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Theorem dvradcnv 26374

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 12815	. 2 ⊢ ℕ₀ = (ℤ_≥‘0)
2		1nn0 12442	. . 3 ⊢ 1 ∈ ℕ₀
3	2	a1i 11	. 2 ⊢ (𝜑 → 1 ∈ ℕ₀)
4		ax-1cn 11085	. . . . 5 ⊢ 1 ∈ ℂ
5		nn0cn 12436	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
6	5	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℂ)
7		nn0ex 12432	. . . . . . 7 ⊢ ℕ₀ ∈ V
8	7	mptex 7167	. . . . . 6 ⊢ (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) ∈ V
9	8	shftval4 15028	. . . . 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 588	. . . 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 11321	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (1 + 𝑘) = (𝑘 + 1))
12	4, 6, 11	sylancr 588	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (1 + 𝑘) = (𝑘 + 1))
13	12	fveq2d 6833	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(1 + 𝑘)) = ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(𝑘 + 1)))
14		peano2nn0 12466	. . . . . . 7 ⊢ (𝑘 ∈ ℕ₀ → (𝑘 + 1) ∈ ℕ₀)
15	14	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℕ₀)
16		id 22	. . . . . . . 8 ⊢ (𝑖 = (𝑘 + 1) → 𝑖 = (𝑘 + 1))
17		2fveq3 6834	. . . . . . . 8 ⊢ (𝑖 = (𝑘 + 1) → (abs‘((𝐺‘𝑋)‘𝑖)) = (abs‘((𝐺‘𝑋)‘(𝑘 + 1))))
18	16, 17	oveq12d 7374	. . . . . . 7 ⊢ (𝑖 = (𝑘 + 1) → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) = ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))))
19		eqid 2735	. . . . . . 7 ⊢ (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) = (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))
20		ovex 7389	. . . . . . 7 ⊢ ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))) ∈ V
21	18, 19, 20	fvmpt 6936	. . . . . 6 ⊢ ((𝑘 + 1) ∈ ℕ₀ → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(𝑘 + 1)) = ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))))
22	15, 21	syl 17	. . . . 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 26364	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 + 1) ∈ ℕ₀) → ((𝐺‘𝑋)‘(𝑘 + 1)) = ((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))
26	23, 14, 25	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐺‘𝑋)‘(𝑘 + 1)) = ((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))
27	26	fveq2d 6833	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝐺‘𝑋)‘(𝑘 + 1))) = (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))
28	27	oveq2d 7372	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
29	22, 28	eqtrd 2770	. . . 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 2774	. . 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 12488	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℝ)
32		dvradcnv.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
33		ffvelcdm 7022	. . . . . . 7 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ (𝑘 + 1) ∈ ℕ₀) → (𝐴‘(𝑘 + 1)) ∈ ℂ)
34	32, 14, 33	syl2an 597	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐴‘(𝑘 + 1)) ∈ ℂ)
35		expcl 14030	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 + 1) ∈ ℕ₀) → (𝑋↑(𝑘 + 1)) ∈ ℂ)
36	23, 14, 35	syl2an 597	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑋↑(𝑘 + 1)) ∈ ℂ)
37	34, 36	mulcld 11154	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) ∈ ℂ)
38	37	abscld 15390	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) ∈ ℝ)
39	31, 38	remulcld 11164	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))) ∈ ℝ)
40	30, 39	eqeltrd 2835	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘) ∈ ℝ)
41		oveq1 7363	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
42	41	fveq2d 6833	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐴‘(𝑛 + 1)) = (𝐴‘(𝑘 + 1)))
43	41, 42	oveq12d 7374	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑛 + 1) · (𝐴‘(𝑛 + 1))) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
44		oveq2 7364	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑋↑𝑛) = (𝑋↑𝑘))
45	43, 44	oveq12d 7374	. . . . 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 7389	. . . . 5 ⊢ (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) ∈ V
48	45, 46, 47	fvmpt 6936	. . . 4 ⊢ (𝑘 ∈ ℕ₀ → (𝐻‘𝑘) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)))
49	48	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐻‘𝑘) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)))
50	15	nn0cnd 12489	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℂ)
51	50, 34	mulcld 11154	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) ∈ ℂ)
52		expcl 14030	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑋↑𝑘) ∈ ℂ)
53	23, 52	sylan 581	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑋↑𝑘) ∈ ℂ)
54	51, 53	mulcld 11154	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) ∈ ℂ)
55	49, 54	eqeltrd 2835	. 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 22	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → 𝑖 = 𝑘)
