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Theorem dvply1 24872

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 7171	. 2 ⊢ (𝜑 → (ℂ D 𝐹) = (ℂ D (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))))
3		eqid 2821	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
4	3	cnfldtopon 23390	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
5	4	toponrestid 21528	. . 3 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
6		cnelprrecn 10629	. . . 4 ⊢ ℂ ∈ {ℝ, ℂ}
7	6	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
8	3	cnfldtop 23391	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ Top
9		unicntop 23393	. . . . 5 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
10	9	topopn 21513	. . . 4 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ℂ ∈ (TopOpen‘ℂ_fld))
11	8, 10	mp1i 13	. . 3 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂ_fld))
12		fzfid 13340	. . 3 ⊢ (𝜑 → (0...𝑁) ∈ Fin)
13		dvply1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
14		elfznn0 12999	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
15		ffvelrn 6848	. . . . . . 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 483	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → (𝐴‘𝑘) ∈ ℂ)
18		simpr 487	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
19	14	ad2antlr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → 𝑘 ∈ ℕ₀)
20	18, 19	expcld 13509	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → (𝑧↑𝑘) ∈ ℂ)
21	17, 20	mulcld 10660	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
22	21	3impa 1106	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
23	16	3adant3 1128	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → (𝐴‘𝑘) ∈ ℂ)
24		0cnd 10633	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 = 0) → 0 ∈ ℂ)
25		simpl2 1188	. . . . . . . 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 11956	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ ℂ)
28		simpl3 1189	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑧 ∈ ℂ)
29		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → ¬ 𝑘 = 0)
30		elnn0 11898	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
31	26, 30	sylib 220	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑘 ∈ ℕ ∨ 𝑘 = 0))
32		orel2 887	. . . . . . . . 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 11937	. . . . . . . 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 13509	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑧↑(𝑘 − 1)) ∈ ℂ)
37	27, 36	mulcld 10660	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑘 · (𝑧↑(𝑘 − 1))) ∈ ℂ)
38	24, 37	ifclda 4500	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ ℂ)
39	23, 38	mulcld 10660	. . 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 10634	. . . . . 6 ⊢ 0 ∈ V
42		ovex 7188	. . . . . 6 ⊢ (𝑘 · (𝑧↑(𝑘 − 1))) ∈ V
43	41, 42	ifex 4514	. . . . 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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
46		dvexp2 24550	. . . . 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 24560	. . 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 24571	. 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 12935	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
51	50	nnne0d 11686	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ≠ 0)
52	51	neneqd 3021	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → ¬ 𝑘 = 0)
53	52	adantl 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ¬ 𝑘 = 0)
54	53	iffalsed 4477	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) = (𝑘 · (𝑧↑(𝑘 − 1))))
55	54	oveq2d 7171	. . . . . 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 15059	. . . . 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 12291	. . . . . . 7 ⊢ 1 ∈ (ℤ_≥‘0)
58		fzss1 12945	. . . . . . 7 ⊢ (1 ∈ (ℤ_≥‘0) → (1...𝑁) ⊆ (0...𝑁))
59	57, 58	mp1i 13	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (1...𝑁) ⊆ (0...𝑁))
60	13	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐴:ℕ₀⟶ℂ)
61	50	nnnn0d 11954	. . . . . . . 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 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ≠ 0)
64	63	neneqd 3021	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ¬ 𝑘 = 0)
65	64	iffalsed 4477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) = (𝑘 · (𝑧↑(𝑘 − 1))))
66	61	adantl 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ₀)
67	66	nn0cnd 11956	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℂ)
68		simplr 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑧 ∈ ℂ)
69	50, 34	syl 17	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → (𝑘 − 1) ∈ ℕ₀)
70	69	adantl 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 − 1) ∈ ℕ₀)
71	68, 70	expcld 13509	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑧↑(𝑘 − 1)) ∈ ℂ)
72	67, 71	mulcld 10660	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · (𝑧↑(𝑘 − 1))) ∈ ℂ)
73	65, 72	eqeltrd 2913	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ ℂ)
74	62, 73	mulcld 10660	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) ∈ ℂ)
75		eldifn 4103	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑁) ∖ (1...𝑁)) → ¬ 𝑘 ∈ (1...𝑁))
76		0p1e1 11758	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
77	76	oveq1i 7165	. . . . . . . . . . . . 13 ⊢ ((0 + 1)...𝑁) = (1...𝑁)
78	77	eleq2i 2904	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0 + 1)...𝑁) ↔ 𝑘 ∈ (1...𝑁))
79	75, 78	sylnibr 331	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((0...𝑁) ∖ (1...𝑁)) → ¬ 𝑘 ∈ ((0 + 1)...𝑁))
80	79	adantl 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ¬ 𝑘 ∈ ((0 + 1)...𝑁))
81		eldifi 4102	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑁) ∖ (1...𝑁)) → 𝑘 ∈ (0...𝑁))
82	81	adantl 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → 𝑘 ∈ (0...𝑁))
83		dvply1.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
84		nn0uz 12279	. . . . . . . . . . . . . 14 ⊢ ℕ₀ = (ℤ_≥‘0)
85	83, 84	eleqtrdi 2923	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
86	85	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → 𝑁 ∈ (ℤ_≥‘0))
87		elfzp12 12985	. . . . . . . . . . . 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 234	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → (𝑘 = 0 ∨ 𝑘 ∈ ((0 + 1)...𝑁)))
