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Theorem dvexp3 24581

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 11983	. 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
2		cnelprrecn 10619	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
3	2	a1i 11	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → ℂ ∈ {ℝ, ℂ})
4		expcl 13443	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (𝑥↑𝑁) ∈ ℂ)
5	4	ancoms 462	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑁) ∈ ℂ)
6		c0ex 10624	. . . . . . 7 ⊢ 0 ∈ V
7		ovex 7168	. . . . . . 7 ⊢ (𝑁 · (𝑥↑(𝑁 − 1))) ∈ V
8	6, 7	ifex 4473	. . . . . 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 24557	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
11		difssd 4060	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (ℂ ∖ {0}) ⊆ ℂ)
12		eqid 2798	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	cnfldtopon 23388	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
14	13	toponrestid 21526	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
15	12	cnfldhaus 23390	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ Haus
16		0cn 10622	. . . . . . . 8 ⊢ 0 ∈ ℂ
17		unicntop 23391	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
18	17	sncld 21976	. . . . . . . 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 21636	. . . . . . 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 24566	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
24		ifid 4464	. . . . . 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 7142	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝑁 − 1) = (0 − 1))
27	26	oveq2d 7151	. . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑥↑(𝑁 − 1)) = (𝑥↑(0 − 1)))
28	25, 27	oveq12d 7153	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁 · (𝑥↑(𝑁 − 1))) = (0 · (𝑥↑(0 − 1))))
29		eldifsn 4680	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
30		0z 11980	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
31		peano2zm 12013	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℤ → (0 − 1) ∈ ℤ)
32	30, 31	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0 − 1) ∈ ℤ
33		expclz 13450	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0 ∧ (0 − 1) ∈ ℤ) → (𝑥↑(0 − 1)) ∈ ℂ)
34	32, 33	mp3an3 1447	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (𝑥↑(0 − 1)) ∈ ℂ)
35	29, 34	sylbi 220	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → (𝑥↑(0 − 1)) ∈ ℂ)
36	35	adantl 485	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(0 − 1)) ∈ ℂ)
37	36	mul02d 10827	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) → (0 · (𝑥↑(0 − 1))) = 0)
38	28, 37	sylan9eqr 2855	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑁 = 0) → (𝑁 · (𝑥↑(𝑁 − 1))) = 0)
39	38	ifeq1da 4455	. . . . . 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	syl5eqr 2847	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑁 · (𝑥↑(𝑁 − 1))) = if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))))
41	40	mpteq2dva 5125	. . . 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 2836	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
43		eldifi 4054	. . . . . . . 8 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ∈ ℂ)
44	43	adantl 485	. . . . . . 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 10658	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑁 ∈ ℂ)
47		nnnn0 11892	. . . . . . . 8 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℕ₀)
48	47	ad2antlr 726	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -𝑁 ∈ ℕ₀)
49		expneg2 13434	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ₀) → (𝑥↑𝑁) = (1 / (𝑥↑-𝑁)))
50	44, 46, 48, 49	syl3anc 1368	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑𝑁) = (1 / (𝑥↑-𝑁)))
51	50	mpteq2dva 5125	. . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / (𝑥↑-𝑁))))
52	51	oveq2d 7151	. . . 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 4683	. . . . . . . 8 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ≠ 0)
55	54	adantl 485	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0)
56		nnz 11992	. . . . . . . 8 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
57	56	ad2antlr 726	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -𝑁 ∈ ℤ)
58	44, 55, 57	expclzd 13511	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑-𝑁) ∈ ℂ)
59	44, 55, 57	expne0d 13512	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑-𝑁) ≠ 0)
60		eldifsn 4680	. . . . . 6 ⊢ ((𝑥↑-𝑁) ∈ (ℂ ∖ {0}) ↔ ((𝑥↑-𝑁) ∈ ℂ ∧ (𝑥↑-𝑁) ≠ 0))
61	58, 59, 60	sylanbrc 586	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑-𝑁) ∈ (ℂ ∖ {0}))
62		ovex 7168	. . . . . 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 488	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ (ℂ ∖ {0}))
65		eldifsn 4680	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
66	64, 65	sylib 221	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
67		reccl 11294	. . . . . 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 10873	. . . . . 6 ⊢ -(1 / (𝑦↑2)) ∈ V
70	69	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(1 / (𝑦↑2)) ∈ V)
71		simpr 488	. . . . . . 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 13506	. . . . . 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 24556	. . . . . . 7 ⊢ (-𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑-𝑁))) = (𝑥 ∈ ℂ ↦ (-𝑁 · (𝑥↑(-𝑁 − 1)))))
