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Theorem radcnvlem1 26161

Description: Lemma for radcnvlt1 26166, radcnvle 26168. If 𝑋 is a point closer to zero than 𝑌 and the power series converges at 𝑌, then it converges absolutely at 𝑋, even if the terms in the sequence are multiplied by 𝑛. (Contributed by Mario Carneiro, 31-Mar-2015.)

**Hypotheses**
Ref	Expression
pser.g	⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))
radcnv.a	⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
psergf.x	⊢ (𝜑 → 𝑋 ∈ ℂ)
radcnvlem2.y	⊢ (𝜑 → 𝑌 ∈ ℂ)
radcnvlem2.a	⊢ (𝜑 → (abs‘𝑋) < (abs‘𝑌))
radcnvlem2.c	⊢ (𝜑 → seq0( + , (𝐺‘𝑌)) ∈ dom ⇝ )
radcnvlem1.h	⊢ 𝐻 = (𝑚 ∈ ℕ₀ ↦ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))))

**Assertion**
Ref	Expression
radcnvlem1	⊢ (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ )

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

Proof of Theorem radcnvlem1 Dummy variables 𝑖 𝑘 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 12868	. . 3 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12574	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
3		1rp 12982	. . . 4 ⊢ 1 ∈ ℝ⁺
4	3	a1i 11	. . 3 ⊢ (𝜑 → 1 ∈ ℝ⁺)
5		radcnvlem2.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ℂ)
6		pser.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))
7	6	pserval2 26159	. . . 4 ⊢ ((𝑌 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((𝐺‘𝑌)‘𝑘) = ((𝐴‘𝑘) · (𝑌↑𝑘)))
8	5, 7	sylan 578	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐺‘𝑌)‘𝑘) = ((𝐴‘𝑘) · (𝑌↑𝑘)))
9		fvexd 6905	. . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ V)
10		radcnvlem2.c	. . . 4 ⊢ (𝜑 → seq0( + , (𝐺‘𝑌)) ∈ dom ⇝ )
11		radcnv.a	. . . . . 6 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
12	6, 11, 5	psergf 26160	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑌):ℕ₀⟶ℂ)
13	12	ffvelcdmda 7085	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐺‘𝑌)‘𝑘) ∈ ℂ)
14	1, 2, 9, 10, 13	serf0 15631	. . 3 ⊢ (𝜑 → (𝐺‘𝑌) ⇝ 0)
15	1, 2, 4, 8, 14	climi0 15460	. 2 ⊢ (𝜑 → ∃𝑗 ∈ ℕ₀ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)
16		simprl 767	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → 𝑗 ∈ ℕ₀)
17		nn0re 12485	. . . . . . 7 ⊢ (𝑖 ∈ ℕ₀ → 𝑖 ∈ ℝ)
18	17	adantl 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℝ)
19		psergf.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℂ)
20	19	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → 𝑋 ∈ ℂ)
21	20	abscld 15387	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → (abs‘𝑋) ∈ ℝ)
22	5	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → 𝑌 ∈ ℂ)
23	22	abscld 15387	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → (abs‘𝑌) ∈ ℝ)
24		0red 11221	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℝ)
25	19	abscld 15387	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘𝑋) ∈ ℝ)
26	5	abscld 15387	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘𝑌) ∈ ℝ)
27	19	absge0d 15395	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (abs‘𝑋))
28		radcnvlem2.a	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘𝑋) < (abs‘𝑌))
29	24, 25, 26, 27, 28	lelttrd 11376	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (abs‘𝑌))
30	29	gt0ne0d 11782	. . . . . . . . 9 ⊢ (𝜑 → (abs‘𝑌) ≠ 0)
31	30	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → (abs‘𝑌) ≠ 0)
32	21, 23, 31	redivcld 12046	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
33		reexpcl 14048	. . . . . . 7 ⊢ ((((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ ∧ 𝑖 ∈ ℕ₀) → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) ∈ ℝ)
34	32, 33	sylan 578	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑖 ∈ ℕ₀) → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) ∈ ℝ)
