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Theorem dvply1 26251

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 7376	. 2 ⊢ (𝜑 → (ℂ D 𝐹) = (ℂ D (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))))
3		eqid 2737	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
4	3	cnfldtopon 24730	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
5	4	toponrestid 22869	. . 3 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
6		cnelprrecn 11123	. . . 4 ⊢ ℂ ∈ {ℝ, ℂ}
7	6	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
8	3	cnfldtop 24731	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ Top
9		unicntop 24733	. . . . 5 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
10	9	topopn 22854	. . . 4 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ℂ ∈ (TopOpen‘ℂ_fld))
11	8, 10	mp1i 13	. . 3 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂ_fld))
12		fzfid 13900	. . 3 ⊢ (𝜑 → (0...𝑁) ∈ Fin)
13		dvply1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
14		elfznn0 13540	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
15		ffvelcdm 7028	. . . . . . 7 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
16	13, 14, 15	syl2an 597	. . . . . 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 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → 𝑘 ∈ ℕ₀)
20	18, 19	expcld 14073	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → (𝑧↑𝑘) ∈ ℂ)
21	17, 20	mulcld 11156	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
22	21	3impa 1110	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
23	16	3adant3 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → (𝐴‘𝑘) ∈ ℂ)
24		0cnd 11129	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 = 0) → 0 ∈ ℂ)
25		simpl2 1194	. . . . . . . 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 12468	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ ℂ)
28		simpl3 1195	. . . . . . 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 12407	. . . . . . . . . 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 891	. . . . . . . . 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 12446	. . . . . . . 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 14073	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑧↑(𝑘 − 1)) ∈ ℂ)
37	27, 36	mulcld 11156	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑘 · (𝑧↑(𝑘 − 1))) ∈ ℂ)
38	24, 37	ifclda 4516	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ ℂ)
39	23, 38	mulcld 11156	. . 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 11130	. . . . . 6 ⊢ 0 ∈ V
42		ovex 7393	. . . . . 6 ⊢ (𝑘 · (𝑧↑(𝑘 − 1))) ∈ V
43	41, 42	ifex 4531	. . . . 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 25918	. . . . 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 25928	. . 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 25939	. 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 13473	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
51	50	nnne0d 12199	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ≠ 0)
52	51	neneqd 2938	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → ¬ 𝑘 = 0)
53	52	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ¬ 𝑘 = 0)
54	53	iffalsed 4491	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) = (𝑘 · (𝑧↑(𝑘 − 1))))
55	54	oveq2d 7376	. . . . . 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 15629	. . . . 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 12797	. . . . . . 7 ⊢ 1 ∈ (ℤ_≥‘0)
58		fzss1 13483	. . . . . . 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 12466	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ₀)
62	60, 61, 15	syl2an 597	. . . . . . 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 2938	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ¬ 𝑘 = 0)
65	64	iffalsed 4491	. . . . . . . 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 12468	. . . . . . . . 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 14073	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑧↑(𝑘 − 1)) ∈ ℂ)
72	67, 71	mulcld 11156	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · (𝑧↑(𝑘 − 1))) ∈ ℂ)
73	65, 72	eqeltrd 2837	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ ℂ)
74	62, 73	mulcld 11156	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) ∈ ℂ)
75		eldifn 4085	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑁) ∖ (1...𝑁)) → ¬ 𝑘 ∈ (1...𝑁))
76		0p1e1 12266	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
77	76	oveq1i 7370	. . . . . . . . . . . . 13 ⊢ ((0 + 1)...𝑁) = (1...𝑁)
78	77	eleq2i 2829	. . . . . . . . . . . 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 4084	. . . . . . . . . . . 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 12793	. . . . . . . . . . . . . 14 ⊢ ℕ₀ = (ℤ_≥‘0)
85	83, 84	eleqtrdi 2847	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
86	85	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → 𝑁 ∈ (ℤ_≥‘0))
87		elfzp12 13523	. . . . . . . . . . . 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 891	. . . . . . . . . 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 4488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) = 0)
93	92	oveq2d 7376	. . . . . . 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 597	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
95	94	mul01d 11336	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴‘𝑘) · 0) = 0)
96	81, 95	sylan2 594	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ((𝐴‘𝑘) · 0) = 0)
97	93, 96	eqtrd 2772	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = 0)