59		2fveq3 6834	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (abs‘((𝐺‘𝑋)‘𝑖)) = (abs‘((𝐺‘𝑋)‘𝑘)))
60	58, 59	oveq12d 7374	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) = (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘))))
61	60	cbvmptv 5178	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) = (𝑘 ∈ ℕ₀ ↦ (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘))))
62	24, 32, 56, 23, 57, 61	radcnvlt1 26371	. . . . . . 7 ⊢ (𝜑 → (seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))) ∈ dom ⇝ ∧ seq0( + , (abs ∘ (𝐺‘𝑋))) ∈ dom ⇝ ))
63	62	simpld 494	. . . . . 6 ⊢ (𝜑 → seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))) ∈ dom ⇝ )
64		climdm 15505	. . . . . 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 218	. . . . 5 ⊢ (𝜑 → seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))) ⇝ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))))))
66		0z 12524	. . . . . 6 ⊢ 0 ∈ ℤ
67		neg1z 12552	. . . . . 6 ⊢ -1 ∈ ℤ
68	8	isershft 15615	. . . . . 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 693	. . . . 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 218	. . . 4 ⊢ (𝜑 → seq(0 + -1)( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ⇝ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))))))
71		seqex 13954	. . . . 5 ⊢ seq(0 + -1)( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ∈ V
72		fvex 6842	. . . . 5 ⊢ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))))) ∈ V
73	71, 72	breldm 5852	. . . 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 17	. . 3 ⊢ (𝜑 → seq(0 + -1)( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ∈ dom ⇝ )
75		eqid 2735	. . . 4 ⊢ (ℤ_≥‘(0 + -1)) = (ℤ_≥‘(0 + -1))
76		neg1cn 12133	. . . . . . . 8 ⊢ -1 ∈ ℂ
77	76	addlidi 11323	. . . . . . 7 ⊢ (0 + -1) = -1
78		0le1 11662	. . . . . . . 8 ⊢ 0 ≤ 1
79		1re 11133	. . . . . . . . 9 ⊢ 1 ∈ ℝ
80		le0neg2 11648	. . . . . . . . 9 ⊢ (1 ∈ ℝ → (0 ≤ 1 ↔ -1 ≤ 0))
81	79, 80	ax-mp 5	. . . . . . . 8 ⊢ (0 ≤ 1 ↔ -1 ≤ 0)
82	78, 81	mpbi 230	. . . . . . 7 ⊢ -1 ≤ 0
83	77, 82	eqbrtri 5095	. . . . . 6 ⊢ (0 + -1) ≤ 0
84	77, 67	eqeltri 2831	. . . . . . 7 ⊢ (0 + -1) ∈ ℤ
85	84	eluz1i 12785	. . . . . 6 ⊢ (0 ∈ (ℤ_≥‘(0 + -1)) ↔ (0 ∈ ℤ ∧ (0 + -1) ≤ 0))
86	66, 83, 85	mpbir2an 712	. . . . 5 ⊢ 0 ∈ (ℤ_≥‘(0 + -1))
87	86	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ (ℤ_≥‘(0 + -1)))
88		eluzelcn 12789	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → 𝑘 ∈ ℂ)
89	88	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(0 + -1))) → 𝑘 ∈ ℂ)
90	4, 89, 9	sylancr 588	. . . . 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 12435	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℕ₀ → 𝑖 ∈ ℝ)
92	91	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℝ)
93	24, 32, 23	psergf 26365	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺‘𝑋):ℕ₀⟶ℂ)
94	93	ffvelcdmda 7025	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → ((𝐺‘𝑋)‘𝑖) ∈ ℂ)
95	94	abscld 15390	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (abs‘((𝐺‘𝑋)‘𝑖)) ∈ ℝ)
96	92, 95	remulcld 11164	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) ∈ ℝ)
97	96	recnd 11162	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) ∈ ℂ)
98	97	fmpttd 7056	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))):ℕ₀⟶ℂ)
99	4, 88, 11	sylancr 588	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (1 + 𝑘) = (𝑘 + 1))
100		eluzp1p1 12805	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (𝑘 + 1) ∈ (ℤ_≥‘((0 + -1) + 1)))
101	77	oveq1i 7366	. . . . . . . . . . 11 ⊢ ((0 + -1) + 1) = (-1 + 1)
102		1pneg1e0 12284	. . . . . . . . . . . 12 ⊢ (1 + -1) = 0
103	4, 76, 102	addcomli 11327	. . . . . . . . . . 11 ⊢ (-1 + 1) = 0
104	101, 103	eqtri 2758	. . . . . . . . . 10 ⊢ ((0 + -1) + 1) = 0
105	104	fveq2i 6832	. . . . . . . . 9 ⊢ (ℤ_≥‘((0 + -1) + 1)) = (ℤ_≥‘0)
106	1, 105	eqtr4i 2761	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘((0 + -1) + 1))
107	100, 106	eleqtrrdi 2846	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (𝑘 + 1) ∈ ℕ₀)
108	99, 107	eqeltrd 2835	. . . . . 6 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (1 + 𝑘) ∈ ℕ₀)
109		ffvelcdm 7022	. . . . . 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 597	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(0 + -1))) → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(1 + 𝑘)) ∈ ℂ)
111	90, 110	eqeltrd 2835	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(0 + -1))) → (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘) ∈ ℂ)
112	75, 87, 111	iserex 15608	. . 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 232	. 2 ⊢ (𝜑 → seq0( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ∈ dom ⇝ )
114		1red 11134	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0) → 1 ∈ ℝ)
115		neqne 2938	. . . . 5 ⊢ (¬ 𝑋 = 0 → 𝑋 ≠ 0)
116		absrpcl 15239	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℝ⁺)
117	23, 115, 116	syl2an 597	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 0) → (abs‘𝑋) ∈ ℝ⁺)
