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Theorem dvradcnv 26328

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 12777	. 2 ⊢ ℕ₀ = (ℤ_≥‘0)
2		1nn0 12400	. . 3 ⊢ 1 ∈ ℕ₀
3	2	a1i 11	. 2 ⊢ (𝜑 → 1 ∈ ℕ₀)
4		ax-1cn 11067	. . . . 5 ⊢ 1 ∈ ℂ
5		nn0cn 12394	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
6	5	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℂ)
7		nn0ex 12390	. . . . . . 7 ⊢ ℕ₀ ∈ V
8	7	mptex 7159	. . . . . 6 ⊢ (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) ∈ V
9	8	shftval4 14984	. . . . 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 587	. . . 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 11302	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (1 + 𝑘) = (𝑘 + 1))
12	4, 6, 11	sylancr 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (1 + 𝑘) = (𝑘 + 1))
13	12	fveq2d 6826	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(1 + 𝑘)) = ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(𝑘 + 1)))
14		peano2nn0 12424	. . . . . . 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 6827	. . . . . . . 8 ⊢ (𝑖 = (𝑘 + 1) → (abs‘((𝐺‘𝑋)‘𝑖)) = (abs‘((𝐺‘𝑋)‘(𝑘 + 1))))
18	16, 17	oveq12d 7367	. . . . . . 7 ⊢ (𝑖 = (𝑘 + 1) → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) = ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))))
19		eqid 2729	. . . . . . 7 ⊢ (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) = (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))
20		ovex 7382	. . . . . . 7 ⊢ ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))) ∈ V
21	18, 19, 20	fvmpt 6930	. . . . . 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 26318	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 + 1) ∈ ℕ₀) → ((𝐺‘𝑋)‘(𝑘 + 1)) = ((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))
26	23, 14, 25	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐺‘𝑋)‘(𝑘 + 1)) = ((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))
27	26	fveq2d 6826	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝐺‘𝑋)‘(𝑘 + 1))) = (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))
28	27	oveq2d 7365	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
29	22, 28	eqtrd 2764	. . . 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 2768	. . 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 12446	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℝ)
32		dvradcnv.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
33		ffvelcdm 7015	. . . . . . 7 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ (𝑘 + 1) ∈ ℕ₀) → (𝐴‘(𝑘 + 1)) ∈ ℂ)
34	32, 14, 33	syl2an 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐴‘(𝑘 + 1)) ∈ ℂ)
35		expcl 13986	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 + 1) ∈ ℕ₀) → (𝑋↑(𝑘 + 1)) ∈ ℂ)
36	23, 14, 35	syl2an 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑋↑(𝑘 + 1)) ∈ ℂ)
37	34, 36	mulcld 11135	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) ∈ ℂ)
38	37	abscld 15346	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) ∈ ℝ)
39	31, 38	remulcld 11145	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))) ∈ ℝ)
40	30, 39	eqeltrd 2828	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘) ∈ ℝ)
41		oveq1 7356	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
42	41	fveq2d 6826	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐴‘(𝑛 + 1)) = (𝐴‘(𝑘 + 1)))
43	41, 42	oveq12d 7367	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑛 + 1) · (𝐴‘(𝑛 + 1))) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
44		oveq2 7357	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑋↑𝑛) = (𝑋↑𝑘))
45	43, 44	oveq12d 7367	. . . . 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 7382	. . . . 5 ⊢ (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) ∈ V
48	45, 46, 47	fvmpt 6930	. . . 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 12447	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℂ)
51	50, 34	mulcld 11135	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) ∈ ℂ)
52		expcl 13986	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑋↑𝑘) ∈ ℂ)
53	23, 52	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑋↑𝑘) ∈ ℂ)
54	51, 53	mulcld 11135	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) ∈ ℂ)
55	49, 54	eqeltrd 2828	. 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 6827	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (abs‘((𝐺‘𝑋)‘𝑖)) = (abs‘((𝐺‘𝑋)‘𝑘)))
60	58, 59	oveq12d 7367	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) = (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘))))
61	60	cbvmptv 5196	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) = (𝑘 ∈ ℕ₀ ↦ (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘))))
62	24, 32, 56, 23, 57, 61	radcnvlt1 26325	. . . . . . 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 15461	. . . . . 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 12482	. . . . . 6 ⊢ 0 ∈ ℤ
67		neg1z 12511	. . . . . 6 ⊢ -1 ∈ ℤ
68	8	isershft 15571	. . . . . 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 692	. . . . 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 13910	. . . . 5 ⊢ seq(0 + -1)( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ∈ V