35	18, 34	remulcld 11248	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑖 ∈ ℕ₀) → (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)) ∈ ℝ)
36		eqid 2730	. . . . 5 ⊢ (𝑖 ∈ ℕ₀ ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖))) = (𝑖 ∈ ℕ₀ ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))
37	35, 36	fmptd 7114	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → (𝑖 ∈ ℕ₀ ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖))):ℕ₀⟶ℝ)
38	37	ffvelcdmda 7085	. . 3 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ ℕ₀) → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚) ∈ ℝ)
39		nn0re 12485	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℝ)
40	39	adantl 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℝ)
41	6, 11, 19	psergf 26160	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘𝑋):ℕ₀⟶ℂ)
42	41	ffvelcdmda 7085	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((𝐺‘𝑋)‘𝑚) ∈ ℂ)
43	42	abscld 15387	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (abs‘((𝐺‘𝑋)‘𝑚)) ∈ ℝ)
44	40, 43	remulcld 11248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))) ∈ ℝ)
45		radcnvlem1.h	. . . . . . 7 ⊢ 𝐻 = (𝑚 ∈ ℕ₀ ↦ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))))
46	44, 45	fmptd 7114	. . . . . 6 ⊢ (𝜑 → 𝐻:ℕ₀⟶ℝ)
47	46	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → 𝐻:ℕ₀⟶ℝ)
48	47	ffvelcdmda 7085	. . . 4 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ ℕ₀) → (𝐻‘𝑚) ∈ ℝ)
49	48	recnd 11246	. . 3 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ ℕ₀) → (𝐻‘𝑚) ∈ ℂ)
50	25, 26, 30	redivcld 12046	. . . . . 6 ⊢ (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
51	50	recnd 11246	. . . . 5 ⊢ (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℂ)
52		divge0 12087	. . . . . . . 8 ⊢ ((((abs‘𝑋) ∈ ℝ ∧ 0 ≤ (abs‘𝑋)) ∧ ((abs‘𝑌) ∈ ℝ ∧ 0 < (abs‘𝑌))) → 0 ≤ ((abs‘𝑋) / (abs‘𝑌)))
53	25, 27, 26, 29, 52	syl22anc 835	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((abs‘𝑋) / (abs‘𝑌)))
54	50, 53	absidd 15373	. . . . . 6 ⊢ (𝜑 → (abs‘((abs‘𝑋) / (abs‘𝑌))) = ((abs‘𝑋) / (abs‘𝑌)))
55	26	recnd 11246	. . . . . . . . 9 ⊢ (𝜑 → (abs‘𝑌) ∈ ℂ)
56	55	mulridd 11235	. . . . . . . 8 ⊢ (𝜑 → ((abs‘𝑌) · 1) = (abs‘𝑌))
57	28, 56	breqtrrd 5175	. . . . . . 7 ⊢ (𝜑 → (abs‘𝑋) < ((abs‘𝑌) · 1))
58		1red 11219	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
59		ltdivmul 12093	. . . . . . . 8 ⊢ (((abs‘𝑋) ∈ ℝ ∧ 1 ∈ ℝ ∧ ((abs‘𝑌) ∈ ℝ ∧ 0 < (abs‘𝑌))) → (((abs‘𝑋) / (abs‘𝑌)) < 1 ↔ (abs‘𝑋) < ((abs‘𝑌) · 1)))
60	25, 58, 26, 29, 59	syl112anc 1372	. . . . . . 7 ⊢ (𝜑 → (((abs‘𝑋) / (abs‘𝑌)) < 1 ↔ (abs‘𝑋) < ((abs‘𝑌) · 1)))
61	57, 60	mpbird 256	. . . . . 6 ⊢ (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) < 1)
62	54, 61	eqbrtrd 5169	. . . . 5 ⊢ (𝜑 → (abs‘((abs‘𝑋) / (abs‘𝑌))) < 1)
63	36	geomulcvg 15826	. . . . 5 ⊢ ((((abs‘𝑋) / (abs‘𝑌)) ∈ ℂ ∧ (abs‘((abs‘𝑋) / (abs‘𝑌))) < 1) → seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))) ∈ dom ⇝ )
64	51, 62, 63	syl2anc 582	. . . 4 ⊢ (𝜑 → seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))) ∈ dom ⇝ )
65	64	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → seq0( + , (𝑖 ∈ ℕ₀ ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))) ∈ dom ⇝ )
66		1red 11219	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → 1 ∈ ℝ)
67	41	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (𝐺‘𝑋):ℕ₀⟶ℂ)
68		eluznn0 12905	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ₀ ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 𝑚 ∈ ℕ₀)
69	16, 68	sylan 578	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 𝑚 ∈ ℕ₀)
70	67, 69	ffvelcdmd 7086	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((𝐺‘𝑋)‘𝑚) ∈ ℂ)
