1 / (𝑦↑2)) ∈ V
64	63	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(1 / (𝑦↑2)) ∈ V)
65		simpr 484	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
66	41	ad2antlr 727	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → -𝑁 ∈ ℕ₀)
67	65, 66	expcld 14087	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → (𝑥↑-𝑁) ∈ ℂ)
68	56	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → (-𝑁 · (𝑥↑(-𝑁 − 1))) ∈ V)
69		dvexp 25833	. . . . . . 7 ⊢ (-𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑-𝑁))) = (𝑥 ∈ ℂ ↦ (-𝑁 · (𝑥↑(-𝑁 − 1)))))
70	69	adantl 481	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑-𝑁))) = (𝑥 ∈ ℂ ↦ (-𝑁 · (𝑥↑(-𝑁 − 1)))))
71		difssd 4096	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ ∖ {0}) ⊆ ℂ)
72	15	a1i 11	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ ∖ {0}) ∈ (TopOpen‘ℂ_fld))
73	47, 67, 68, 70, 71, 14, 12, 72	dvmptres 25843	. . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑-𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (-𝑁 · (𝑥↑(-𝑁 − 1)))))
74		ax-1cn 11102	. . . . . 6 ⊢ 1 ∈ ℂ
75		dvrec 25835	. . . . . 6 ⊢ (1 ∈ ℂ → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
76	74, 75	mp1i 13	. . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
77		oveq2 7377	. . . . 5 ⊢ (𝑦 = (𝑥↑-𝑁) → (1 / 𝑦) = (1 / (𝑥↑-𝑁)))
78		oveq1 7376	. . . . . . 7 ⊢ (𝑦 = (𝑥↑-𝑁) → (𝑦↑2) = ((𝑥↑-𝑁)↑2))
79	78	oveq2d 7385	. . . . . 6 ⊢ (𝑦 = (𝑥↑-𝑁) → (1 / (𝑦↑2)) = (1 / ((𝑥↑-𝑁)↑2)))
80	79	negeqd 11391	. . . . 5 ⊢ (𝑦 = (𝑥↑-𝑁) → -(1 / (𝑦↑2)) = -(1 / ((𝑥↑-𝑁)↑2)))
81	47, 47, 55, 57, 62, 64, 73, 76, 77, 80	dvmptco 25852	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / (𝑥↑-𝑁)))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (-(1 / ((𝑥↑-𝑁)↑2)) · (-𝑁 · (𝑥↑(-𝑁 − 1))))))
82		2z 12541	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
83	82	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 2 ∈ ℤ)
84		expmulz 14049	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (-𝑁 ∈ ℤ ∧ 2 ∈ ℤ)) → (𝑥↑(-𝑁 · 2)) = ((𝑥↑-𝑁)↑2))
85	38, 49, 51, 83, 84	syl22anc 838	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 · 2)) = ((𝑥↑-𝑁)↑2))
86	85	eqcomd 2735	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((𝑥↑-𝑁)↑2) = (𝑥↑(-𝑁 · 2)))
87	86	oveq2d 7385	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (1 / ((𝑥↑-𝑁)↑2)) = (1 / (𝑥↑(-𝑁 · 2))))
88	87	negeqd 11391	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -(1 / ((𝑥↑-𝑁)↑2)) = -(1 / (𝑥↑(-𝑁 · 2))))
89		peano2zm 12552	. . . . . . . . . 10 ⊢ (-𝑁 ∈ ℤ → (-𝑁 − 1) ∈ ℤ)
90	51, 89	syl 17	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − 1) ∈ ℤ)
91	38, 49, 90	expclzd 14092	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 − 1)) ∈ ℂ)
92	40, 91	mulneg1d 11607	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · (𝑥↑(-𝑁 − 1))) = -(𝑁 · (𝑥↑(-𝑁 − 1))))
93	88, 92	oveq12d 7387	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-(1 / ((𝑥↑-𝑁)↑2)) · (-𝑁 · (𝑥↑(-𝑁 − 1)))) = (-(1 / (𝑥↑(-𝑁 · 2))) · -(𝑁 · (𝑥↑(-𝑁 − 1)))))
94		zmulcl 12558	. . . . . . . . . 10 ⊢ ((-𝑁 ∈ ℤ ∧ 2 ∈ ℤ) → (-𝑁 · 2) ∈ ℤ)
95	51, 82, 94	sylancl 586	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) ∈ ℤ)
96	38, 49, 95	expclzd 14092	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 · 2)) ∈ ℂ)
97	38, 49, 95	expne0d 14093	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 · 2)) ≠ 0)
98	96, 97	reccld 11927	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (1 / (𝑥↑(-𝑁 · 2))) ∈ ℂ)
99	40, 91	mulcld 11170	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑁 · (𝑥↑(-𝑁 − 1))) ∈ ℂ)
