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Theorem dvradcnv 26479

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 12918	. 2 ⊢ ℕ₀ = (ℤ_≥‘0)
2		1nn0 12540	. . 3 ⊢ 1 ∈ ℕ₀
3	2	a1i 11	. 2 ⊢ (𝜑 → 1 ∈ ℕ₀)
4		ax-1cn 11211	. . . . 5 ⊢ 1 ∈ ℂ
5		nn0cn 12534	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
6	5	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℂ)
7		nn0ex 12530	. . . . . . 7 ⊢ ℕ₀ ∈ V
8	7	mptex 7243	. . . . . 6 ⊢ (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) ∈ V
9	8	shftval4 15113	. . . . 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 11445	. . . . . 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 6911	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(1 + 𝑘)) = ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(𝑘 + 1)))
14		peano2nn0 12564	. . . . . . 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 6912	. . . . . . . 8 ⊢ (𝑖 = (𝑘 + 1) → (abs‘((𝐺‘𝑋)‘𝑖)) = (abs‘((𝐺‘𝑋)‘(𝑘 + 1))))
18	16, 17	oveq12d 7449	. . . . . . 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 7464	. . . . . . 7 ⊢ ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))) ∈ V
21	18, 19, 20	fvmpt 7016	. . . . . 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 26469	. . . . . . . 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 6911	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝐺‘𝑋)‘(𝑘 + 1))) = (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))
28	27	oveq2d 7447	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))))
29	22, 28	eqtrd 2775	. . . 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 2779	. . 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 12586	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℝ)
32		dvradcnv.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
33		ffvelcdm 7101	. . . . . . 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 14117	. . . . . . 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 11279	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) ∈ ℂ)
38	37	abscld 15472	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) ∈ ℝ)
39	31, 38	remulcld 11289	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))) ∈ ℝ)
40	30, 39	eqeltrd 2839	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘) ∈ ℝ)
41		oveq1 7438	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
42	41	fveq2d 6911	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐴‘(𝑛 + 1)) = (𝐴‘(𝑘 + 1)))
43	41, 42	oveq12d 7449	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑛 + 1) · (𝐴‘(𝑛 + 1))) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
44		oveq2 7439	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑋↑𝑛) = (𝑋↑𝑘))
45	43, 44	oveq12d 7449	. . . . 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 7464	. . . . 5 ⊢ (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) ∈ V
48	45, 46, 47	fvmpt 7016	. . . 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 12587	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℂ)
51	50, 34	mulcld 11279	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) ∈ ℂ)
52		expcl 14117	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑋↑𝑘) ∈ ℂ)
53	23, 52	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑋↑𝑘) ∈ ℂ)
54	51, 53	mulcld 11279	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) ∈ ℂ)
55	49, 54	eqeltrd 2839	. 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 6912	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (abs‘((𝐺‘𝑋)‘𝑖)) = (abs‘((𝐺‘𝑋)‘𝑘)))
60	58, 59	oveq12d 7449	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) = (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘))))
61	60	cbvmptv 5261	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) = (𝑘 ∈ ℕ₀ ↦ (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘))))
62	24, 32, 56, 23, 57, 61	radcnvlt1 26476	. . . . . . 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 15587	. . . . . 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 12622	. . . . . 6 ⊢ 0 ∈ ℤ
67		neg1z 12651	. . . . . 6 ⊢ -1 ∈ ℤ
68	8	isershft 15697	. . . . . 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 14041	. . . . 5 ⊢ seq(0 + -1)( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ∈ V
72		fvex 6920	. . . . 5 ⊢ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))))) ∈ V
73	71, 72	breldm 5922	. . . 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 12378	. . . . . . . 8 ⊢ -1 ∈ ℂ
77	76	addlidi 11447	. . . . . . 7 ⊢ (0 + -1) = -1
78		0le1 11784	. . . . . . . 8 ⊢ 0 ≤ 1
79		1re 11259	. . . . . . . . 9 ⊢ 1 ∈ ℝ
80		le0neg2 11770	. . . . . . . . 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 5169	. . . . . 6 ⊢ (0 + -1) ≤ 0
84	77, 67	eqeltri 2835	. . . . . . 7 ⊢ (0 + -1) ∈ ℤ
85	84	eluz1i 12884	. . . . . 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 12888	. . . . . . 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 12533	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℕ₀ → 𝑖 ∈ ℝ)
92	91	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℝ)
93	24, 32, 23	psergf 26470	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺‘𝑋):ℕ₀⟶ℂ)
94	93	ffvelcdmda 7104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → ((𝐺‘𝑋)‘𝑖) ∈ ℂ)
95	94	abscld 15472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (abs‘((𝐺‘𝑋)‘𝑖)) ∈ ℝ)
96	92, 95	remulcld 11289	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) ∈ ℝ)
97	96	recnd 11287	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) ∈ ℂ)
98	97	fmpttd 7135	. . . . . 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 12904	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (𝑘 + 1) ∈ (ℤ_≥‘((0 + -1) + 1)))
101	77	oveq1i 7441	. . . . . . . . . . 11 ⊢ ((0 + -1) + 1) = (-1 + 1)
102		1pneg1e0 12383	. . . . . . . . . . . 12 ⊢ (1 + -1) = 0
103	4, 76, 102	addcomli 11451	. . . . . . . . . . 11 ⊢ (-1 + 1) = 0
104	101, 103	eqtri 2763	. . . . . . . . . 10 ⊢ ((0 + -1) + 1) = 0
105	104	fveq2i 6910	. . . . . . . . 9 ⊢ (ℤ_≥‘((0 + -1) + 1)) = (ℤ_≥‘0)
106	1, 105	eqtr4i 2766	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘((0 + -1) + 1))
107	100, 106	eleqtrrdi 2850	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (𝑘 + 1) ∈ ℕ₀)
108	99, 107	eqeltrd 2839	. . . . . 6 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (1 + 𝑘) ∈ ℕ₀)
