dvply1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvply1		Structured version Visualization version GIF version

Theorem dvply1 26605

Description: Derivative of a polynomial, explicit sum version. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

**Hypotheses**
Ref	Expression
dvply1.f	⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))
dvply1.g	⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵‘𝑘) · (𝑧↑𝑘))))
dvply1.a	⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
dvply1.b	⊢ 𝐵 = (𝑘 ∈ ℕ₀ ↦ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
dvply1.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)

**Assertion**
Ref	Expression
dvply1	⊢ (𝜑 → (ℂ D 𝐹) = 𝐺)

Distinct variable groups: 𝜑,𝑧,𝑘 𝑧,𝐴,𝑘 𝑧,𝐵 𝑘,𝑁,𝑧 Allowed substitution hints: 𝐵(𝑘) 𝐹(𝑧, 𝑘) 𝐺(𝑧, 𝑘)

Proof of Theorem dvply1 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvply1.f	. . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))
2	1	oveq2d 7436	. 2 ⊢ (𝜑 → (ℂ D 𝐹) = (ℂ D (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))))
3		eqid 2760	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
4	3	cnfldtopon 25101	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
5	4	toponrestid 23239	. . 3 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
6		cnelprrecn 11293	. . . 4 ⊢ ℂ ∈ {ℝ, ℂ}
7	6	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
8	3	cnfldtop 25102	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ Top
9		unicntop 25104	. . . . 5 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
10	9	topopn 23224	. . . 4 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ℂ ∈ (TopOpen‘ℂ_fld))
11	8, 10	mp1i 14	. . 3 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂ_fld))
12		fzfid 14116	. . 3 ⊢ (𝜑 → (0...𝑁) ∈ Fin)
13		dvply1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
14		elfznn0 13754	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
15		ffvelcdm 7081	. . . . . . 7 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
16	13, 14, 15	syl2an 608	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
17	16	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → (𝐴‘𝑘) ∈ ℂ)
18		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
19	14	ad2antlr 740	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → 𝑘 ∈ ℕ₀)
20	18, 19	expcld 14289	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → (𝑧↑𝑘) ∈ ℂ)
21	17, 20	mulcld 11329	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
22	21	3impa 1127	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
23	16	3adant3 1150	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → (𝐴‘𝑘) ∈ ℂ)
24		0cnd 11299	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 = 0) → 0 ∈ ℂ)
25		simpl2 1211	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ (0...𝑁))
26	25, 14	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ ℕ₀)
27	26	nn0cnd 12669	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ ℂ)
28		simpl3 1212	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑧 ∈ ℂ)
29		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → ¬ 𝑘 = 0)
30		elnn0 12608	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
31	26, 30	sylib 221	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑘 ∈ ℕ ∨ 𝑘 = 0))
32		orel2 904	. . . . . . . . 9 ⊢ (¬ 𝑘 = 0 → ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → 𝑘 ∈ ℕ))
33	29, 31, 32	sylc 66	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ ℕ)
34		nnm1nn0 12647	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
35	33, 34	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑘 − 1) ∈ ℕ₀)
36	28, 35	expcld 14289	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑧↑(𝑘 − 1)) ∈ ℂ)
37	27, 36	mulcld 11329	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑘 · (𝑧↑(𝑘 − 1))) ∈ ℂ)
38	24, 37	ifclda 4517	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ ℂ)
39	23, 38	mulcld 11329	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) ∈ ℂ)
40	6	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ℂ ∈ {ℝ, ℂ})
41		c0ex 11300	. . . . . 6 ⊢ 0 ∈ V