90		orel2 887	. . . . . . . . . 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 4474	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) = 0)
93	92	oveq2d 7171	. . . . . . 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 10838	. . . . . . . 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 2856	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = 0)
98		fzfid 13340	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑁) ∈ Fin)
99	59, 74, 97, 98	fsumss 15081	. . . . 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 12999	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℕ₀)
101	100	adantl 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ ℕ₀)
102	101	nn0cnd 11956	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ ℂ)
103		ax-1cn 10594	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
104		pncan 10891	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 1) = 𝑗)
105	102, 103, 104	sylancl 588	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑗 + 1) − 1) = 𝑗)
106	105	oveq2d 7171	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑧↑((𝑗 + 1) − 1)) = (𝑧↑𝑗))
107	106	oveq2d 7171	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1))) = ((𝑗 + 1) · (𝑧↑𝑗)))
108	107	oveq2d 7171	. . . . . . . . 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 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝐴:ℕ₀⟶ℂ)
110		peano2nn0 11936	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ₀)
111	100, 110	syl 17	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → (𝑗 + 1) ∈ ℕ₀)
112	111	adantl 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑗 + 1) ∈ ℕ₀)
113	109, 112	ffvelrnd 6851	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝐴‘(𝑗 + 1)) ∈ ℂ)
114	112	nn0cnd 11956	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑗 + 1) ∈ ℂ)
115		simplr 767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑧 ∈ ℂ)
116	115, 101	expcld 13509	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑧↑𝑗) ∈ ℂ)
117	113, 114, 116	mulassd 10663	. . . . . . . . 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 10661	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝐴‘(𝑗 + 1)) · (𝑗 + 1)) = ((𝑗 + 1) · (𝐴‘(𝑗 + 1))))
119	118	oveq1d 7170	. . . . . . . . 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 2862	. . . . . . . 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 15059	. . . . . . 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 11708	. . . . . . . . 9 ⊢ (1 − 1) = 0
123	122	oveq1i 7165	. . . . . . . 8 ⊢ ((1 − 1)...(𝑁 − 1)) = (0...(𝑁 − 1))
124	123	sumeq1i 15054	. . . . . . 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 7162	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝑘 + 1) = (𝑗 + 1))
126		fvoveq1 7178	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐴‘(𝑘 + 1)) = (𝐴‘(𝑗 + 1)))
127	125, 126	oveq12d 7173	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) = ((𝑗 + 1) · (𝐴‘(𝑗 + 1))))
128		oveq2 7163	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑧↑𝑘) = (𝑧↑𝑗))
129	127, 128	oveq12d 7173	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘)) = (((𝑗 + 1) · (𝐴‘(𝑗 + 1))) · (𝑧↑𝑗)))
130	129	cbvsumv 15052	. . . . . . 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 2881	. . . . . 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 12012	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 1 ∈ ℤ)
133	83	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑁 ∈ ℕ₀)
134	133	nn0zd 12084	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑁 ∈ ℤ)
135	62, 72	mulcld 10660	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ ℂ)
136		fveq2 6669	. . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑗 + 1)))
137		id 22	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → 𝑘 = (𝑗 + 1))
138		oveq1 7162	. . . . . . . . . 10 ⊢ (𝑘 = (𝑗 + 1) → (𝑘 − 1) = ((𝑗 + 1) − 1))
139	138	oveq2d 7171	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → (𝑧↑(𝑘 − 1)) = (𝑧↑((𝑗 + 1) − 1)))
140	137, 139	oveq12d 7173	. . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → (𝑘 · (𝑧↑(𝑘 − 1))) = ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1))))
141	136, 140	oveq12d 7173	. . . . . . 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 15135	. . . . . 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 12999	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
144	143	adantl 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ₀)
145		ovex 7188	. . . . . . . . 9 ⊢ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) ∈ V
146		dvply1.b	. . . . . . . . . 10 ⊢ 𝐵 = (𝑘 ∈ ℕ₀ ↦ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
147	146	fvmpt2 6778	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) ∈ V) → (𝐵‘𝑘) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
148	144, 145, 147	sylancl 588	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐵‘𝑘) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
149	148	oveq1d 7170	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘)))
150	149	sumeq2dv 15059	. . . . . 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 2866	. . . . 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 2864	. . . 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 5160	. . 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 2859	. 2 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))))) = 𝐺)
156	2, 49, 155	3eqtrd 2860	1 ⊢ (𝜑 → (ℂ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∖ cdif 3932 ⊆ wss 3935 ifcif 4466 {cpr 4568 ↦ cmpt 5145 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ℝcr 10535 0cc0 10536 1c1 10537 + caddc 10539 · cmul 10541 − cmin 10869 ℕcn 11637 ℕ₀cn0 11896 ℤ_≥cuz 12242 ...cfz 12891 ↑cexp 13428 Σcsu 15041 TopOpenctopn 16694 ℂ_fldccnfld 20544 Topctop 21500 D cdv 24460

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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-icc 12744 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-sum 15042 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-fbas 20541 df-fg 20542 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-lp 21743 df-perf 21744 df-cn 21834 df-cnp 21835 df-haus 21922 df-tx 22169 df-hmeo 22362 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-xms 22929 df-ms 22930 df-tms 22931 df-cncf 23485 df-limc 24463 df-dv 24464

This theorem is referenced by: dvply2g 24873

W3C validator