118	117	rprecred 12986	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = 0) → (1 / (abs‘𝑋)) ∈ ℝ)
119	114, 118	ifclda 4492	. 2 ⊢ (𝜑 → if(𝑋 = 0, 1, (1 / (abs‘𝑋))) ∈ ℝ)
120		oveq1 7363	. . . . 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 5086	. . . 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 7363	. . . . 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 5086	. . . 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 12817	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
125		nnnn0 12433	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
126	124, 125	sylbir 235	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘1) → 𝑘 ∈ ℕ₀)
127	15	nn0ge0d 12490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 0 ≤ (𝑘 + 1))
128	37	absge0d 15398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 0 ≤ (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))
129	31, 38, 127, 128	mulge0d 11716	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 0 ≤ ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
130	126, 129	sylan2 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → 0 ≤ ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
131	130	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → 0 ≤ ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
132		oveq1 7363	. . . . . . . . 9 ⊢ (𝑋 = 0 → (𝑋↑𝑘) = (0↑𝑘))
133		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → 𝑘 ∈ (ℤ_≥‘1))
134	133, 124	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → 𝑘 ∈ ℕ)
135	134	0expd 14090	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (0↑𝑘) = 0)
136	132, 135	sylan9eqr 2792	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (𝑋↑𝑘) = 0)
137	136	oveq2d 7372	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · 0))
138	51	mul01d 11334	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · 0) = 0)
139	126, 138	sylan2 594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · 0) = 0)
140	139	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · 0) = 0)
141	137, 140	eqtrd 2770	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) = 0)
142	141	abs00bd 15242	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) = 0)
143	39	recnd 11162	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))) ∈ ℂ)
144	143	mullidd 11152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (1 · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
145	126, 144	sylan2 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (1 · ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
146	145	adantr 480	. . . . 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))))))
147	131, 142, 146	3brtr4d 5106	. . . 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)))))))
148		df-ne 2931	. . . . 5 ⊢ (𝑋 ≠ 0 ↔ ¬ 𝑋 = 0)
149	54	abscld 15390	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ∈ ℝ)
150	50, 34, 53	mulassd 11157	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) = ((𝑘 + 1) · ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘))))
151	150	fveq2d 6833	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) = (abs‘((𝑘 + 1) · ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
152	34, 53	mulcld 11154	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)) ∈ ℂ)
153	50, 152	absmuld 15408	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝑘 + 1) · ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))) = ((abs‘(𝑘 + 1)) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
154	31, 127	absidd 15374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝑘 + 1)) = (𝑘 + 1))
155	154	oveq1d 7371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((abs‘(𝑘 + 1)) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
156	151, 153, 155	3eqtrd 2774	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
157	149, 156	eqled 11238	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ≤ ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
158	157	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ≤ ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
159	23	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝑋 ∈ ℂ)
160	116	rpreccld 12985	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (1 / (abs‘𝑋)) ∈ ℝ⁺)
161	159, 160	sylan 581	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (1 / (abs‘𝑋)) ∈ ℝ⁺)
162	161	rpcnd 12977	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (1 / (abs‘𝑋)) ∈ ℂ)
163	50	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑘 + 1) ∈ ℂ)
164	38	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) ∈ ℝ)
165	164	recnd 11162	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) ∈ ℂ)
166	162, 163, 165	mul12d 11344	. . . . . . . 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)))))))
167	37	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) ∈ ℂ)
