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Theorem dvply1 26340

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 7447	. 2 ⊢ (𝜑 → (ℂ D 𝐹) = (ℂ D (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))))
3		eqid 2735	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
4	3	cnfldtopon 24819	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
5	4	toponrestid 22943	. . 3 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
6		cnelprrecn 11246	. . . 4 ⊢ ℂ ∈ {ℝ, ℂ}
7	6	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
8	3	cnfldtop 24820	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ Top
9		unicntop 24822	. . . . 5 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
10	9	topopn 22928	. . . 4 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ℂ ∈ (TopOpen‘ℂ_fld))
11	8, 10	mp1i 13	. . 3 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂ_fld))
12		fzfid 14011	. . 3 ⊢ (𝜑 → (0...𝑁) ∈ Fin)
13		dvply1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
14		elfznn0 13657	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
15		ffvelcdm 7101	. . . . . . 7 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
16	13, 14, 15	syl2an 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
17	16	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → (𝐴‘𝑘) ∈ ℂ)
18		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
19	14	ad2antlr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → 𝑘 ∈ ℕ₀)
20	18, 19	expcld 14183	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → (𝑧↑𝑘) ∈ ℂ)
21	17, 20	mulcld 11279	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
22	21	3impa 1109	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
23	16	3adant3 1131	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → (𝐴‘𝑘) ∈ ℂ)
24		0cnd 11252	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 = 0) → 0 ∈ ℂ)
25		simpl2 1191	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ (0...𝑁))
26	25, 14	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ ℕ₀)
27	26	nn0cnd 12587	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ ℂ)
28		simpl3 1192	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑧 ∈ ℂ)
29		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → ¬ 𝑘 = 0)
30		elnn0 12526	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
31	26, 30	sylib 218	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑘 ∈ ℕ ∨ 𝑘 = 0))
32		orel2 890	. . . . . . . . 9 ⊢ (¬ 𝑘 = 0 → ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → 𝑘 ∈ ℕ))
33	29, 31, 32	sylc 65	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ ℕ)
34		nnm1nn0 12565	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
35	33, 34	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑘 − 1) ∈ ℕ₀)
36	28, 35	expcld 14183	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑧↑(𝑘 − 1)) ∈ ℂ)
37	27, 36	mulcld 11279	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑘 · (𝑧↑(𝑘 − 1))) ∈ ℂ)
38	24, 37	ifclda 4566	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ ℂ)
39	23, 38	mulcld 11279	. . 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 11253	. . . . . 6 ⊢ 0 ∈ V
42		ovex 7464	. . . . . 6 ⊢ (𝑘 · (𝑧↑(𝑘 − 1))) ∈ V
43	41, 42	ifex 4581	. . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
46		dvexp2 26007	. . . . 5 ⊢ (𝑘 ∈ ℕ₀ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))))
47	45, 46	syl 17	. . . 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 26017	. . 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 26028	. 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 13590	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
51	50	nnne0d 12314	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ≠ 0)
52	51	neneqd 2943	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → ¬ 𝑘 = 0)
53	52	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ¬ 𝑘 = 0)
54	53	iffalsed 4542	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) = (𝑘 · (𝑧↑(𝑘 − 1))))
55	54	oveq2d 7447	. . . . . 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 15735	. . . . 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 12932	. . . . . . 7 ⊢ 1 ∈ (ℤ_≥‘0)
58		fzss1 13600	. . . . . . 7 ⊢ (1 ∈ (ℤ_≥‘0) → (1...𝑁) ⊆ (0...𝑁))
59	57, 58	mp1i 13	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (1...𝑁) ⊆ (0...𝑁))
60	13	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐴:ℕ₀⟶ℂ)
61	50	nnnn0d 12585	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ₀)
62	60, 61, 15	syl2an 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
63	51	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ≠ 0)
64	63	neneqd 2943	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ¬ 𝑘 = 0)
65	64	iffalsed 4542	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) = (𝑘 · (𝑧↑(𝑘 − 1))))
66	61	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ₀)
67	66	nn0cnd 12587	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℂ)
68		simplr 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑧 ∈ ℂ)
69	50, 34	syl 17	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → (𝑘 − 1) ∈ ℕ₀)
70	69	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 − 1) ∈ ℕ₀)
71	68, 70	expcld 14183	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑧↑(𝑘 − 1)) ∈ ℂ)
72	67, 71	mulcld 11279	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · (𝑧↑(𝑘 − 1))) ∈ ℂ)
73	65, 72	eqeltrd 2839	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ ℂ)
74	62, 73	mulcld 11279	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) ∈ ℂ)
75		eldifn 4142	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑁) ∖ (1...𝑁)) → ¬ 𝑘 ∈ (1...𝑁))
76		0p1e1 12386	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
77	76	oveq1i 7441	. . . . . . . . . . . . 13 ⊢ ((0 + 1)...𝑁) = (1...𝑁)
78	77	eleq2i 2831	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0 + 1)...𝑁) ↔ 𝑘 ∈ (1...𝑁))
79	75, 78	sylnibr 329	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((0...𝑁) ∖ (1...𝑁)) → ¬ 𝑘 ∈ ((0 + 1)...𝑁))
80	79	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ¬ 𝑘 ∈ ((0 + 1)...𝑁))
81		eldifi 4141	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑁) ∖ (1...𝑁)) → 𝑘 ∈ (0...𝑁))
82	81	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → 𝑘 ∈ (0...𝑁))
83		dvply1.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
84		nn0uz 12918	. . . . . . . . . . . . . 14 ⊢ ℕ₀ = (ℤ_≥‘0)
85	83, 84	eleqtrdi 2849	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