72		fvex 6835	. . . . 5 ⊢ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))))) ∈ V
73	71, 72	breldm 5851	. . . 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 2729	. . . 4 ⊢ (ℤ_≥‘(0 + -1)) = (ℤ_≥‘(0 + -1))
76		neg1cn 12113	. . . . . . . 8 ⊢ -1 ∈ ℂ
77	76	addlidi 11304	. . . . . . 7 ⊢ (0 + -1) = -1
78		0le1 11643	. . . . . . . 8 ⊢ 0 ≤ 1
79		1re 11115	. . . . . . . . 9 ⊢ 1 ∈ ℝ
80		le0neg2 11629	. . . . . . . . 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 5113	. . . . . 6 ⊢ (0 + -1) ≤ 0
84	77, 67	eqeltri 2824	. . . . . . 7 ⊢ (0 + -1) ∈ ℤ
85	84	eluz1i 12743	. . . . . 6 ⊢ (0 ∈ (ℤ_≥‘(0 + -1)) ↔ (0 ∈ ℤ ∧ (0 + -1) ≤ 0))
86	66, 83, 85	mpbir2an 711	. . . . 5 ⊢ 0 ∈ (ℤ_≥‘(0 + -1))
87	86	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ (ℤ_≥‘(0 + -1)))
88		eluzelcn 12747	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → 𝑘 ∈ ℂ)
89	88	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(0 + -1))) → 𝑘 ∈ ℂ)
90	4, 89, 9	sylancr 587	. . . . 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 12393	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℕ₀ → 𝑖 ∈ ℝ)
92	91	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℝ)
93	24, 32, 23	psergf 26319	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺‘𝑋):ℕ₀⟶ℂ)
94	93	ffvelcdmda 7018	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → ((𝐺‘𝑋)‘𝑖) ∈ ℂ)
95	94	abscld 15346	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (abs‘((𝐺‘𝑋)‘𝑖)) ∈ ℝ)
96	92, 95	remulcld 11145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) ∈ ℝ)
97	96	recnd 11143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) ∈ ℂ)
98	97	fmpttd 7049	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))):ℕ₀⟶ℂ)
99	4, 88, 11	sylancr 587	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (1 + 𝑘) = (𝑘 + 1))
100		eluzp1p1 12763	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (𝑘 + 1) ∈ (ℤ_≥‘((0 + -1) + 1)))
101	77	oveq1i 7359	. . . . . . . . . . 11 ⊢ ((0 + -1) + 1) = (-1 + 1)
102		1pneg1e0 12242	. . . . . . . . . . . 12 ⊢ (1 + -1) = 0
103	4, 76, 102	addcomli 11308	. . . . . . . . . . 11 ⊢ (-1 + 1) = 0
104	101, 103	eqtri 2752	. . . . . . . . . 10 ⊢ ((0 + -1) + 1) = 0
105	104	fveq2i 6825	. . . . . . . . 9 ⊢ (ℤ_≥‘((0 + -1) + 1)) = (ℤ_≥‘0)
106	1, 105	eqtr4i 2755	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘((0 + -1) + 1))
107	100, 106	eleqtrrdi 2839	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (𝑘 + 1) ∈ ℕ₀)
108	99, 107	eqeltrd 2828	. . . . . 6 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (1 + 𝑘) ∈ ℕ₀)
109		ffvelcdm 7015	. . . . . 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 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(0 + -1))) → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(1 + 𝑘)) ∈ ℂ)
111	90, 110	eqeltrd 2828	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(0 + -1))) → (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘) ∈ ℂ)
112	75, 87, 111	iserex 15564	. . 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 11116	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0) → 1 ∈ ℝ)
115		neqne 2933	. . . . 5 ⊢ (¬ 𝑋 = 0 → 𝑋 ≠ 0)
116		absrpcl 15195	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℝ⁺)
117	23, 115, 116	syl2an 596	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 0) → (abs‘𝑋) ∈ ℝ⁺)
118	117	rprecred 12948	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = 0) → (1 / (abs‘𝑋)) ∈ ℝ)
119	114, 118	ifclda 4512	. 2 ⊢ (𝜑 → if(𝑋 = 0, 1, (1 / (abs‘𝑋))) ∈ ℝ)
120		oveq1 7356	. . . . 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 5104	. . . 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 7356	. . . . 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 5104	. . . 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 12779	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
125		nnnn0 12391	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
126	124, 125	sylbir 235	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘1) → 𝑘 ∈ ℕ₀)
127	15	nn0ge0d 12448	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 0 ≤ (𝑘 + 1))
128	37	absge0d 15354	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 0 ≤ (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))
129	31, 38, 127, 128	mulge0d 11697	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 0 ≤ ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
130	126, 129	sylan2 593	. . . . . 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 7356	. . . . . . . . 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 14046	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (0↑𝑘) = 0)
136	132, 135	sylan9eqr 2786	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (𝑋↑𝑘) = 0)
137	136	oveq2d 7365	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · 0))
138	51	mul01d 11315	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · 0) = 0)
139	126, 138	sylan2 593	. . . . . . . 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 2764	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) = 0)
142	141	abs00bd 15198	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) = 0)
143	39	recnd 11143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))) ∈ ℂ)