168	23	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → 𝑋 ∈ ℂ)
169		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → 𝑋 ≠ 0)
170	167, 168, 169	absdivd 15409	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘(((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) / 𝑋)) = ((abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) / (abs‘𝑋)))
171	34	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝐴‘(𝑘 + 1)) ∈ ℂ)
172	36	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑(𝑘 + 1)) ∈ ℂ)
173	171, 172, 168, 169	divassd 11955	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) / 𝑋) = ((𝐴‘(𝑘 + 1)) · ((𝑋↑(𝑘 + 1)) / 𝑋)))
174	6	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → 𝑘 ∈ ℂ)
175		pncan 11388	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
176	174, 4, 175	sylancl 587	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((𝑘 + 1) − 1) = 𝑘)
177	176	oveq2d 7372	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑((𝑘 + 1) − 1)) = (𝑋↑𝑘))
178	15	nn0zd 12538	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℤ)
179	178	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑘 + 1) ∈ ℤ)
180	168, 169, 179	expm1d 14107	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑((𝑘 + 1) − 1)) = ((𝑋↑(𝑘 + 1)) / 𝑋))
181	177, 180	eqtr3d 2772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑𝑘) = ((𝑋↑(𝑘 + 1)) / 𝑋))
182	181	oveq2d 7372	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)) = ((𝐴‘(𝑘 + 1)) · ((𝑋↑(𝑘 + 1)) / 𝑋)))
183	173, 182	eqtr4d 2773	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) / 𝑋) = ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))
184	183	fveq2d 6833	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘(((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) / 𝑋)) = (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘))))
185	23	abscld 15390	. . . . . . . . . . . . 13 ⊢ (𝜑 → (abs‘𝑋) ∈ ℝ)
186	185	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℝ)
187	186	recnd 11162	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℂ)
188	159, 116	sylan 581	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℝ⁺)
189	188	rpne0d 12980	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ≠ 0)
190	165, 187, 189	divrec2d 11924	. . . . . . . . . 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))))))
191	170, 184, 190	3eqtr3rd 2779	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((1 / (abs‘𝑋)) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))) = (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘))))
192	191	oveq2d 7372	. . . . . . . 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)))))
193	166, 192	eqtrd 2770	. . . . . . 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)))))
194	158, 193	breqtrrd 5102	. . . . . 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)))))))
195	126, 194	sylanl2 682	. . . . 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)))))))
196	148, 195	sylan2br 596	. . . 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)))))))
197	121, 123, 147, 196	ifbothda 4495	. . 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)))))))
198	49	fveq2d 6833	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝐻‘𝑘)) = (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))))
199	126, 198	sylan2 594	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (abs‘(𝐻‘𝑘)) = (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))))
200	30	oveq2d 7372	. . . 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)))))))
201	126, 200	sylan2 594	. . 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)))))))
202	197, 199, 201	3brtr4d 5106	. 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)))
203	1, 3, 40, 55, 113, 119, 202	cvgcmpce 15770	1 ⊢ (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ )

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2930 {crab 3387 ifcif 4456 class class class wbr 5074 ↦ cmpt 5155 dom cdm 5620 ∘ ccom 5624 ⟶wf 6483 ‘cfv 6487 (class class class)co 7356 supcsup 9342 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 · cmul 11032 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 − cmin 11366 -cneg 11367 / cdiv 11796 ℕcn 12163 ℕ₀cn0 12426 ℤcz 12513 ℤ_≥cuz 12777 ℝ⁺crp 12931 seqcseq 13952 ↑cexp 14012 shift cshi 15017 abscabs 15185 ⇝ cli 15435

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 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8632 df-pm 8765 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-sup 9344 df-inf 9345 df-oi 9414 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-n0 12427 df-z 12514 df-uz 12778 df-rp 12932 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-seq 13953 df-exp 14013 df-hash 14282 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15638

This theorem is referenced by: pserdvlem2 26381 dvradcnv2 44762

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