86	85	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → 𝑁 ∈ (ℤ_≥‘0))
87		elfzp12 13640	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘0) → (𝑘 ∈ (0...𝑁) ↔ (𝑘 = 0 ∨ 𝑘 ∈ ((0 + 1)...𝑁))))
88	86, 87	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → (𝑘 ∈ (0...𝑁) ↔ (𝑘 = 0 ∨ 𝑘 ∈ ((0 + 1)...𝑁))))
89	82, 88	mpbid 232	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → (𝑘 = 0 ∨ 𝑘 ∈ ((0 + 1)...𝑁)))
90		orel2 890	. . . . . . . . . 10 ⊢ (¬ 𝑘 ∈ ((0 + 1)...𝑁) → ((𝑘 = 0 ∨ 𝑘 ∈ ((0 + 1)...𝑁)) → 𝑘 = 0))
91	80, 89, 90	sylc 65	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → 𝑘 = 0)
92	91	iftrued 4539	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) = 0)
93	92	oveq2d 7447	. . . . . . 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 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
95	94	mul01d 11458	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴‘𝑘) · 0) = 0)
96	81, 95	sylan2 593	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ((𝐴‘𝑘) · 0) = 0)
97	93, 96	eqtrd 2775	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = 0)
98		fzfid 14011	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑁) ∈ Fin)
99	59, 74, 97, 98	fsumss 15758	. . . . 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 13657	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℕ₀)
101	100	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ ℕ₀)
102	101	nn0cnd 12587	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ ℂ)
103		ax-1cn 11211	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
104		pncan 11512	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 1) = 𝑗)
105	102, 103, 104	sylancl 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑗 + 1) − 1) = 𝑗)
106	105	oveq2d 7447	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑧↑((𝑗 + 1) − 1)) = (𝑧↑𝑗))
107	106	oveq2d 7447	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1))) = ((𝑗 + 1) · (𝑧↑𝑗)))
108	107	oveq2d 7447	. . . . . . . . 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 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝐴:ℕ₀⟶ℂ)
110		peano2nn0 12564	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ₀)
111	100, 110	syl 17	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → (𝑗 + 1) ∈ ℕ₀)
112	111	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑗 + 1) ∈ ℕ₀)
113	109, 112	ffvelcdmd 7105	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝐴‘(𝑗 + 1)) ∈ ℂ)
114	112	nn0cnd 12587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑗 + 1) ∈ ℂ)
115		simplr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑧 ∈ ℂ)
116	115, 101	expcld 14183	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑧↑𝑗) ∈ ℂ)
117	113, 114, 116	mulassd 11282	. . . . . . . . 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 11280	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝐴‘(𝑗 + 1)) · (𝑗 + 1)) = ((𝑗 + 1) · (𝐴‘(𝑗 + 1))))
119	118	oveq1d 7446	. . . . . . . . 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 2781	. . . . . . . 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 15735	. . . . . . 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 12336	. . . . . . . . 9 ⊢ (1 − 1) = 0
123	122	oveq1i 7441	. . . . . . . 8 ⊢ ((1 − 1)...(𝑁 − 1)) = (0...(𝑁 − 1))
124	123	sumeq1i 15730	. . . . . . 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 7438	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝑘 + 1) = (𝑗 + 1))
126		fvoveq1 7454	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐴‘(𝑘 + 1)) = (𝐴‘(𝑗 + 1)))
127	125, 126	oveq12d 7449	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) = ((𝑗 + 1) · (𝐴‘(𝑗 + 1))))
128		oveq2 7439	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑧↑𝑘) = (𝑧↑𝑗))
129	127, 128	oveq12d 7449	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘)) = (((𝑗 + 1) · (𝐴‘(𝑗 + 1))) · (𝑧↑𝑗)))
130	129	cbvsumv 15729	. . . . . . 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 2800	. . . . . 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 12646	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 1 ∈ ℤ)
133	83	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑁 ∈ ℕ₀)
134	133	nn0zd 12637	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑁 ∈ ℤ)
135	62, 72	mulcld 11279	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ ℂ)
136		fveq2 6907	. . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑗 + 1)))
137		id 22	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → 𝑘 = (𝑗 + 1))
138		oveq1 7438	. . . . . . . . . 10 ⊢ (𝑘 = (𝑗 + 1) → (𝑘 − 1) = ((𝑗 + 1) − 1))
139	138	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → (𝑧↑(𝑘 − 1)) = (𝑧↑((𝑗 + 1) − 1)))
140	137, 139	oveq12d 7449	. . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → (𝑘 · (𝑧↑(𝑘 − 1))) = ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1))))
141	136, 140	oveq12d 7449	. . . . . . 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 15814	. . . . . 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 13657	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
144	143	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ₀)
145		ovex 7464	. . . . . . . . 9 ⊢ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) ∈ V
146		dvply1.b	. . . . . . . . . 10 ⊢ 𝐵 = (𝑘 ∈ ℕ₀ ↦ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
147	146	fvmpt2 7027	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) ∈ V) → (𝐵‘𝑘) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
148	144, 145, 147	sylancl 586	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐵‘𝑘) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
149	148	oveq1d 7446	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘)))
150	149	sumeq2dv 15735	. . . . . 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 2785	. . . . 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 2783	. . . 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 5248	. . 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 2778	. 2 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))))) = 𝐺)
156	2, 49, 155	3eqtrd 2779	1 ⊢ (𝜑 → (ℂ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 ifcif 4531 {cpr 4633 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 ℕcn 12264 ℕ₀cn0 12524 ℤ_≥cuz 12876 ...cfz 13544 ↑cexp 14099 Σcsu 15719 TopOpenctopn 17468 ℂ_fldccnfld 21382 Topctop 22915 D cdv 25913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917

This theorem is referenced by: dvply2g 26341 dvply2gOLD 26342

W3C validator