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Theorem radcnvlem1 26471
Description: Lemma for radcnvlt1 26476, radcnvle 26478. 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 ↦ ((𝐴𝑛) · (𝑥𝑛))))
radcnv.a (𝜑𝐴:ℕ0⟶ℂ)
psergf.x (𝜑𝑋 ∈ ℂ)
radcnvlem2.y (𝜑𝑌 ∈ ℂ)
radcnvlem2.a (𝜑 → (abs‘𝑋) < (abs‘𝑌))
radcnvlem2.c (𝜑 → seq0( + , (𝐺𝑌)) ∈ dom ⇝ )
radcnvlem1.h 𝐻 = (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺𝑋)‘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.
StepHypRef Expression
1 nn0uz 12918 . . 3 0 = (ℤ‘0)
2 0zd 12623 . . 3 (𝜑 → 0 ∈ ℤ)
3 1rp 13036 . . . 4 1 ∈ ℝ+
43a1i 11 . . 3 (𝜑 → 1 ∈ ℝ+)
5 radcnvlem2.y . . . 4 (𝜑𝑌 ∈ ℂ)
6 pser.g . . . . 5 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))
76pserval2 26469 . . . 4 ((𝑌 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) = ((𝐴𝑘) · (𝑌𝑘)))
85, 7sylan 580 . . 3 ((𝜑𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) = ((𝐴𝑘) · (𝑌𝑘)))
9 fvexd 6922 . . . 4 (𝜑 → (𝐺𝑌) ∈ V)
10 radcnvlem2.c . . . 4 (𝜑 → seq0( + , (𝐺𝑌)) ∈ dom ⇝ )
11 radcnv.a . . . . . 6 (𝜑𝐴:ℕ0⟶ℂ)
126, 11, 5psergf 26470 . . . . 5 (𝜑 → (𝐺𝑌):ℕ0⟶ℂ)
1312ffvelcdmda 7104 . . . 4 ((𝜑𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) ∈ ℂ)
141, 2, 9, 10, 13serf0 15714 . . 3 (𝜑 → (𝐺𝑌) ⇝ 0)
151, 2, 4, 8, 14climi0 15545 . 2 (𝜑 → ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)
16 simprl 771 . . 3 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝑗 ∈ ℕ0)
17 nn0re 12533 . . . . . . 7 (𝑖 ∈ ℕ0𝑖 ∈ ℝ)
1817adantl 481 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℝ)
19 psergf.x . . . . . . . . . 10 (𝜑𝑋 ∈ ℂ)
2019adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝑋 ∈ ℂ)
2120abscld 15472 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑋) ∈ ℝ)
225adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝑌 ∈ ℂ)
2322abscld 15472 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑌) ∈ ℝ)
24 0red 11262 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℝ)
2519abscld 15472 . . . . . . . . . . 11 (𝜑 → (abs‘𝑋) ∈ ℝ)
265abscld 15472 . . . . . . . . . . 11 (𝜑 → (abs‘𝑌) ∈ ℝ)
2719absge0d 15480 . . . . . . . . . . 11 (𝜑 → 0 ≤ (abs‘𝑋))
28 radcnvlem2.a . . . . . . . . . . 11 (𝜑 → (abs‘𝑋) < (abs‘𝑌))
2924, 25, 26, 27, 28lelttrd 11417 . . . . . . . . . 10 (𝜑 → 0 < (abs‘𝑌))
3029gt0ne0d 11825 . . . . . . . . 9 (𝜑 → (abs‘𝑌) ≠ 0)
3130adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑌) ≠ 0)
3221, 23, 31redivcld 12093 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
33 reexpcl 14116 . . . . . . 7 ((((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ ∧ 𝑖 ∈ ℕ0) → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) ∈ ℝ)
3432, 33sylan 580 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑖 ∈ ℕ0) → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) ∈ ℝ)
3518, 34remulcld 11289 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑖 ∈ ℕ0) → (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)) ∈ ℝ)
36 eqid 2735 . . . . 5 (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖))) = (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))
3735, 36fmptd 7134 . . . 4 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖))):ℕ0⟶ℝ)
3837ffvelcdmda 7104 . . 3 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → ((𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚) ∈ ℝ)
39 nn0re 12533 . . . . . . . . 9 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
4039adantl 481 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ0) → 𝑚 ∈ ℝ)
416, 11, 19psergf 26470 . . . . . . . . . 10 (𝜑 → (𝐺𝑋):ℕ0⟶ℂ)
4241ffvelcdmda 7104 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ0) → ((𝐺𝑋)‘𝑚) ∈ ℂ)
4342abscld 15472 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ0) → (abs‘((𝐺𝑋)‘𝑚)) ∈ ℝ)
4440, 43remulcld 11289 . . . . . . 7 ((𝜑𝑚 ∈ ℕ0) → (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ ℝ)
45 radcnvlem1.h . . . . . . 7 𝐻 = (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
4644, 45fmptd 7134 . . . . . 6 (𝜑𝐻:ℕ0⟶ℝ)
