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Theorem dvexp3 26036

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 12653	. 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
2		cnelprrecn 11277	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
3	2	a1i 11	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → ℂ ∈ {ℝ, ℂ})
4		expcl 14130	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (𝑥↑𝑁) ∈ ℂ)
5	4	ancoms 458	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑁) ∈ ℂ)
6		c0ex 11284	. . . . . . 7 ⊢ 0 ∈ V
7		ovex 7481	. . . . . . 7 ⊢ (𝑁 · (𝑥↑(𝑁 − 1))) ∈ V
8	6, 7	ifex 4598	. . . . . 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 26012	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
11		difssd 4160	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (ℂ ∖ {0}) ⊆ ℂ)
12		eqid 2740	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	cnfldtopon 24824	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
14	13	toponrestid 22948	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
15	12	cnfldhaus 24826	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ Haus
16		0cn 11282	. . . . . . . 8 ⊢ 0 ∈ ℂ
17		unicntop 24827	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
18	17	sncld 23400	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Haus ∧ 0 ∈ ℂ) → {0} ∈ (Clsd‘(TopOpen‘ℂ_fld)))
19	15, 16, 18	mp2an 691	. . . . . . 7 ⊢ {0} ∈ (Clsd‘(TopOpen‘ℂ_fld))
20	17	cldopn 23060	. . . . . . 7 ⊢ ({0} ∈ (Clsd‘(TopOpen‘ℂ_fld)) → (ℂ ∖ {0}) ∈ (TopOpen‘ℂ_fld))
21	19, 20	ax-mp 5	. . . . . 6 ⊢ (ℂ ∖ {0}) ∈ (TopOpen‘ℂ_fld)
22	21	a1i 11	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (ℂ ∖ {0}) ∈ (TopOpen‘ℂ_fld))
23	3, 5, 9, 10, 11, 14, 12, 22	dvmptres 26021	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
24		ifid 4588	. . . . . 6 ⊢ if(𝑁 = 0, (𝑁 · (𝑥↑(𝑁 − 1))), (𝑁 · (𝑥↑(𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1)))
25		id 22	. . . . . . . . 9 ⊢ (𝑁 = 0 → 𝑁 = 0)
26		oveq1 7455	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝑁 − 1) = (0 − 1))
27	26	oveq2d 7464	. . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑥↑(𝑁 − 1)) = (𝑥↑(0 − 1)))
28	25, 27	oveq12d 7466	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁 · (𝑥↑(𝑁 − 1))) = (0 · (𝑥↑(0 − 1))))
29		eldifsn 4811	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
30		0z 12650	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
31		peano2zm 12686	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℤ → (0 − 1) ∈ ℤ)
32	30, 31	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0 − 1) ∈ ℤ
33		expclz 14135	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0 ∧ (0 − 1) ∈ ℤ) → (𝑥↑(0 − 1)) ∈ ℂ)
34	32, 33	mp3an3 1450	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (𝑥↑(0 − 1)) ∈ ℂ)
35	29, 34	sylbi 217	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → (𝑥↑(0 − 1)) ∈ ℂ)
36	35	adantl 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(0 − 1)) ∈ ℂ)
37	36	mul02d 11488	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) → (0 · (𝑥↑(0 − 1))) = 0)
38	28, 37	sylan9eqr 2802	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑁 = 0) → (𝑁 · (𝑥↑(𝑁 − 1))) = 0)
39	38	ifeq1da 4579	. . . . . 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)))))
40	24, 39	eqtr3id 2794	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑁 · (𝑥↑(𝑁 − 1))) = if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))))
41	40	mpteq2dva 5266	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
42	23, 41	eqtr4d 2783	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
43		eldifi 4154	. . . . . . . 8 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ∈ ℂ)
44	43	adantl 481	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
45		simpll 766	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑁 ∈ ℝ)
46	45	recnd 11318	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑁 ∈ ℂ)
47		nnnn0 12560	. . . . . . . 8 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℕ₀)
48	47	ad2antlr 726	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -𝑁 ∈ ℕ₀)
49		expneg2 14121	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ₀) → (𝑥↑𝑁) = (1 / (𝑥↑-𝑁)))
50	44, 46, 48, 49	syl3anc 1371	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑𝑁) = (1 / (𝑥↑-𝑁)))
