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Theorem dvradcnv 26365

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 12814	. 2 ⊢ ℕ₀ = (ℤ_≥‘0)
2		1nn0 12437	. . 3 ⊢ 1 ∈ ℕ₀
3	2	a1i 11	. 2 ⊢ (𝜑 → 1 ∈ ℕ₀)
4		ax-1cn 11105	. . . . 5 ⊢ 1 ∈ ℂ
5		nn0cn 12431	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
6	5	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℂ)
7		nn0ex 12427	. . . . . . 7 ⊢ ℕ₀ ∈ V
8	7	mptex 7180	. . . . . 6 ⊢ (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) ∈ V
9	8	shftval4 15021	. . . . 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 11339	. . . . . 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 6845	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(1 + 𝑘)) = ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))))‘(𝑘 + 1)))
14		peano2nn0 12461	. . . . . . 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 6846	. . . . . . . 8 ⊢ (𝑖 = (𝑘 + 1) → (abs‘((𝐺‘𝑋)‘𝑖)) = (abs‘((𝐺‘𝑋)‘(𝑘 + 1))))
18	16, 17	oveq12d 7388	. . . . . . 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 7403	. . . . . . 7 ⊢ ((𝑘 + 1) · (abs‘((𝐺‘𝑋)‘(𝑘 + 1)))) ∈ V
21	18, 19, 20	fvmpt 6951	. . . . . 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 26355	. . . . . . . 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 6845	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝐺‘𝑋)‘(𝑘 + 1))) = (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))
28	27	oveq2d 7386	. . . . 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 12483	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℝ)
32		dvradcnv.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
33		ffvelcdm 7036	. . . . . . 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 14023	. . . . . . 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 11173	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) ∈ ℂ)
38	37	abscld 15383	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) ∈ ℝ)
39	31, 38	remulcld 11183	. . 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 7377	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
42	41	fveq2d 6845	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐴‘(𝑛 + 1)) = (𝐴‘(𝑘 + 1)))
43	41, 42	oveq12d 7388	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑛 + 1) · (𝐴‘(𝑛 + 1))) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))
44		oveq2 7378	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑋↑𝑛) = (𝑋↑𝑘))
45	43, 44	oveq12d 7388	. . . . 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 7403	. . . . 5 ⊢ (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) ∈ V
48	45, 46, 47	fvmpt 6951	. . . 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 12484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℂ)
51	50, 34	mulcld 11173	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) ∈ ℂ)
52		expcl 14023	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑋↑𝑘) ∈ ℂ)
53	23, 52	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑋↑𝑘) ∈ ℂ)
54	51, 53	mulcld 11173	. . 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 6846	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (abs‘((𝐺‘𝑋)‘𝑖)) = (abs‘((𝐺‘𝑋)‘𝑘)))
60	58, 59	oveq12d 7388	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) = (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘))))
61	60	cbvmptv 5206	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) = (𝑘 ∈ ℕ₀ ↦ (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘))))
62	24, 32, 56, 23, 57, 61	radcnvlt1 26362	. . . . . . 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 15498	. . . . . 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 12519	. . . . . 6 ⊢ 0 ∈ ℤ
67		neg1z 12548	. . . . . 6 ⊢ -1 ∈ ℤ
68	8	isershft 15608	. . . . . 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 13947	. . . . 5 ⊢ seq(0 + -1)( + , ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))) shift -1)) ∈ V
72		fvex 6854	. . . . 5 ⊢ ( ⇝ ‘seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖)))))) ∈ V
73	71, 72	breldm 5863	. . . 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 12150	. . . . . . . 8 ⊢ -1 ∈ ℂ
77	76	addlidi 11341	. . . . . . 7 ⊢ (0 + -1) = -1
78		0le1 11680	. . . . . . . 8 ⊢ 0 ≤ 1
79		1re 11153	. . . . . . . . 9 ⊢ 1 ∈ ℝ
80		le0neg2 11666	. . . . . . . . 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 5123	. . . . . 6 ⊢ (0 + -1) ≤ 0
84	77, 67	eqeltri 2824	. . . . . . 7 ⊢ (0 + -1) ∈ ℤ
85	84	eluz1i 12780	. . . . . 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 12784	. . . . . . 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 12430	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℕ₀ → 𝑖 ∈ ℝ)
92	91	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℝ)
93	24, 32, 23	psergf 26356	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺‘𝑋):ℕ₀⟶ℂ)
94	93	ffvelcdmda 7039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → ((𝐺‘𝑋)‘𝑖) ∈ ℂ)
95	94	abscld 15383	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (abs‘((𝐺‘𝑋)‘𝑖)) ∈ ℝ)
96	92, 95	remulcld 11183	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) ∈ ℝ)
97	96	recnd 11181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑖 · (abs‘((𝐺‘𝑋)‘𝑖))) ∈ ℂ)
98	97	fmpttd 7070	. . . . . 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 12800	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ_≥‘(0 + -1)) → (𝑘 + 1) ∈ (ℤ_≥‘((0 + -1) + 1)))
101	77	oveq1i 7380	. . . . . . . . . . 11 ⊢ ((0 + -1) + 1) = (-1 + 1)
102		1pneg1e0 12279	. . . . . . . . . . . 12 ⊢ (1 + -1) = 0
103	4, 76, 102	addcomli 11345	. . . . . . . . . . 11 ⊢ (-1 + 1) = 0
104	101, 103	eqtri 2752	. . . . . . . . . 10 ⊢ ((0 + -1) + 1) = 0
105	104	fveq2i 6844	. . . . . . . . 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 7036	. . . . . 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 15601	. . 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 11154	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0) → 1 ∈ ℝ)
115		neqne 2933	. . . . 5 ⊢ (¬ 𝑋 = 0 → 𝑋 ≠ 0)