144	143	mullidd 11133	. . . . . . 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 593	. . . . . 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 5124	. . . 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 2926	. . . . 5 ⊢ (𝑋 ≠ 0 ↔ ¬ 𝑋 = 0)
149	54	abscld 15346	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ∈ ℝ)
150	50, 34, 53	mulassd 11138	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) = ((𝑘 + 1) · ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘))))
151	150	fveq2d 6826	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) = (abs‘((𝑘 + 1) · ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
152	34, 53	mulcld 11135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)) ∈ ℂ)
153	50, 152	absmuld 15364	. . . . . . . . . 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 15330	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝑘 + 1)) = (𝑘 + 1))
155	154	oveq1d 7364	. . . . . . . . . 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 2768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
157	149, 156	eqled 11219	. . . . . . . 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 12947	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (1 / (abs‘𝑋)) ∈ ℝ⁺)
161	159, 160	sylan 580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (1 / (abs‘𝑋)) ∈ ℝ⁺)
162	161	rpcnd 12939	. . . . . . . . 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 11143	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) ∈ ℂ)
166	162, 163, 165	mul12d 11325	. . . . . . . 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 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → 𝑋 ∈ ℂ)
169		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → 𝑋 ≠ 0)
170	167, 168, 169	absdivd 15365	. . . . . . . . . 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 11935	. . . . . . . . . . . 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 11369	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
176	174, 4, 175	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((𝑘 + 1) − 1) = 𝑘)
177	176	oveq2d 7365	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑((𝑘 + 1) − 1)) = (𝑋↑𝑘))
178	15	nn0zd 12497	. . . . . . . . . . . . . . . 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 14063	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑((𝑘 + 1) − 1)) = ((𝑋↑(𝑘 + 1)) / 𝑋))
181	177, 180	eqtr3d 2766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑𝑘) = ((𝑋↑(𝑘 + 1)) / 𝑋))
182	181	oveq2d 7365	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)) = ((𝐴‘(𝑘 + 1)) · ((𝑋↑(𝑘 + 1)) / 𝑋)))
183	173, 182	eqtr4d 2767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) / 𝑋) = ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))
184	183	fveq2d 6826	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘(((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) / 𝑋)) = (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘))))
185	23	abscld 15346	. . . . . . . . . . . . 13 ⊢ (𝜑 → (abs‘𝑋) ∈ ℝ)
186	185	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℝ)
187	186	recnd 11143	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℂ)
188	159, 116	sylan 580	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℝ⁺)
189	188	rpne0d 12942	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ≠ 0)
190	165, 187, 189	divrec2d 11904	. . . . . . . . . 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 2773	. . . . . . . . 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 7365	. . . . . . . 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 2764	. . . . . . 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 5120	. . . . . 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 681	. . . . 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 595	. . . 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 4515	. . 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 6826	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝐻‘𝑘)) = (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))))
199	126, 198	sylan2 593	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (abs‘(𝐻‘𝑘)) = (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))))
200	30	oveq2d 7365	. . . 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 593	. . 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 5124	. 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 15725	1 ⊢ (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ )

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3394 ifcif 4476 class class class wbr 5092 ↦ cmpt 5173 dom cdm 5619 ∘ ccom 5623 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 supcsup 9330 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 ℝ^*cxr 11148 < clt 11149 ≤ cle 11150 − cmin 11347 -cneg 11348 / cdiv 11777 ℕcn 12128 ℕ₀cn0 12384 ℤcz 12471 ℤ_≥cuz 12735 ℝ⁺crp 12893 seqcseq 13908 ↑cexp 13968 shift cshi 14973 abscabs 15141 ⇝ cli 15391

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-pm 8756 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-n0 12385 df-z 12472 df-uz 12736 df-rp 12894 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594

This theorem is referenced by: pserdvlem2 26336 dvradcnv2 44340

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