42		ovex 7453	. . . . . 6 ⊢ (𝑘 · (𝑧↑(𝑘 − 1))) ∈ V
43	41, 42	ifex 4532	. . . . 5 ⊢ if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ V
44	43	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ V)
45	14	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
46		dvexp2 26274	. . . . 5 ⊢ (𝑘 ∈ ℕ₀ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))))
47	45, 46	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))))
48	40, 20, 44, 47, 16	dvmptcmul 26284	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (ℂ D (𝑧 ∈ ℂ ↦ ((𝐴‘𝑘) · (𝑧↑𝑘)))) = (𝑧 ∈ ℂ ↦ ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))))))
49	5, 3, 7, 11, 12, 22, 39, 48	dvmptfsum 26295	. 2 ⊢ (𝜑 → (ℂ D (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))))))
50		elfznn 13687	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
51	50	nnne0d 12388	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ≠ 0)
52	51	neneqd 2960	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → ¬ 𝑘 = 0)
53	52	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ¬ 𝑘 = 0)
54	53	iffalsed 4492	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) = (𝑘 · (𝑧↑(𝑘 − 1))))
55	54	oveq2d 7436	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = ((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1)))))
56	55	sumeq2dv 15869	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (1...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = Σ𝑘 ∈ (1...𝑁)((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1)))))
57		1eluzge0 13007	. . . . . . 7 ⊢ 1 ∈ (ℤ_≥‘0)
58		fzss1 13697	. . . . . . 7 ⊢ (1 ∈ (ℤ_≥‘0) → (1...𝑁) ⊆ (0...𝑁))
59	57, 58	mp1i 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (1...𝑁) ⊆ (0...𝑁))
60	13	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐴:ℕ₀⟶ℂ)
61	50	nnnn0d 12667	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ₀)
62	60, 61, 15	syl2an 608	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
63	51	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ≠ 0)
64	63	neneqd 2960	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ¬ 𝑘 = 0)
65	64	iffalsed 4492	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) = (𝑘 · (𝑧↑(𝑘 − 1))))
66	61	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ₀)
67	66	nn0cnd 12669	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℂ)
68		simplr 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑧 ∈ ℂ)
69	50, 34	syl 18	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → (𝑘 − 1) ∈ ℕ₀)
70	69	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 − 1) ∈ ℕ₀)
71	68, 70	expcld 14289	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑧↑(𝑘 − 1)) ∈ ℂ)
72	67, 71	mulcld 11329	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · (𝑧↑(𝑘 − 1))) ∈ ℂ)
73	65, 72	eqeltrd 2860	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ ℂ)
74	62, 73	mulcld 11329	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) ∈ ℂ)
75		eldifn 4068	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑁) ∖ (1...𝑁)) → ¬ 𝑘 ∈ (1...𝑁))
76		0p1e1 12463	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
77	76	oveq1i 7430	. . . . . . . . . . . . 13 ⊢ ((0 + 1)...𝑁) = (1...𝑁)
78	77	eleq2i 2852	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0 + 1)...𝑁) ↔ 𝑘 ∈ (1...𝑁))
79	75, 78	sylnibr 332	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((0...𝑁) ∖ (1...𝑁)) → ¬ 𝑘 ∈ ((0 + 1)...𝑁))
80	79	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ¬ 𝑘 ∈ ((0 + 1)...𝑁))
81		eldifi 4067	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑁) ∖ (1...𝑁)) → 𝑘 ∈ (0...𝑁))
82	81	adantl 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → 𝑘 ∈ (0...𝑁))
83		dvply1.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
84		nn0uz 13003	. . . . . . . . . . . . . 14 ⊢ ℕ₀ = (ℤ_≥‘0)
85	83, 84	eleqtrdi 2870	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
86	85	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → 𝑁 ∈ (ℤ_≥‘0))
87		elfzp12 13737	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘0) → (𝑘 ∈ (0...𝑁) ↔ (𝑘 = 0 ∨ 𝑘 ∈ ((0 + 1)...𝑁))))