109		ffvelcdm 7101	. . . . . 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 2839	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(0 + -1))) → (((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)‘𝑘) ∈ ℂ)
112	75, 87, 111	iserex 15690	. . 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 11260	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0) → 1 ∈ ℝ)
115		neqne 2946	. . . . 5 ⊢ (¬ 𝑋 = 0 → 𝑋 ≠ 0)
116		absrpcl 15324	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℝ⁺)
117	23, 115, 116	syl2an 596	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 0) → (abs‘𝑋) ∈ ℝ⁺)
118	117	rprecred 13086	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = 0) → (1 / (abs‘𝑋)) ∈ ℝ)
119	114, 118	ifclda 4566	. 2 ⊢ (𝜑 → if(𝑋 = 0, 1, (1 / (abs‘𝑋))) ∈ ℝ)
120		oveq1 7438	. . . . 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 5160	. . . 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 7438	. . . . 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 5160	. . . 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 12920	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
125		nnnn0 12531	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
126	124, 125	sylbir 235	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘1) → 𝑘 ∈ ℕ₀)
127	15	nn0ge0d 12588	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 0 ≤ (𝑘 + 1))
128	37	absge0d 15480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 0 ≤ (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))
129	31, 38, 127, 128	mulge0d 11838	. . . . . . 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 7438	. . . . . . . . 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 14176	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (0↑𝑘) = 0)
136	132, 135	sylan9eqr 2797	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (𝑋↑𝑘) = 0)
137	136	oveq2d 7447	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · 0))
138	51	mul01d 11458	. . . . . . . . 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 2775	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) = 0)
142	141	abs00bd 15327	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) = 0)
143	39	recnd 11287	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))) ∈ ℂ)
144	143	mullidd 11277	. . . . . . 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 5180	. . . 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 2939	. . . . 5 ⊢ (𝑋 ≠ 0 ↔ ¬ 𝑋 = 0)
149	54	abscld 15472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ∈ ℝ)
150	50, 34, 53	mulassd 11282	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) = ((𝑘 + 1) · ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘))))
151	150	fveq2d 6911	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) = (abs‘((𝑘 + 1) · ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
152	34, 53	mulcld 11279	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)) ∈ ℂ)
153	50, 152	absmuld 15490	. . . . . . . . . 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 15458	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝑘 + 1)) = (𝑘 + 1))
155	154	oveq1d 7446	. . . . . . . . . 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 2779	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) = ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
157	149, 156	eqled 11362	. . . . . . . 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 13085	. . . . . . . . . . 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 13077	. . . . . . . . 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 11287	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) ∈ ℂ)
166	162, 163, 165	mul12d 11468	. . . . . . . 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 15491	. . . . . . . . . 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 12076	. . . . . . . . . . . 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 11512	. . . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑((𝑘 + 1) − 1)) = (𝑋↑𝑘))
178	15	nn0zd 12637	. . . . . . . . . . . . . . . 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 14193	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑((𝑘 + 1) − 1)) = ((𝑋↑(𝑘 + 1)) / 𝑋))
181	177, 180	eqtr3d 2777	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑𝑘) = ((𝑋↑(𝑘 + 1)) / 𝑋))
182	181	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)) = ((𝐴‘(𝑘 + 1)) · ((𝑋↑(𝑘 + 1)) / 𝑋)))
183	173, 182	eqtr4d 2778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) / 𝑋) = ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))
184	183	fveq2d 6911	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘(((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) / 𝑋)) = (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘))))
185	23	abscld 15472	. . . . . . . . . . . . 13 ⊢ (𝜑 → (abs‘𝑋) ∈ ℝ)
186	185	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℝ)
187	186	recnd 11287	. . . . . . . . . . 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 13080	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ≠ 0)
190	165, 187, 189	divrec2d 12045	. . . . . . . . . 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 2784	. . . . . . . . 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 7447	. . . . . . . 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 2775	. . . . . . 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 5176	. . . . . 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 4569	. . 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 6911	. . . 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 7447	. . . 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 5180	. 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 15851	1 ⊢ (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ )

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 {crab 3433 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 supcsup 9478 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 -cneg 11491 / cdiv 11918 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ℝ⁺crp 13032 seqcseq 14039 ↑cexp 14099 shift cshi 15102 abscabs 15270 ⇝ cli 15517

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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720

This theorem is referenced by: pserdvlem2 26487 dvradcnv2 44343

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