98		fzfid 13900	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑁) ∈ Fin)
99	59, 74, 97, 98	fsumss 15652	. . . . 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 13540	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℕ₀)
101	100	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ ℕ₀)
102	101	nn0cnd 12468	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ ℂ)
103		ax-1cn 11088	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
104		pncan 11390	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 1) = 𝑗)
105	102, 103, 104	sylancl 587	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑗 + 1) − 1) = 𝑗)
106	105	oveq2d 7376	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑧↑((𝑗 + 1) − 1)) = (𝑧↑𝑗))
107	106	oveq2d 7376	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1))) = ((𝑗 + 1) · (𝑧↑𝑗)))
108	107	oveq2d 7376	. . . . . . . . 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 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝐴:ℕ₀⟶ℂ)
110		peano2nn0 12445	. . . . . . . . . . . . 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 7032	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝐴‘(𝑗 + 1)) ∈ ℂ)
114	112	nn0cnd 12468	. . . . . . . . . 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 14073	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑧↑𝑗) ∈ ℂ)
117	113, 114, 116	mulassd 11159	. . . . . . . . 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 11157	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝐴‘(𝑗 + 1)) · (𝑗 + 1)) = ((𝑗 + 1) · (𝐴‘(𝑗 + 1))))
119	118	oveq1d 7375	. . . . . . . . 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 2778	. . . . . . . 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 15629	. . . . . . 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 12221	. . . . . . . . 9 ⊢ (1 − 1) = 0
123	122	oveq1i 7370	. . . . . . . 8 ⊢ ((1 − 1)...(𝑁 − 1)) = (0...(𝑁 − 1))
124	123	sumeq1i 15624	. . . . . . 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 7367	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝑘 + 1) = (𝑗 + 1))
126		fvoveq1 7383	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐴‘(𝑘 + 1)) = (𝐴‘(𝑗 + 1)))
127	125, 126	oveq12d 7378	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) = ((𝑗 + 1) · (𝐴‘(𝑗 + 1))))
128		oveq2 7368	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑧↑𝑘) = (𝑧↑𝑗))
129	127, 128	oveq12d 7378	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘)) = (((𝑗 + 1) · (𝐴‘(𝑗 + 1))) · (𝑧↑𝑗)))
130	129	cbvsumv 15623	. . . . . . 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 2797	. . . . . 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 12526	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 1 ∈ ℤ)
133	83	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑁 ∈ ℕ₀)
134	133	nn0zd 12517	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑁 ∈ ℤ)
135	62, 72	mulcld 11156	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ ℂ)
136		fveq2 6835	. . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑗 + 1)))
137		id 22	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → 𝑘 = (𝑗 + 1))
138		oveq1 7367	. . . . . . . . . 10 ⊢ (𝑘 = (𝑗 + 1) → (𝑘 − 1) = ((𝑗 + 1) − 1))
139	138	oveq2d 7376	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → (𝑧↑(𝑘 − 1)) = (𝑧↑((𝑗 + 1) − 1)))
140	137, 139	oveq12d 7378	. . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → (𝑘 · (𝑧↑(𝑘 − 1))) = ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1))))
141	136, 140	oveq12d 7378	. . . . . . 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 15708	. . . . . 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 13540	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
144	143	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ₀)
145		ovex 7393	. . . . . . . . 9 ⊢ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) ∈ V
146		dvply1.b	. . . . . . . . . 10 ⊢ 𝐵 = (𝑘 ∈ ℕ₀ ↦ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
147	146	fvmpt2 6954	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) ∈ V) → (𝐵‘𝑘) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
148	144, 145, 147	sylancl 587	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐵‘𝑘) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
149	148	oveq1d 7375	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘)))
150	149	sumeq2dv 15629	. . . . . 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 2782	. . . . 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 2780	. . . 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 5192	. . 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 2775	. 2 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))))) = 𝐺)
156	2, 49, 155	3eqtrd 2776	1 ⊢ (𝜑 → (ℂ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3441 ∖ cdif 3899 ⊆ wss 3902 ifcif 4480 {cpr 4583 ↦ cmpt 5180 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 ℂcc 11028 ℝcr 11029 0cc0 11030 1c1 11031 + caddc 11033 · cmul 11035 − cmin 11368 ℕcn 12149 ℕ₀cn0 12405 ℤ_≥cuz 12755 ...cfz 13427 ↑cexp 13988 Σcsu 15613 TopOpenctopn 17345 ℂ_fldccnfld 21313 Topctop 22841 D cdv 25824

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-icc 13272 df-fz 13428 df-fzo 13575 df-seq 13929 df-exp 13989 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-clim 15415 df-sum 15614 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24268 df-ms 24269 df-tms 24270 df-cncf 24831 df-limc 25827 df-dv 25828

This theorem is referenced by: dvply2g 26252 dvply2gOLD 26253

W3C validator