88	86, 87	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → (𝑘 ∈ (0...𝑁) ↔ (𝑘 = 0 ∨ 𝑘 ∈ ((0 + 1)...𝑁))))
89	82, 88	mpbid 235	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → (𝑘 = 0 ∨ 𝑘 ∈ ((0 + 1)...𝑁)))
90		orel2 904	. . . . . . . . . 10 ⊢ (¬ 𝑘 ∈ ((0 + 1)...𝑁) → ((𝑘 = 0 ∨ 𝑘 ∈ ((0 + 1)...𝑁)) → 𝑘 = 0))
91	80, 89, 90	sylc 66	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → 𝑘 = 0)
92	91	iftrued 4489	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) = 0)
93	92	oveq2d 7436	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = ((𝐴‘𝑘) · 0))
94	60, 14, 15	syl2an 608	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
95	94	mul01d 11509	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴‘𝑘) · 0) = 0)
96	81, 95	sylan2 605	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ((𝐴‘𝑘) · 0) = 0)
97	93, 96	eqtrd 2795	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = 0)
98		fzfid 14116	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑁) ∈ Fin)
99	59, 74, 97, 98	fsumss 15891	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (1...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))))
100		elfznn0 13754	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℕ₀)
101	100	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ ℕ₀)
102	101	nn0cnd 12669	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ ℂ)
103		ax-1cn 11258	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
104		pncan 11563	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 1) = 𝑗)
105	102, 103, 104	sylancl 598	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑗 + 1) − 1) = 𝑗)
106	105	oveq2d 7436	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑧↑((𝑗 + 1) − 1)) = (𝑧↑𝑗))
107	106	oveq2d 7436	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1))) = ((𝑗 + 1) · (𝑧↑𝑗)))
108	107	oveq2d 7436	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1)))) = ((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑𝑗))))
109	13	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝐴:ℕ₀⟶ℂ)
110		peano2nn0 12646	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ₀)
111	100, 110	syl 18	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → (𝑗 + 1) ∈ ℕ₀)
112	111	adantl 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑗 + 1) ∈ ℕ₀)
113	109, 112	ffvelcdmd 7085	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝐴‘(𝑗 + 1)) ∈ ℂ)
114	112	nn0cnd 12669	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑗 + 1) ∈ ℂ)
115		simplr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑧 ∈ ℂ)
116	115, 101	expcld 14289	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑧↑𝑗) ∈ ℂ)
117	113, 114, 116	mulassd 11332	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (((𝐴‘(𝑗 + 1)) · (𝑗 + 1)) · (𝑧↑𝑗)) = ((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑𝑗))))
118	113, 114	mulcomd 11330	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝐴‘(𝑗 + 1)) · (𝑗 + 1)) = ((𝑗 + 1) · (𝐴‘(𝑗 + 1))))
119	118	oveq1d 7435	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (((𝐴‘(𝑗 + 1)) · (𝑗 + 1)) · (𝑧↑𝑗)) = (((𝑗 + 1) · (𝐴‘(𝑗 + 1))) · (𝑧↑𝑗)))
120	108, 117, 119	3eqtr2d 2801	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1)))) = (((𝑗 + 1) · (𝐴‘(𝑗 + 1))) · (𝑧↑𝑗)))
121	120	sumeq2dv 15869	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑗 ∈ (0...(𝑁 − 1))((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1)))) = Σ𝑗 ∈ (0...(𝑁 − 1))(((𝑗 + 1) · (𝐴‘(𝑗 + 1))) · (𝑧↑𝑗)))
122		1m1e0 12415	. . . . . . . . 9 ⊢ (1 − 1) = 0
123	122	oveq1i 7430	. . . . . . . 8 ⊢ ((1 − 1)...(𝑁 − 1)) = (0...(𝑁 − 1))
124	123	sumeq1i 15864	. . . . . . 7 ⊢ Σ𝑗 ∈ ((1 − 1)...(𝑁 − 1))((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1)))) = Σ𝑗 ∈ (0...(𝑁 − 1))((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1))))
125		oveq1 7427	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝑘 + 1) = (𝑗 + 1))
126		fvoveq1 7443	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐴‘(𝑘 + 1)) = (𝐴‘(𝑗 + 1)))
127	125, 126	oveq12d 7438	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) = ((𝑗 + 1) · (𝐴‘(𝑗 + 1))))