71	70	abscld 15387	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐺‘𝑋)‘𝑚)) ∈ ℝ)
72	32	adantr 479	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
73	72, 69	reexpcld 14132	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (((abs‘𝑋) / (abs‘𝑌))↑𝑚) ∈ ℝ)
74	69	nn0red 12537	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 𝑚 ∈ ℝ)
75	69	nn0ge0d 12539	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 0 ≤ 𝑚)
76	11	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 𝐴:ℕ₀⟶ℂ)
77	76, 69	ffvelcdmd 7086	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (𝐴‘𝑚) ∈ ℂ)
78	5	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 𝑌 ∈ ℂ)
79	78, 69	expcld 14115	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (𝑌↑𝑚) ∈ ℂ)
80	77, 79	mulcld 11238	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((𝐴‘𝑚) · (𝑌↑𝑚)) ∈ ℂ)
81	80	abscld 15387	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) ∈ ℝ)
82		1red 11219	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 1 ∈ ℝ)
83	19	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 𝑋 ∈ ℂ)
84	83	abscld 15387	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘𝑋) ∈ ℝ)
85	84, 69	reexpcld 14132	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((abs‘𝑋)↑𝑚) ∈ ℝ)
86	83	absge0d 15395	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 0 ≤ (abs‘𝑋))
87	84, 69, 86	expge0d 14133	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 0 ≤ ((abs‘𝑋)↑𝑚))
88		simprr 769	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)
89		fveq2 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (𝐴‘𝑘) = (𝐴‘𝑚))
90		oveq2 7419	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (𝑌↑𝑘) = (𝑌↑𝑚))
91	89, 90	oveq12d 7429	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → ((𝐴‘𝑘) · (𝑌↑𝑘)) = ((𝐴‘𝑚) · (𝑌↑𝑚)))
92	91	fveq2d 6894	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) = (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))))
93	92	breq1d 5157	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ((abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1 ↔ (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) < 1))
94	93	rspccva 3610	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1 ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) < 1)
95	88, 94	sylan 578	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) < 1)
96		1re 11218	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
97		ltle 11306	. . . . . . . . . . . 12 ⊢ (((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) < 1 → (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) ≤ 1))
98	81, 96, 97	sylancl 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) < 1 → (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) ≤ 1))
99	95, 98	mpd 15	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) ≤ 1)
100	81, 82, 85, 87, 99	lemul1ad 12157	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) · ((abs‘𝑋)↑𝑚)) ≤ (1 · ((abs‘𝑋)↑𝑚)))
101	83, 69	expcld 14115	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (𝑋↑𝑚) ∈ ℂ)
102	77, 101	mulcld 11238	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((𝐴‘𝑚) · (𝑋↑𝑚)) ∈ ℂ)
103	102, 79	absmuld 15405	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘(((𝐴‘𝑚) · (𝑋↑𝑚)) · (𝑌↑𝑚))) = ((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) · (abs‘(𝑌↑𝑚))))
104	80, 101	absmuld 15405	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘(((𝐴‘𝑚) · (𝑌↑𝑚)) · (𝑋↑𝑚))) = ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) · (abs‘(𝑋↑𝑚))))
105	77, 79, 101	mul32d 11428	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (((𝐴‘𝑚) · (𝑌↑𝑚)) · (𝑋↑𝑚)) = (((𝐴‘𝑚) · (𝑋↑𝑚)) · (𝑌↑𝑚)))