100	98, 99	mul2negd 11609	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-(1 / (𝑥↑(-𝑁 · 2))) · -(𝑁 · (𝑥↑(-𝑁 − 1)))) = ((1 / (𝑥↑(-𝑁 · 2))) · (𝑁 · (𝑥↑(-𝑁 − 1)))))
101	98, 40, 91	mul12d 11359	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((1 / (𝑥↑(-𝑁 · 2))) · (𝑁 · (𝑥↑(-𝑁 − 1)))) = (𝑁 · ((1 / (𝑥↑(-𝑁 · 2))) · (𝑥↑(-𝑁 − 1)))))
102	38, 49, 95, 90	expsubd 14098	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑((-𝑁 − 1) − (-𝑁 · 2))) = ((𝑥↑(-𝑁 − 1)) / (𝑥↑(-𝑁 · 2))))
103		nncn 12170	. . . . . . . . . . . . 13 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℂ)
104	103	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -𝑁 ∈ ℂ)
105	74	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 1 ∈ ℂ)
106	95	zcnd 12615	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) ∈ ℂ)
107	104, 105, 106	sub32d 11541	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((-𝑁 − 1) − (-𝑁 · 2)) = ((-𝑁 − (-𝑁 · 2)) − 1))
108	104	times2d 12402	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) = (-𝑁 + -𝑁))
109	104, 40	negsubd 11515	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 + -𝑁) = (-𝑁 − 𝑁))
110	108, 109	eqtrd 2764	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) = (-𝑁 − 𝑁))
111	110	oveq2d 7385	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − (-𝑁 · 2)) = (-𝑁 − (-𝑁 − 𝑁)))
112	104, 40	nncand 11514	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − (-𝑁 − 𝑁)) = 𝑁)
113	111, 112	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − (-𝑁 · 2)) = 𝑁)
114	113	oveq1d 7384	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((-𝑁 − (-𝑁 · 2)) − 1) = (𝑁 − 1))
115	107, 114	eqtrd 2764	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((-𝑁 − 1) − (-𝑁 · 2)) = (𝑁 − 1))
116	115	oveq2d 7385	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑((-𝑁 − 1) − (-𝑁 · 2))) = (𝑥↑(𝑁 − 1)))
117	91, 96, 97	divrec2d 11938	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((𝑥↑(-𝑁 − 1)) / (𝑥↑(-𝑁 · 2))) = ((1 / (𝑥↑(-𝑁 · 2))) · (𝑥↑(-𝑁 − 1))))
118	102, 116, 117	3eqtr3rd 2773	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((1 / (𝑥↑(-𝑁 · 2))) · (𝑥↑(-𝑁 − 1))) = (𝑥↑(𝑁 − 1)))
119	118	oveq2d 7385	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑁 · ((1 / (𝑥↑(-𝑁 · 2))) · (𝑥↑(-𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1))))
120	101, 119	eqtrd 2764	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((1 / (𝑥↑(-𝑁 · 2))) · (𝑁 · (𝑥↑(-𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1))))
121	93, 100, 120	3eqtrd 2768	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-(1 / ((𝑥↑-𝑁)↑2)) · (-𝑁 · (𝑥↑(-𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1))))
122	121	mpteq2dva 5195	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ (ℂ ∖ {0}) ↦ (-(1 / ((𝑥↑-𝑁)↑2)) · (-𝑁 · (𝑥↑(-𝑁 − 1))))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
123	46, 81, 122	3eqtrd 2768	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
124	36, 123	jaoi 857	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
125	1, 124	sylbi 217	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 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 Vcvv 3444 ∖ cdif 3908 ifcif 4484 {csn 4585 {cpr 4587 ↦ cmpt 5183 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 − cmin 11381 -cneg 11382 / cdiv 11811 ℕcn 12162 2c2 12217 ℕ₀cn0 12418 ℤcz 12505 ↑cexp 14002 TopOpenctopn 17360 ℂ_fldccnfld 21240 D cdv 25740

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-t1 23177 df-haus 23178 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-limc 25743 df-dv 25744

This theorem is referenced by: (None)

W3C validator