128		oveq2 7428	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑧↑𝑘) = (𝑧↑𝑗))
129	127, 128	oveq12d 7438	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘)) = (((𝑗 + 1) · (𝐴‘(𝑗 + 1))) · (𝑧↑𝑗)))
130	129	cbvsumv 15863	. . . . . . 7 ⊢ Σ𝑘 ∈ (0...(𝑁 − 1))(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...(𝑁 − 1))(((𝑗 + 1) · (𝐴‘(𝑗 + 1))) · (𝑧↑𝑗))
131	121, 124, 130	3eqtr4g 2820	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑗 ∈ ((1 − 1)...(𝑁 − 1))((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1)))) = Σ𝑘 ∈ (0...(𝑁 − 1))(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘)))
132		1zzd 12727	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 1 ∈ ℤ)
133	83	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑁 ∈ ℕ₀)
134	133	nn0zd 12718	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑁 ∈ ℤ)
135	62, 72	mulcld 11329	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ ℂ)
136		fveq2 6885	. . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑗 + 1)))
137		id 23	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → 𝑘 = (𝑗 + 1))
138		oveq1 7427	. . . . . . . . . 10 ⊢ (𝑘 = (𝑗 + 1) → (𝑘 − 1) = ((𝑗 + 1) − 1))
139	138	oveq2d 7436	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → (𝑧↑(𝑘 − 1)) = (𝑧↑((𝑗 + 1) − 1)))
140	137, 139	oveq12d 7438	. . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → (𝑘 · (𝑧↑(𝑘 − 1))) = ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1))))
141	136, 140	oveq12d 7438	. . . . . . 7 ⊢ (𝑘 = (𝑗 + 1) → ((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1)))) = ((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1)))))
142	132, 132, 134, 135, 141	fsumshftm 15947	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (1...𝑁)((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1)))) = Σ𝑗 ∈ ((1 − 1)...(𝑁 − 1))((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1)))))
143		elfznn0 13754	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
144	143	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ₀)
145		ovex 7453	. . . . . . . . 9 ⊢ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) ∈ V
146		dvply1.b	. . . . . . . . . 10 ⊢ 𝐵 = (𝑘 ∈ ℕ₀ ↦ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
147	146	fvmpt2 7005	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) ∈ V) → (𝐵‘𝑘) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
148	144, 145, 147	sylancl 598	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐵‘𝑘) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
149	148	oveq1d 7435	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘)))
150	149	sumeq2dv 15869	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(𝑁 − 1))(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘)))
151	131, 142, 150	3eqtr4d 2805	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (1...𝑁)((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1)))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵‘𝑘) · (𝑧↑𝑘)))
152	56, 99, 151	3eqtr3d 2803	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵‘𝑘) · (𝑧↑𝑘)))
153	152	mpteq2dva 5197	. . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵‘𝑘) · (𝑧↑𝑘))))
154		dvply1.g	. . 3 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵‘𝑘) · (𝑧↑𝑘))))
155	153, 154	eqtr4d 2798	. 2 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))))) = 𝐺)
156	2, 49, 155	3eqtrd 2799	1 ⊢ (𝜑 → (ℂ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 Vcvv 3450 ⊆ wss 3895 ∖ cdif 4045 ifcif 4481 {cpr 4585 ↦ cmpt 5185 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 − cmin 11541 ℕcn 12335 ℕ₀cn0 12606 ℤ_≥cuz 12965 ...cfz 13639 ↑cexp 14204 Σcsu 15853 TopOpenctopn 17592 ℂ_fldccnfld 21678 Topctop 23211 D cdv 26183

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-icc 13483 df-fz 13640 df-fzo 13789 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-sum 15854 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-limc 26186 df-dv 26187

This theorem is used by: dvply2g 26606

W3C validator