dvradcnv - Metamath Proof Explorer

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Theorem dvradcnv 26654

Description: The radius of convergence of the (formal) derivative 𝐻 of the power series 𝐺 is at least as large as the radius of convergence of 𝐺. (In fact they are equal, but we don't have as much use for the negative side of this claim.) (Contributed by Mario Carneiro, 31-Mar-2015.)

**Hypotheses**
Ref	Expression
dvradcnv.g	⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))
dvradcnv.r	⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ^*, < )
dvradcnv.h	⊢ 𝐻 = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 1) · (𝐴‘(𝑛 + 1))) · (𝑋↑𝑛)))
dvradcnv.a	⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
dvradcnv.x	⊢ (𝜑 → 𝑋 ∈ ℂ)
dvradcnv.l	⊢ (𝜑 → (abs‘𝑋) < 𝑅)

**Assertion**
Ref	Expression
dvradcnv	⊢ (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ )

Distinct variable groups: 𝑥,𝑛,𝐴 𝐺,𝑟 𝑛,𝑟,𝑋,𝑥 Allowed substitution hints: 𝜑(𝑥, 𝑛, 𝑟) 𝐴(𝑟) 𝑅(𝑥, 𝑛, 𝑟) 𝐺(𝑥, 𝑛) 𝐻(𝑥, 𝑛, 𝑟)