106	105	fveq2d 6894	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘(((𝐴‘𝑚) · (𝑌↑𝑚)) · (𝑋↑𝑚))) = (abs‘(((𝐴‘𝑚) · (𝑋↑𝑚)) · (𝑌↑𝑚))))
107	83, 69	absexpd 15403	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘(𝑋↑𝑚)) = ((abs‘𝑋)↑𝑚))
108	107	oveq2d 7427	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) · (abs‘(𝑋↑𝑚))) = ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) · ((abs‘𝑋)↑𝑚)))
109	104, 106, 108	3eqtr3d 2778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘(((𝐴‘𝑚) · (𝑋↑𝑚)) · (𝑌↑𝑚))) = ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) · ((abs‘𝑋)↑𝑚)))
110	78, 69	absexpd 15403	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘(𝑌↑𝑚)) = ((abs‘𝑌)↑𝑚))
111	110	oveq2d 7427	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) · (abs‘(𝑌↑𝑚))) = ((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) · ((abs‘𝑌)↑𝑚)))
112	103, 109, 111	3eqtr3d 2778	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) · ((abs‘𝑋)↑𝑚)) = ((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) · ((abs‘𝑌)↑𝑚)))
113	85	recnd 11246	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((abs‘𝑋)↑𝑚) ∈ ℂ)
114	113	mullidd 11236	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (1 · ((abs‘𝑋)↑𝑚)) = ((abs‘𝑋)↑𝑚))
115	100, 112, 114	3brtr3d 5178	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) · ((abs‘𝑌)↑𝑚)) ≤ ((abs‘𝑋)↑𝑚))
116	102	abscld 15387	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) ∈ ℝ)
117	23	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘𝑌) ∈ ℝ)
118	117, 69	reexpcld 14132	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((abs‘𝑌)↑𝑚) ∈ ℝ)
119		eluzelz 12836	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ_≥‘𝑗) → 𝑚 ∈ ℤ)
120	119	adantl 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 𝑚 ∈ ℤ)
121	29	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 0 < (abs‘𝑌))
122		expgt0 14065	. . . . . . . . . 10 ⊢ (((abs‘𝑌) ∈ ℝ ∧ 𝑚 ∈ ℤ ∧ 0 < (abs‘𝑌)) → 0 < ((abs‘𝑌)↑𝑚))
123	117, 120, 121, 122	syl3anc 1369	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 0 < ((abs‘𝑌)↑𝑚))
124		lemuldiv 12098	. . . . . . . . 9 ⊢ (((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) ∈ ℝ ∧ ((abs‘𝑋)↑𝑚) ∈ ℝ ∧ (((abs‘𝑌)↑𝑚) ∈ ℝ ∧ 0 < ((abs‘𝑌)↑𝑚))) → (((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) · ((abs‘𝑌)↑𝑚)) ≤ ((abs‘𝑋)↑𝑚) ↔ (abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) ≤ (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚))))
125	116, 85, 118, 123, 124	syl112anc 1372	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) · ((abs‘𝑌)↑𝑚)) ≤ ((abs‘𝑋)↑𝑚) ↔ (abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) ≤ (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚))))
126	115, 125	mpbid 231	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) ≤ (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚)))
127	6	pserval2 26159	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → ((𝐺‘𝑋)‘𝑚) = ((𝐴‘𝑚) · (𝑋↑𝑚)))
128	83, 69, 127	syl2anc 582	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((𝐺‘𝑋)‘𝑚) = ((𝐴‘𝑚) · (𝑋↑𝑚)))
129	128	fveq2d 6894	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐺‘𝑋)‘𝑚)) = (abs‘((𝐴‘𝑚) · (𝑋↑𝑚))))
130	21	recnd 11246	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → (abs‘𝑋) ∈ ℂ)
131	130	adantr 479	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘𝑋) ∈ ℂ)
132	23	recnd 11246	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → (abs‘𝑌) ∈ ℂ)
133	132	adantr 479	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘𝑌) ∈ ℂ)
134	30	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘𝑌) ≠ 0)
135	131, 133, 134, 69	expdivd 14129	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (((abs‘𝑋) / (abs‘𝑌))↑𝑚) = (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚)))