4746adantr 480 . . . . 5 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝐻:ℕ0⟶ℝ)
4847ffvelcdmda 7104 . . . 4 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → (𝐻𝑚) ∈ ℝ)
4948recnd 11287 . . 3 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → (𝐻𝑚) ∈ ℂ)
5025, 26, 30redivcld 12093 . . . . . 6 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
5150recnd 11287 . . . . 5 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℂ)
52 divge0 12135 . . . . . . . 8 ((((abs‘𝑋) ∈ ℝ ∧ 0 ≤ (abs‘𝑋)) ∧ ((abs‘𝑌) ∈ ℝ ∧ 0 < (abs‘𝑌))) → 0 ≤ ((abs‘𝑋) / (abs‘𝑌)))
5325, 27, 26, 29, 52syl22anc 839 . . . . . . 7 (𝜑 → 0 ≤ ((abs‘𝑋) / (abs‘𝑌)))
5450, 53absidd 15458 . . . . . 6 (𝜑 → (abs‘((abs‘𝑋) / (abs‘𝑌))) = ((abs‘𝑋) / (abs‘𝑌)))
5526recnd 11287 . . . . . . . . 9 (𝜑 → (abs‘𝑌) ∈ ℂ)
5655mulridd 11276 . . . . . . . 8 (𝜑 → ((abs‘𝑌) · 1) = (abs‘𝑌))
5728, 56breqtrrd 5176 . . . . . . 7 (𝜑 → (abs‘𝑋) < ((abs‘𝑌) · 1))
58 1red 11260 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
59 ltdivmul 12141 . . . . . . . 8 (((abs‘𝑋) ∈ ℝ ∧ 1 ∈ ℝ ∧ ((abs‘𝑌) ∈ ℝ ∧ 0 < (abs‘𝑌))) → (((abs‘𝑋) / (abs‘𝑌)) < 1 ↔ (abs‘𝑋) < ((abs‘𝑌) · 1)))
6025, 58, 26, 29, 59syl112anc 1373 . . . . . . 7 (𝜑 → (((abs‘𝑋) / (abs‘𝑌)) < 1 ↔ (abs‘𝑋) < ((abs‘𝑌) · 1)))
6157, 60mpbird 257 . . . . . 6 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) < 1)
6254, 61eqbrtrd 5170 . . . . 5 (𝜑 → (abs‘((abs‘𝑋) / (abs‘𝑌))) < 1)
6336geomulcvg 15909 . . . . 5 ((((abs‘𝑋) / (abs‘𝑌)) ∈ ℂ ∧ (abs‘((abs‘𝑋) / (abs‘𝑌))) < 1) → seq0( + , (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))) ∈ dom ⇝ )
6451, 62, 63syl2anc 584 . . . 4 (𝜑 → seq0( + , (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))) ∈ dom ⇝ )
6564adantr 480 . . 3 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → seq0( + , (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))) ∈ dom ⇝ )
66 1red 11260 . . 3 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 1 ∈ ℝ)
6741ad2antrr 726 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝐺𝑋):ℕ0⟶ℂ)
68 eluznn0 12957 . . . . . . . . 9 ((𝑗 ∈ ℕ0𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℕ0)
6916, 68sylan 580 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℕ0)
7067, 69ffvelcdmd 7105 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐺𝑋)‘𝑚) ∈ ℂ)
7170abscld 15472 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐺𝑋)‘𝑚)) ∈ ℝ)
7232adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
7372, 69reexpcld 14200 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (((abs‘𝑋) / (abs‘𝑌))↑𝑚) ∈ ℝ)
7469nn0red 12586 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℝ)
7569nn0ge0d 12588 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ 𝑚)
7611ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝐴:ℕ0⟶ℂ)
7776, 69ffvelcdmd 7105 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝐴𝑚) ∈ ℂ)
785ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑌 ∈ ℂ)
7978, 69expcld 14183 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑌𝑚) ∈ ℂ)
8077, 79mulcld 11279 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐴𝑚) · (𝑌𝑚)) ∈ ℂ)
8180abscld 15472 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑌𝑚))) ∈ ℝ)
82 1red 11260 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 1 ∈ ℝ)
8319ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑋 ∈ ℂ)
8483abscld 15472 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑋) ∈ ℝ)
8584, 69reexpcld 14200 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑋)↑𝑚) ∈ ℝ)
8683absge0d 15480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (abs‘𝑋))
8784, 69, 86expge0d 14201 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ ((abs‘𝑋)↑𝑚))
88 simprr 773 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)
89 fveq2 6907 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝐴𝑘) = (𝐴𝑚))
90 oveq2 7439 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑌𝑘) = (𝑌𝑚))
9189, 90oveq12d 7449 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((𝐴𝑘) · (𝑌𝑘)) = ((𝐴𝑚) · (𝑌𝑚)))