116		absrpcl 15232	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℝ⁺)
117	23, 115, 116	syl2an 596	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 0) → (abs‘𝑋) ∈ ℝ⁺)
118	117	rprecred 12985	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = 0) → (1 / (abs‘𝑋)) ∈ ℝ)
119	114, 118	ifclda 4520	. 2 ⊢ (𝜑 → if(𝑋 = 0, 1, (1 / (abs‘𝑋))) ∈ ℝ)
120		oveq1 7377	. . . . 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 5114	. . . 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 7377	. . . . 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 5114	. . . 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 12816	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
125		nnnn0 12428	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
126	124, 125	sylbir 235	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘1) → 𝑘 ∈ ℕ₀)
127	15	nn0ge0d 12485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 0 ≤ (𝑘 + 1))
128	37	absge0d 15391	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 0 ≤ (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))))
129	31, 38, 127, 128	mulge0d 11734	. . . . . . 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 7377	. . . . . . . . 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 14083	. . . . . . . . 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 7386	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · 0))
138	51	mul01d 11352	. . . . . . . . 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 15235	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑋 = 0) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) = 0)
143	39	recnd 11181	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) · (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))))) ∈ ℂ)
144	143	mullidd 11171	. . . . . . 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 5134	. . . 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 15383	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) ∈ ℝ)
150	50, 34, 53	mulassd 11176	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘)) = ((𝑘 + 1) · ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘))))
151	150	fveq2d 6845	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑋↑𝑘))) = (abs‘((𝑘 + 1) · ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)))))
152	34, 53	mulcld 11173	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘)) ∈ ℂ)
153	50, 152	absmuld 15401	. . . . . . . . . 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 15367	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝑘 + 1)) = (𝑘 + 1))
155	154	oveq1d 7385	. . . . . . . . . 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 11256	. . . . . . . 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 12984	. . . . . . . . . . 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 12976	. . . . . . . . 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 11181	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1)))) ∈ ℂ)
166	162, 163, 165	mul12d 11362	. . . . . . . 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 15402	. . . . . . . . . 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 11972	. . . . . . . . . . . 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 11406	. . . . . . . . . . . . . . . 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 7386	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (𝑋↑((𝑘 + 1) − 1)) = (𝑋↑𝑘))
178	15	nn0zd 12534	. . . . . . . . . . . . . . . 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 14100	. . . . . . . . . . . . . 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 7386	. . . . . . . . . . . 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 6845	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘(((𝐴‘(𝑘 + 1)) · (𝑋↑(𝑘 + 1))) / 𝑋)) = (abs‘((𝐴‘(𝑘 + 1)) · (𝑋↑𝑘))))
185	23	abscld 15383	. . . . . . . . . . . . 13 ⊢ (𝜑 → (abs‘𝑋) ∈ ℝ)
186	185	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ∈ ℝ)
187	186	recnd 11181	. . . . . . . . . . 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 12979	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑋 ≠ 0) → (abs‘𝑋) ≠ 0)
190	165, 187, 189	divrec2d 11941	. . . . . . . . . 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 7386	. . . . . . . 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 5130	. . . . . 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 4523	. . 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 6845	. . . 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 7386	. . . 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 5134	. 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 15762	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 3402 ifcif 4484 class class class wbr 5102 ↦ cmpt 5183 dom cdm 5631 ∘ ccom 5635 ⟶wf 6496 ‘cfv 6500 (class class class)co 7370 supcsup 9368 ℂcc 11045 ℝcr 11046 0cc0 11047 1c1 11048 + caddc 11050 · cmul 11052 ℝ^*cxr 11186 < clt 11187 ≤ cle 11188 − cmin 11384 -cneg 11385 / cdiv 11814 ℕcn 12165 ℕ₀cn0 12421 ℤcz 12508 ℤ_≥cuz 12772 ℝ⁺crp 12930 seqcseq 13945 ↑cexp 14005 shift cshi 15010 abscabs 15178 ⇝ cli 15428

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7692 ax-inf2 9573 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6453 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-om 7824 df-1st 7948 df-2nd 7949 df-frecs 8238 df-wrecs 8269 df-recs 8318 df-rdg 8356 df-1o 8412 df-er 8649 df-pm 8780 df-en 8897 df-dom 8898 df-sdom 8899 df-fin 8900 df-sup 9370 df-inf 9371 df-oi 9440 df-card 9871 df-pnf 11189 df-mnf 11190 df-xr 11191 df-ltxr 11192 df-le 11193 df-sub 11386 df-neg 11387 df-div 11815 df-nn 12166 df-2 12228 df-3 12229 df-n0 12422 df-z 12509 df-uz 12773 df-rp 12931 df-ico 13291 df-icc 13292 df-fz 13448 df-fzo 13595 df-fl 13733 df-seq 13946 df-exp 14006 df-hash 14275 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15631

This theorem is referenced by: pserdvlem2 26373 dvradcnv2 44331

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