Proof of Theorem dvradcnv Dummy variables 𝑘 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 12928	. 2 ⊢ ℕ₀ = (ℤ_≥‘0)
2		1nn0 12547	. . 3 ⊢ 1 ∈ ℕ₀
3	2	a1i 11	. 2 ⊢ (𝜑 → 1 ∈ ℕ₀)
4		ax-1cn 11185	. . . . 5 ⊢ 1 ∈ ℂ
5		nn0cn 12541	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
6	5	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℂ)
7		nn0ex 12537	. . . . . . 7 ⊢ ℕ₀ ∈ V
8	7	mptex 7225	. . . . . 6 ⊢ (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) ∈ V
9	8	shftval4 15152	. . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘) = ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(1 + 𝑘)))
10	4, 6, 9	sylancr 599	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘) = ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(1 + 𝑘)))
11		addcom 11423	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (1 + 𝑘) = (𝑘 + 1))
12	4, 6, 11	sylancr 599	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (1 + 𝑘) = (𝑘 + 1))
13	12	fveq2d 6886	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(1 + 𝑘)) = ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(𝑘 + 1)))
14		peano2nn0 12571	. . . . . . 7 ⊢ (𝑘 ∈ ℕ₀ → (𝑘 + 1) ∈ ℕ₀)
15	14	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℕ₀)
16		id 23	. . . . . . . 8 ⊢ (𝑖 = (𝑘 + 1) → 𝑖 = (𝑘 + 1))
17		2fveq3 6887	. . . . . . . 8 ⊢ (𝑖 = (𝑘 + 1) → (abs‘((𝐺‘𝑋)‘𝑖)) = (abs‘((𝐺‘𝑋)‘(𝑘 + 1))))
18	16, 17	oveq12d 7434	. . . . . . 7 ⊢ (𝑖 = (𝑘 + 1) → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) = ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))))
19		eqid 2762	. . . . . . 7 ⊢ (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) = (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))
20		ovex 7449	. . . . . . 7 ⊢ ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))) ∈ V
21	18, 19, 20	fvmpt 6990	. . . . . 6 ⊢ ((𝑘 + 1) ∈ ℕ₀ → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(𝑘 + 1)) = ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))))
22	15, 21	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(𝑘 + 1)) = ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))))
23		dvradcnv.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℂ)
24		dvradcnv.g	. . . . . . . . 9 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))
25	24	pserval2 26644	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 + 1) ∈ ℕ₀) → ((𝐺‘𝑋)‘(𝑘 + 1)) = ((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))
26	23, 14, 25	syl2an 608	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐺‘𝑋)‘(𝑘 + 1)) = ((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))
27	26	fveq2d 6886	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝐺‘𝑋)‘(𝑘 + 1))) = (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))
28	27	oveq2d 7432	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
29	22, 28	eqtrd 2797	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(𝑘 + 1)) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
30	10, 13, 29	3eqtrd 2801	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
31	15	nn0red 12593	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℝ)
32		dvradcnv.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
33		ffvelcdm 7077	. . . . . . 7 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ (𝑘 + 1) ∈ ℕ₀) → (𝐴‘(𝑘 + 1)) ∈ ℂ)
34	32, 14, 33	syl2an 608	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐴‘(𝑘 + 1)) ∈ ℂ)
35		expcl 14145	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 + 1) ∈ ℕ₀) → (𝑋↑(𝑘 + 1)) ∈ ℂ)
36	23, 14, 35	syl2an 608	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑋↑(𝑘 + 1)) ∈ ℂ)
37	34, 36	mulcld 11256	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) ∈ ℂ)
38	37	abscld 15528	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) ∈ ℝ)
39	31, 38	remulcld 11266	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))) ∈ ℝ)
40	30, 39	eqeltrd 2862	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘) ∈ ℝ)
41		oveq1 7423	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
42	41	fveq2d 6886	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐴‘(𝑛 + 1)) = (𝐴‘(𝑘 + 1)))
43	41, 42	oveq12d 7434	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑛 + 1) · (𝐴‘(𝑛 + 1))) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
44		oveq2 7424	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑋↑𝑛) = (𝑋↑𝑘))
45	43, 44	oveq12d 7434	. . . . 5 ⊢ (𝑛 = 𝑘 → (((𝑛 + 1) · (𝐴‘(𝑛 + 1))) · (𝑋↑𝑛)) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)))
46		dvradcnv.h	. . . . 5 ⊢ 𝐻 = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 1) · (𝐴‘(𝑛 + 1))) · (𝑋↑𝑛)))
47		ovex 7449	. . . . 5 ⊢ (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) ∈ V
48	45, 46, 47	fvmpt 6990	. . . 4 ⊢ (𝑘 ∈ ℕ₀ → (𝐻‘𝑘) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)))
49	48	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐻‘𝑘) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)))
50	15	nn0cnd 12594	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℂ)
51	50, 34	mulcld 11256	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) ∈ ℂ)
52		expcl 14145	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑋↑𝑘) ∈ ℂ)
53	23, 52	sylan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑋↑𝑘) ∈ ℂ)
54	51, 53	mulcld 11256	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) ∈ ℂ)
55	49, 54	eqeltrd 2862	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐻‘𝑘) ∈ ℂ)
56		dvradcnv.r	. . . . . . . 8 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ^*, < )
57		dvradcnv.l	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝑋) < 𝑅)
58		id 23	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → 𝑖 = 𝑘)
59		2fveq3 6887	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (abs‘((𝐺‘𝑋)‘𝑖)) = (abs‘((𝐺‘𝑋)‘𝑘)))
60	58, 59	oveq12d 7434	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) = (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘))))