51	50	mpteq2dva 5266	. . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / (𝑥↑-𝑁))))
52	51	oveq2d 7464	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / (𝑥↑-𝑁)))))
53	2	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → ℂ ∈ {ℝ, ℂ})
54		eldifsni 4815	. . . . . . . 8 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ≠ 0)
55	54	adantl 481	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0)
56		nnz 12660	. . . . . . . 8 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
57	56	ad2antlr 726	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -𝑁 ∈ ℤ)
58	44, 55, 57	expclzd 14201	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑-𝑁) ∈ ℂ)
59	44, 55, 57	expne0d 14202	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑-𝑁) ≠ 0)
60		eldifsn 4811	. . . . . 6 ⊢ ((𝑥↑-𝑁) ∈ (ℂ ∖ {0}) ↔ ((𝑥↑-𝑁) ∈ ℂ ∧ (𝑥↑-𝑁) ≠ 0))
61	58, 59, 60	sylanbrc 582	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑-𝑁) ∈ (ℂ ∖ {0}))
62		ovex 7481	. . . . . 6 ⊢ (-𝑁 · (𝑥↑(-𝑁 − 1))) ∈ V
63	62	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · (𝑥↑(-𝑁 − 1))) ∈ V)
64		simpr 484	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ (ℂ ∖ {0}))
65		eldifsn 4811	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
66	64, 65	sylib 218	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
67		reccl 11956	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (1 / 𝑦) ∈ ℂ)
68	66, 67	syl 17	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (1 / 𝑦) ∈ ℂ)
69		negex 11534	. . . . . 6 ⊢ -(1 / (𝑦↑2)) ∈ V
70	69	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(1 / (𝑦↑2)) ∈ V)
71		simpr 484	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
72	47	ad2antlr 726	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → -𝑁 ∈ ℕ₀)
73	71, 72	expcld 14196	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → (𝑥↑-𝑁) ∈ ℂ)
74	62	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → (-𝑁 · (𝑥↑(-𝑁 − 1))) ∈ V)
75		dvexp 26011	. . . . . . 7 ⊢ (-𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑-𝑁))) = (𝑥 ∈ ℂ ↦ (-𝑁 · (𝑥↑(-𝑁 − 1)))))
76	75	adantl 481	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑-𝑁))) = (𝑥 ∈ ℂ ↦ (-𝑁 · (𝑥↑(-𝑁 − 1)))))
77		difssd 4160	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ ∖ {0}) ⊆ ℂ)
78	21	a1i 11	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ ∖ {0}) ∈ (TopOpen‘ℂ_fld))
79	53, 73, 74, 76, 77, 14, 12, 78	dvmptres 26021	. . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑-𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (-𝑁 · (𝑥↑(-𝑁 − 1)))))
80		ax-1cn 11242	. . . . . 6 ⊢ 1 ∈ ℂ
81		dvrec 26013	. . . . . 6 ⊢ (1 ∈ ℂ → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
82	80, 81	mp1i 13	. . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
83		oveq2 7456	. . . . 5 ⊢ (𝑦 = (𝑥↑-𝑁) → (1 / 𝑦) = (1 / (𝑥↑-𝑁)))
84		oveq1 7455	. . . . . . 7 ⊢ (𝑦 = (𝑥↑-𝑁) → (𝑦↑2) = ((𝑥↑-𝑁)↑2))
85	84	oveq2d 7464	. . . . . 6 ⊢ (𝑦 = (𝑥↑-𝑁) → (1 / (𝑦↑2)) = (1 / ((𝑥↑-𝑁)↑2)))
86	85	negeqd 11530	. . . . 5 ⊢ (𝑦 = (𝑥↑-𝑁) → -(1 / (𝑦↑2)) = -(1 / ((𝑥↑-𝑁)↑2)))
87	53, 53, 61, 63, 68, 70, 79, 82, 83, 86	dvmptco 26030	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / (𝑥↑-𝑁)))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (-(1 / ((𝑥↑-𝑁)↑2)) · (-𝑁 · (𝑥↑(-𝑁 − 1))))))
88		2z 12675	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
89	88	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 2 ∈ ℤ)
90		expmulz 14159	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (-𝑁 ∈ ℤ ∧ 2 ∈ ℤ)) → (𝑥↑(-𝑁 · 2)) = ((𝑥↑-𝑁)↑2))
91	44, 55, 57, 89, 90	syl22anc 838	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 · 2)) = ((𝑥↑-𝑁)↑2))
92	91	eqcomd 2746	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((𝑥↑-𝑁)↑2) = (𝑥↑(-𝑁 · 2)))
93	92	oveq2d 7464	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (1 / ((𝑥↑-𝑁)↑2)) = (1 / (𝑥↑(-𝑁 · 2))))
94	93	negeqd 11530	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -(1 / ((𝑥↑-𝑁)↑2)) = -(1 / (𝑥↑(-𝑁 · 2))))
95		peano2zm 12686	. . . . . . . . . 10 ⊢ (-𝑁 ∈ ℤ → (-𝑁 − 1) ∈ ℤ)
96	57, 95	syl 17	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − 1) ∈ ℤ)
97	44, 55, 96	expclzd 14201	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 − 1)) ∈ ℂ)
98	46, 97	mulneg1d 11743	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · (𝑥↑(-𝑁 − 1))) = -(𝑁 · (𝑥↑(-𝑁 − 1))))