136	126, 129, 135	3brtr4d 5179	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐺‘𝑋)‘𝑚)) ≤ (((abs‘𝑋) / (abs‘𝑌))↑𝑚))
137	71, 73, 74, 75, 136	lemul2ad 12158	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))) ≤ (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
138	74, 71	remulcld 11248	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))) ∈ ℝ)
139	70	absge0d 15395	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 0 ≤ (abs‘((𝐺‘𝑋)‘𝑚)))
140	74, 71, 75, 139	mulge0d 11795	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 0 ≤ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))))
141	138, 140	absidd 15373	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘(𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))) = (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))))
142	74, 73	remulcld 11248	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ ℝ)
143	142	recnd 11246	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ ℂ)
144	143	mullidd 11236	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (1 · (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚))) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
145	137, 141, 144	3brtr4d 5179	. . . 4 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘(𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))) ≤ (1 · (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚))))
146		ovex 7444	. . . . . 6 ⊢ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))) ∈ V
147	45	fvmpt2 7008	. . . . . 6 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))) ∈ V) → (𝐻‘𝑚) = (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))))
148	69, 146, 147	sylancl 584	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (𝐻‘𝑚) = (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))))
149	148	fveq2d 6894	. . . 4 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘(𝐻‘𝑚)) = (abs‘(𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))))
150		id 22	. . . . . . . 8 ⊢ (𝑖 = 𝑚 → 𝑖 = 𝑚)
151		oveq2 7419	. . . . . . . 8 ⊢ (𝑖 = 𝑚 → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) = (((abs‘𝑋) / (abs‘𝑌))↑𝑚))
152	150, 151	oveq12d 7429	. . . . . . 7 ⊢ (𝑖 = 𝑚 → (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
153		ovex 7444	. . . . . . 7 ⊢ (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ V
154	152, 36, 153	fvmpt 6997	. . . . . 6 ⊢ (𝑚 ∈ ℕ₀ → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
155	69, 154	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
156	155	oveq2d 7427	. . . 4 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (1 · ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚)) = (1 · (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚))))
157	145, 149, 156	3brtr4d 5179	. . 3 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (abs‘(𝐻‘𝑚)) ≤ (1 · ((𝑖 ∈ ℕ₀ ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚)))
158	1, 16, 38, 49, 65, 66, 157	cvgcmpce 15768	. 2 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → seq0( + , 𝐻) ∈ dom ⇝ )
159	15, 158	rexlimddv 3159	1 ⊢ (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 Vcvv 3472 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 < clt 11252 ≤ cle 11253 / cdiv 11875 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ℝ⁺crp 12978 seqcseq 13970 ↑cexp 14031 abscabs 15185 ⇝ cli 15432

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-ico 13334 df-fz 13489 df-fzo 13632 df-fl 13761 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637

This theorem is referenced by: radcnvlem2 26162 radcnvlt1 26166

W3C validator