61	60	cbvmptv 5213	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) = (𝑘 ∈ ℕ₀ ↦ (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘))))
62	24, 32, 56, 23, 57, 61	radcnvlt1 26651	. . . . . . 7 ⊢ (𝜑 → (seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))) ∈ dom ⇝ ∧ seq0( + , (abs ∘ (𝐺‘𝑋))) ∈ dom ⇝ ))
63	62	simpld 500	. . . . . 6 ⊢ (𝜑 → seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))) ∈ dom ⇝ )
64		climdm 15643	. . . . . 6 ⊢ (seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))) ∈ dom ⇝ ↔ seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))) ⇝ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))))))
65	63, 64	sylib 221	. . . . 5 ⊢ (𝜑 → seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))) ⇝ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))))))
66		0z 12629	. . . . . 6 ⊢ 0 ∈ ℤ
67		neg1z 12657	. . . . . 6 ⊢ -1 ∈ ℤ
68	8	isershft 15753	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ -1 ∈ ℤ) → (seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))) ⇝ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))))) ↔ seq(0 + -1)( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ⇝ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))))))
69	66, 67, 68	mp2an 705	. . . . 5 ⊢ (seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))) ⇝ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))))) ↔ seq(0 + -1)( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ⇝ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))))))
70	65, 69	sylib 221	. . . 4 ⊢ (𝜑 → seq(0 + -1)( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ⇝ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))))))
71		seqex 14069	. . . . 5 ⊢ seq(0 + -1)( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ∈ V
72		fvex 6895	. . . . 5 ⊢ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))))) ∈ V
73	71, 72	breldm 5896	. . . 4 ⊢ (seq(0 + -1)( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ⇝ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))))) → seq(0 + -1)( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ∈ dom ⇝ )
74	70, 73	syl 18	. . 3 ⊢ (𝜑 → seq(0 + -1)( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ∈ dom ⇝ )
75		eqid 2762	. . . 4 ⊢ (ℤ_≥‘(0 + -1)) = (ℤ_≥‘(0 + -1))
76		neg1cn 12230	. . . . . . . 8 ⊢ -1 ∈ ℂ
77	76	addlidi 11425	. . . . . . 7 ⊢ (0 + -1) = -1
78		0le1 11764	. . . . . . . 8 ⊢ 0 ≤ 1
79		1re 11235	. . . . . . . . 9 ⊢ 1 ∈ ℝ
80		le0neg2 11750	. . . . . . . . 9 ⊢ (1 ∈ ℝ → (0 ≤ 1 ↔ -1 ≤ 0))
81	79, 80	ax-mp 5	. . . . . . . 8 ⊢ (0 ≤ 1 ↔ -1 ≤ 0)
82	78, 81	mpbi 233	. . . . . . 7 ⊢ -1 ≤ 0
83	77, 82	eqbrtri 5130	. . . . . 6 ⊢ (0 + -1) ≤ 0
84	77, 67	eqeltri 2858	. . . . . . 7 ⊢ (0 + -1) ∈ ℤ
85	84	eluz1i 12898	. . . . . 6 ⊢ (0 ∈ (ℤ_≥‘(0 + -1)) ↔ (0 ∈ ℤ ∧ (0 + -1) ≤ 0))
86	66, 83, 85	mpbir2an 724	. . . . 5 ⊢ 0 ∈ (ℤ_≥‘(0 + -1))
87	86	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ (ℤ_≥‘(0 + -1)))
88		eluzelcn 12902	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → 𝑘 ∈ ℂ)
89	88	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(0 + -1))) → 𝑘 ∈ ℂ)
90	4, 89, 9	sylancr 599	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(0 + -1))) → (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘) = ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(1 + 𝑘)))
91		nn0re 12540	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℕ₀ → 𝑖 ∈ ℝ)
92	91	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℝ)
93	24, 32, 23	psergf 26645	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺‘𝑋):ℕ₀⟶ℂ)
94	93	ffvelcdmda 7080	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → ((𝐺‘𝑋)‘𝑖) ∈ ℂ)
95	94	abscld 15528	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (abs‘((𝐺‘𝑋)‘𝑖)) ∈ ℝ)
96	92, 95	remulcld 11266	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) ∈ ℝ)
97	96	recnd 11264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) ∈ ℂ)
98	97	fmpttd 7111	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))):ℕ₀⟶ℂ)
99	4, 88, 11	sylancr 599	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (1 + 𝑘) = (𝑘 + 1))
100		eluzp1p1 12918	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (𝑘 + 1) ∈ (ℤ_≥‘((0 + -1) + 1)))
101	77	oveq1i 7426	. . . . . . . . . . 11 ⊢ ((0 + -1) + 1) = (-1 + 1)
102		1pneg1e0 12385	. . . . . . . . . . . 12 ⊢ (1 + -1) = 0
103	4, 76, 102	addcomli 11429	. . . . . . . . . . 11 ⊢ (-1 + 1) = 0
104	101, 103	eqtri 2785	. . . . . . . . . 10 ⊢ ((0 + -1) + 1) = 0
105	104	fveq2i 6885	. . . . . . . . 9 ⊢ (ℤ_≥‘((0 + -1) + 1)) = (ℤ_≥‘0)
106	1, 105	eqtr4i 2788	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘((0 + -1) + 1))
107	100, 106	eleqtrrdi 2873	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (𝑘 + 1) ∈ ℕ₀)
108	99, 107	eqeltrd 2862	. . . . . 6 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (1 + 𝑘) ∈ ℕ₀)
109		ffvelcdm 7077	. . . . . 6 ⊢ (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))):ℕ₀⟶ℂ ∧ (1 + 𝑘) ∈ ℕ₀) → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(1 + 𝑘)) ∈ ℂ)
110	98, 108, 109	syl2an 608	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(0 + -1))) → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(1 + 𝑘)) ∈ ℂ)
111	90, 110	eqeltrd 2862	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(0 + -1))) → (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘) ∈ ℂ)
112	75, 87, 111	iserex 15746	. . 3 ⊢ (𝜑 → (seq(0 + -1)( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ∈ dom ⇝ ↔ seq0( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ∈ dom ⇝ ))
113	74, 112	mpbid 235	. 2 ⊢ (𝜑 → seq0( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ∈ dom ⇝ )
114		1red 11236	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0) → 1 ∈ ℝ)
115		neqne 2965	. . . . 5 ⊢ (¬ 𝑋 = 0 → 𝑋 ≠ 0)
116		absrpcl 15377	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℝ⁺)