76	75	adantl 485	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑-𝑁))) = (𝑥 ∈ ℂ ↦ (-𝑁 · (𝑥↑(-𝑁 − 1)))))
77		difssd 4060	. . . . . 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 24566	. . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑-𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (-𝑁 · (𝑥↑(-𝑁 − 1)))))
80		ax-1cn 10584	. . . . . 6 ⊢ 1 ∈ ℂ
81		dvrec 24558	. . . . . 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 7143	. . . . 5 ⊢ (𝑦 = (𝑥↑-𝑁) → (1 / 𝑦) = (1 / (𝑥↑-𝑁)))
84		oveq1 7142	. . . . . . 7 ⊢ (𝑦 = (𝑥↑-𝑁) → (𝑦↑2) = ((𝑥↑-𝑁)↑2))
85	84	oveq2d 7151	. . . . . 6 ⊢ (𝑦 = (𝑥↑-𝑁) → (1 / (𝑦↑2)) = (1 / ((𝑥↑-𝑁)↑2)))
86	85	negeqd 10869	. . . . 5 ⊢ (𝑦 = (𝑥↑-𝑁) → -(1 / (𝑦↑2)) = -(1 / ((𝑥↑-𝑁)↑2)))
87	53, 53, 61, 63, 68, 70, 79, 82, 83, 86	dvmptco 24575	. . . 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 12002	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
89	88	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → 2 ∈ ℤ)
90		expmulz 13471	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (-𝑁 ∈ ℤ ∧ 2 ∈ ℤ)) → (𝑥↑(-𝑁 · 2)) = ((𝑥↑-𝑁)↑2))
91	44, 55, 57, 89, 90	syl22anc 837	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 · 2)) = ((𝑥↑-𝑁)↑2))
92	91	eqcomd 2804	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((𝑥↑-𝑁)↑2) = (𝑥↑(-𝑁 · 2)))
93	92	oveq2d 7151	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (1 / ((𝑥↑-𝑁)↑2)) = (1 / (𝑥↑(-𝑁 · 2))))
94	93	negeqd 10869	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → -(1 / ((𝑥↑-𝑁)↑2)) = -(1 / (𝑥↑(-𝑁 · 2))))
95		peano2zm 12013	. . . . . . . . . 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 13511	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 − 1)) ∈ ℂ)
98	46, 97	mulneg1d 11082	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · (𝑥↑(-𝑁 − 1))) = -(𝑁 · (𝑥↑(-𝑁 − 1))))
99	94, 98	oveq12d 7153	. . . . . 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 12019	. . . . . . . . . 10 ⊢ ((-𝑁 ∈ ℤ ∧ 2 ∈ ℤ) → (-𝑁 · 2) ∈ ℤ)
101	57, 88, 100	sylancl 589	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) ∈ ℤ)
102	44, 55, 101	expclzd 13511	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 · 2)) ∈ ℂ)
103	44, 55, 101	expne0d 13512	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑(-𝑁 · 2)) ≠ 0)
104	102, 103	reccld 11398	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (1 / (𝑥↑(-𝑁 · 2))) ∈ ℂ)
105	46, 97	mulcld 10650	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑁 · (𝑥↑(-𝑁 − 1))) ∈ ℂ)
106	104, 105	mul2negd 11084	. . . . . 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 10838	. . . . . . 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 13517	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑((-𝑁 − 1) − (-𝑁 · 2))) = ((𝑥↑(-𝑁 − 1)) / (𝑥↑(-𝑁 · 2))))
109		nncn 11633	. . . . . . . . . . . . 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 12076	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) ∈ ℂ)
113	110, 111, 112	sub32d 11018	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((-𝑁 − 1) − (-𝑁 · 2)) = ((-𝑁 − (-𝑁 · 2)) − 1))
114	110	times2d 11869	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) = (-𝑁 + -𝑁))
115	110, 46	negsubd 10992	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 + -𝑁) = (-𝑁 − 𝑁))
116	114, 115	eqtrd 2833	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 · 2) = (-𝑁 − 𝑁))
117	116	oveq2d 7151	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − (-𝑁 · 2)) = (-𝑁 − (-𝑁 − 𝑁)))
118	110, 46	nncand 10991	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − (-𝑁 − 𝑁)) = 𝑁)
119	117, 118	eqtrd 2833	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-𝑁 − (-𝑁 · 2)) = 𝑁)
120	119	oveq1d 7150	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((-𝑁 − (-𝑁 · 2)) − 1) = (𝑁 − 1))
121	113, 120	eqtrd 2833	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((-𝑁 − 1) − (-𝑁 · 2)) = (𝑁 − 1))
122	121	oveq2d 7151	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥↑((-𝑁 − 1) − (-𝑁 · 2))) = (𝑥↑(𝑁 − 1)))
123	97, 102, 103	divrec2d 11409	. . . . . . . . 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 2842	. . . . . . . 8 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((1 / (𝑥↑(-𝑁 · 2))) · (𝑥↑(-𝑁 − 1))) = (𝑥↑(𝑁 − 1)))
125	124	oveq2d 7151	. . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑁 · ((1 / (𝑥↑(-𝑁 · 2))) · (𝑥↑(-𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1))))
126	107, 125	eqtrd 2833	. . . . . 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 2837	. . . . 5 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (-(1 / ((𝑥↑-𝑁)↑2)) · (-𝑁 · (𝑥↑(-𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1))))
128	127	mpteq2dva 5125	. . . 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 2837	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
130	42, 129	jaoi 854	. 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 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 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∖ cdif 3878 ifcif 4425 {csn 4525 {cpr 4527 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 − cmin 10859 -cneg 10860 / cdiv 11286 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 ℤcz 11969 ↑cexp 13425 TopOpenctopn 16687 ℂ_fldccnfld 20091 Clsdccld 21621 Hauscha 21913 D cdv 24466

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-t1 21919 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470

This theorem is referenced by: (None)

W3C validator