99	94, 98	oveq12d 7466	. . . . . 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		zmulcl 12692	. . . . . . . . . 10 ⊢ ((-𝑁 ∈ ℤ ∧ 2 ∈ ℤ) → (-𝑁 · 2) ∈ ℤ)
101	57, 88, 100	sylancl 585	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) ∈ ℤ)
102	44, 55, 101	expclzd 14201	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 · 2)) ∈ ℂ)
103	44, 55, 101	expne0d 14202	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 · 2)) ≠ 0)
104	102, 103	reccld 12063	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (1 / (𝑥↑(-𝑁 · 2))) ∈ ℂ)
105	46, 97	mulcld 11310	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑁 · (𝑥↑(-𝑁 − 1))) ∈ ℂ)
106	104, 105	mul2negd 11745	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-(1 / (𝑥↑(-𝑁 · 2))) · -(𝑁 · (𝑥↑(-𝑁 − 1)))) = ((1 / (𝑥↑(-𝑁 · 2))) · (𝑁 · (𝑥↑(-𝑁 − 1)))))
107	104, 46, 97	mul12d 11499	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((1 / (𝑥↑(-𝑁 · 2))) · (𝑁 · (𝑥↑(-𝑁 − 1)))) = (𝑁 · ((1 / (𝑥↑(-𝑁 · 2))) · (𝑥↑(-𝑁 − 1)))))
108	44, 55, 101, 96	expsubd 14207	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑((-𝑁 − 1) − (-𝑁 · 2))) = ((𝑥↑(-𝑁 − 1)) / (𝑥↑(-𝑁 · 2))))
109		nncn 12301	. . . . . . . . . . . . 13 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℂ)
110	109	ad2antlr 726	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -𝑁 ∈ ℂ)
111	80	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 1 ∈ ℂ)
112	101	zcnd 12748	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) ∈ ℂ)
113	110, 111, 112	sub32d 11679	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((-𝑁 − 1) − (-𝑁 · 2)) = ((-𝑁 − (-𝑁 · 2)) − 1))
114	110	times2d 12537	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) = (-𝑁 + -𝑁))
115	110, 46	negsubd 11653	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 + -𝑁) = (-𝑁 − 𝑁))
116	114, 115	eqtrd 2780	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) = (-𝑁 − 𝑁))
117	116	oveq2d 7464	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − (-𝑁 · 2)) = (-𝑁 − (-𝑁 − 𝑁)))
118	110, 46	nncand 11652	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − (-𝑁 − 𝑁)) = 𝑁)
119	117, 118	eqtrd 2780	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − (-𝑁 · 2)) = 𝑁)
120	119	oveq1d 7463	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((-𝑁 − (-𝑁 · 2)) − 1) = (𝑁 − 1))
121	113, 120	eqtrd 2780	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((-𝑁 − 1) − (-𝑁 · 2)) = (𝑁 − 1))
122	121	oveq2d 7464	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑((-𝑁 − 1) − (-𝑁 · 2))) = (𝑥↑(𝑁 − 1)))
123	97, 102, 103	divrec2d 12074	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((𝑥↑(-𝑁 − 1)) / (𝑥↑(-𝑁 · 2))) = ((1 / (𝑥↑(-𝑁 · 2))) · (𝑥↑(-𝑁 − 1))))
124	108, 122, 123	3eqtr3rd 2789	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((1 / (𝑥↑(-𝑁 · 2))) · (𝑥↑(-𝑁 − 1))) = (𝑥↑(𝑁 − 1)))
125	124	oveq2d 7464	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑁 · ((1 / (𝑥↑(-𝑁 · 2))) · (𝑥↑(-𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1))))
126	107, 125	eqtrd 2780	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((1 / (𝑥↑(-𝑁 · 2))) · (𝑁 · (𝑥↑(-𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1))))
127	99, 106, 126	3eqtrd 2784	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-(1 / ((𝑥↑-𝑁)↑2)) · (-𝑁 · (𝑥↑(-𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1))))
128	127	mpteq2dva 5266	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ (ℂ ∖ {0}) ↦ (-(1 / ((𝑥↑-𝑁)↑2)) · (-𝑁 · (𝑥↑(-𝑁 − 1))))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
129	52, 87, 128	3eqtrd 2784	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
130	42, 129	jaoi 856	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
131	1, 130	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 846 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 Vcvv 3488 ∖ cdif 3973 ifcif 4548 {csn 4648 {cpr 4650 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 − cmin 11520 -cneg 11521 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ↑cexp 14112 TopOpenctopn 17481 ℂ_fldccnfld 21387 Clsdccld 23045 Hauscha 23337 D cdv 25918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-t1 23343 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922

This theorem is referenced by: (None)

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