117	23, 115, 116	syl2an 608	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 0) → (abs‘𝑋) ∈ ℝ⁺)
118	117	rprecred 13099	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = 0) → (1 / (abs‘𝑋)) ∈ ℝ)
119	114, 118	ifclda 4521	. 2 ⊢ (𝜑 → if(𝑋 = 0, 1, (1 / (abs‘𝑋))) ∈ ℝ)
120		oveq1 7423	. . . . 5 ⊢ (1 = if(𝑋 = 0, 1, (1 / (abs‘𝑋))) → (1 · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))) = (if(𝑋 = 0, 1, (1 / (abs‘𝑋))) · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))))
121	120	breq2d 5119	. . . 4 ⊢ (1 = if(𝑋 = 0, 1, (1 / (abs‘𝑋))) → ((abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ≤ (1 · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))) ↔ (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ≤ (if(𝑋 = 0, 1, (1 / (abs‘𝑋))) · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))))
122		oveq1 7423	. . . . 5 ⊢ ((1 / (abs‘𝑋)) = if(𝑋 = 0, 1, (1 / (abs‘𝑋))) → ((1 / (abs‘𝑋)) · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))) = (if(𝑋 = 0, 1, (1 / (abs‘𝑋))) · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))))
123	122	breq2d 5119	. . . 4 ⊢ ((1 / (abs‘𝑋)) = if(𝑋 = 0, 1, (1 / (abs‘𝑋))) → ((abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ≤ ((1 / (abs‘𝑋)) · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))) ↔ (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ≤ (if(𝑋 = 0, 1, (1 / (abs‘𝑋))) · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))))
124		elnnuz 12930	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
125		nnnn0 12538	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
126	124, 125	sylbir 238	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘1) → 𝑘 ∈ ℕ₀)
127	15	nn0ge0d 12595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 0 ≤ (𝑘 + 1))
128	37	absge0d 15536	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 0 ≤ (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))
129	31, 38, 127, 128	mulge0d 11818	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 0 ≤ ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
130	126, 129	sylan2 605	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → 0 ≤ ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
131	130	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → 0 ≤ ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
132		oveq1 7423	. . . . . . . . 9 ⊢ (𝑋 = 0 → (𝑋↑𝑘) = (0↑𝑘))
133	124	bilanri 512	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → 𝑘 ∈ ℕ)
134	133	0expd 14205	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (0↑𝑘) = 0)
135	132, 134	sylan9eqr 2819	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (𝑋↑𝑘) = 0)
136	135	oveq2d 7432	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · 0))
137	51	mul01d 11436	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · 0) = 0)
138	126, 137	sylan2 605	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · 0) = 0)
139	138	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · 0) = 0)
140	136, 139	eqtrd 2797	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) = 0)
141	140	abs00bd 15380	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) = 0)
142	39	recnd 11264	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))) ∈ ℂ)
143	142	mullidd 11254	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (1 · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
144	126, 143	sylan2 605	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (1 · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
145	144	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (1 · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
146	131, 141, 145	3brtr4d 5141	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ≤ (1 · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))))
147		df-ne 2958	. . . . 5 ⊢ (𝑋 ≠ 0 ↔ ¬ 𝑋 = 0)
148	54	abscld 15528	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ∈ ℝ)
149	50, 34, 53	mulassd 11259	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) = ((𝑘 + 1) · ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘))))
150	149	fveq2d 6886	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) = (abs‘((𝑘 + 1) · ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
151	34, 53	mulcld 11256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)) ∈ ℂ)
152	50, 151	absmuld 15546	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝑘 + 1) · ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))) = ((abs‘(𝑘 + 1)) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
153	31, 127	absidd 15512	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝑘 + 1)) = (𝑘 + 1))
154	153	oveq1d 7431	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((abs‘(𝑘 + 1)) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
155	150, 152, 154	3eqtrd 2801	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
156	148, 155	eqled 11340	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ≤ ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
157	156	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ≤ ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
158	23	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝑋 ∈ ℂ)
159	116	rpreccld 13098	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (1 / (abs‘𝑋)) ∈ ℝ⁺)
160	158, 159	sylan 592	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (1 / (abs‘𝑋)) ∈ ℝ⁺)
161	160	rpcnd 13090	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (1 / (abs‘𝑋)) ∈ ℂ)
162	50	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑘 + 1) ∈ ℂ)
163	38	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) ∈ ℝ)
164	163	recnd 11264	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) ∈ ℂ)
165	161, 162, 164	mul12d 11446	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((1 / (abs‘𝑋)) · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))) = ((𝑘 + 1) · ((1 / (abs‘𝑋)) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))))
166	37	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) ∈ ℂ)
167	23	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → 𝑋 ∈ ℂ)
168		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → 𝑋 ≠ 0)
169	166, 167, 168	absdivd 15547	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘(((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) / 𝑋)) = ((abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) / (abs‘𝑋)))
170	34	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝐴‘(𝑘 + 1)) ∈ ℂ)
171	36	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑(𝑘 + 1)) ∈ ℂ)
172	170, 171, 167, 168	divassd 12053	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) / 𝑋) = ((𝐴‘(𝑘 + 1)) · ((𝑋↑(𝑘 + 1)) / 𝑋)))
173	6	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → 𝑘 ∈ ℂ)
174		pncan 11490	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
175	173, 4, 174	sylancl 598	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((𝑘 + 1) − 1) = 𝑘)
176	175	oveq2d 7432	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑((𝑘 + 1) − 1)) = (𝑋↑𝑘))
177	15	nn0zd 12643	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℤ)
178	177	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑘 + 1) ∈ ℤ)
179	167, 168, 178	expm1d 14222	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑((𝑘 + 1) − 1)) = ((𝑋↑(𝑘 + 1)) / 𝑋))
180	176, 179	eqtr3d 2799	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑𝑘) = ((𝑋↑(𝑘 + 1)) / 𝑋))
181	180	oveq2d 7432	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)) = ((𝐴‘(𝑘 + 1)) · ((𝑋↑(𝑘 + 1)) / 𝑋)))
182	172, 181	eqtr4d 2800	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) / 𝑋) = ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))
183	182	fveq2d 6886	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘(((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) / 𝑋)) = (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘))))
184	23	abscld 15528	. . . . . . . . . . . . 13 ⊢ (𝜑 → (abs‘𝑋) ∈ ℝ)
185	184	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℝ)
186	185	recnd 11264	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℂ)
187	158, 116	sylan 592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℝ⁺)
188	187	rpne0d 13093	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ≠ 0)
189	164, 186, 188	divrec2d 12022	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) / (abs‘𝑋)) = ((1 / (abs‘𝑋)) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
190	169, 183, 189	3eqtr3rd 2806	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((1 / (abs‘𝑋)) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))) = (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘))))
191	190	oveq2d 7432	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((𝑘 + 1) · ((1 / (abs‘𝑋)) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
192	165, 191	eqtrd 2797	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((1 / (abs‘𝑋)) · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
193	157, 192	breqtrrd 5137	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ≤ ((1 / (abs‘𝑋)) · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))))
194	126, 193	sylanl2 694	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 ≠ 0) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ≤ ((1 / (abs‘𝑋)) · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))))
195	147, 194	sylan2br 607	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑋 = 0) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ≤ ((1 / (abs‘𝑋)) · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))))
196	121, 123, 146, 195	ifbothda 4524	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ≤ (if(𝑋 = 0, 1, (1 / (abs‘𝑋))) · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))))
197	49	fveq2d 6886	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝐻‘𝑘)) = (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))))
198	126, 197	sylan2 605	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (abs‘(𝐻‘𝑘)) = (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))))
199	30	oveq2d 7432	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (if(𝑋 = 0, 1, (1 / (abs‘𝑋))) · (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘)) = (if(𝑋 = 0, 1, (1 / (abs‘𝑋))) · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))))
200	126, 199	sylan2 605	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (if(𝑋 = 0, 1, (1 / (abs‘𝑋))) · (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘)) = (if(𝑋 = 0, 1, (1 / (abs‘𝑋))) · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))))
201	196, 198, 200	3brtr4d 5141	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (abs‘(𝐻‘𝑘)) ≤ (if(𝑋 = 0, 1, (1 / (abs‘𝑋))) · (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘)))
202	1, 3, 40, 55, 113, 119, 201	cvgcmpce 15907	1 ⊢ (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ )

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 {crab 3414 ifcif 4485 class class class wbr 5107 ↦ cmpt 5190 dom cdm 5659 ∘ ccom 5663 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 supcsup 9413 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 − cmin 11468 -cneg 11469 / cdiv 11898 ℕcn 12260 ℕ₀cn0 12531 ℤcz 12618 ℤ_≥cuz 12890 ℝ⁺crp 13044 seqcseq 14067 ↑cexp 14127 shift cshi 15141 abscabs 15323 ⇝ cli 15573

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-er 8699 df-pm 8832 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-n0 12532 df-z 12619 df-uz 12891 df-rp 13045 df-ico 13406 df-icc 13407 df-fz 13564 df-fzo 13712 df-fl 13855 df-seq 14068 df-exp 14128 df-hash 14397 df-shft 15142 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-limsup 15560 df-clim 15577 df-rlim 15578 df-sum 15776

This theorem is used by: pserdvlem2 26661 dvradcnv2 45158

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