9291fveq2d 6911 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → (abs‘((𝐴𝑘) · (𝑌𝑘))) = (abs‘((𝐴𝑚) · (𝑌𝑚))))
9392breq1d 5158 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → ((abs‘((𝐴𝑘) · (𝑌𝑘))) < 1 ↔ (abs‘((𝐴𝑚) · (𝑌𝑚))) < 1))
9493rspccva 3621 . . . . . . . . . . . 12 ((∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1 ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑌𝑚))) < 1)
9588, 94sylan 580 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑌𝑚))) < 1)
96 1re 11259 . . . . . . . . . . . 12 1 ∈ ℝ
97 ltle 11347 . . . . . . . . . . . 12 (((abs‘((𝐴𝑚) · (𝑌𝑚))) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘((𝐴𝑚) · (𝑌𝑚))) < 1 → (abs‘((𝐴𝑚) · (𝑌𝑚))) ≤ 1))
9881, 96, 97sylancl 586 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑌𝑚))) < 1 → (abs‘((𝐴𝑚) · (𝑌𝑚))) ≤ 1))
9995, 98mpd 15 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑌𝑚))) ≤ 1)
10081, 82, 85, 87, 99lemul1ad 12205 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑌𝑚))) · ((abs‘𝑋)↑𝑚)) ≤ (1 · ((abs‘𝑋)↑𝑚)))
10183, 69expcld 14183 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑋𝑚) ∈ ℂ)
10277, 101mulcld 11279 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐴𝑚) · (𝑋𝑚)) ∈ ℂ)
103102, 79absmuld 15490 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))) = ((abs‘((𝐴𝑚) · (𝑋𝑚))) · (abs‘(𝑌𝑚))))
10480, 101absmuld 15490 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚))) = ((abs‘((𝐴𝑚) · (𝑌𝑚))) · (abs‘(𝑋𝑚))))
10577, 79, 101mul32d 11469 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚)) = (((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚)))
106105fveq2d 6911 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚))) = (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))))
10783, 69absexpd 15488 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑋𝑚)) = ((abs‘𝑋)↑𝑚))
108107oveq2d 7447 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑌𝑚))) · (abs‘(𝑋𝑚))) = ((abs‘((𝐴𝑚) · (𝑌𝑚))) · ((abs‘𝑋)↑𝑚)))
109104, 106, 1083eqtr3d 2783 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))) = ((abs‘((𝐴𝑚) · (𝑌𝑚))) · ((abs‘𝑋)↑𝑚)))
11078, 69absexpd 15488 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑌𝑚)) = ((abs‘𝑌)↑𝑚))
111110oveq2d 7447 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑋𝑚))) · (abs‘(𝑌𝑚))) = ((abs‘((𝐴𝑚) · (𝑋𝑚))) · ((abs‘𝑌)↑𝑚)))
112103, 109, 1113eqtr3d 2783 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑌𝑚))) · ((abs‘𝑋)↑𝑚)) = ((abs‘((𝐴𝑚) · (𝑋𝑚))) · ((abs‘𝑌)↑𝑚)))
11385recnd 11287 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑋)↑𝑚) ∈ ℂ)
114113mullidd 11277 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (1 · ((abs‘𝑋)↑𝑚)) = ((abs‘𝑋)↑𝑚))
115100, 112, 1143brtr3d 5179 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑋𝑚))) · ((abs‘𝑌)↑𝑚)) ≤ ((abs‘𝑋)↑𝑚))
116102abscld 15472 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑋𝑚))) ∈ ℝ)
11723adantr 480 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑌) ∈ ℝ)
118117, 69reexpcld 14200 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑌)↑𝑚) ∈ ℝ)
119 eluzelz 12886 . . . . . . . . . . 11 (𝑚 ∈ (ℤ𝑗) → 𝑚 ∈ ℤ)
120119adantl 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℤ)
12129ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 < (abs‘𝑌))
122 expgt0 14133 . . . . . . . . . 10 (((abs‘𝑌) ∈ ℝ ∧ 𝑚 ∈ ℤ ∧ 0 < (abs‘𝑌)) → 0 < ((abs‘𝑌)↑𝑚))
123117, 120, 121, 122syl3anc 1370 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 < ((abs‘𝑌)↑𝑚))
124 lemuldiv 12146 . . . . . . . . 9 (((abs‘((𝐴𝑚) · (𝑋𝑚))) ∈ ℝ ∧ ((abs‘𝑋)↑𝑚) ∈ ℝ ∧ (((abs‘𝑌)↑𝑚) ∈ ℝ ∧ 0 < ((abs‘𝑌)↑𝑚))) → (((abs‘((𝐴𝑚) · (𝑋𝑚))) · ((abs‘𝑌)↑𝑚)) ≤ ((abs‘𝑋)↑𝑚) ↔ (abs‘((𝐴𝑚) · (𝑋𝑚))) ≤ (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚))))
125116, 85, 118, 123, 124syl112anc 1373 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (((abs‘((𝐴𝑚) · (𝑋𝑚))) · ((abs‘𝑌)↑𝑚)) ≤ ((abs‘𝑋)↑𝑚) ↔ (abs‘((𝐴𝑚) · (𝑋𝑚))) ≤ (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚))))
126115, 125mpbid 232 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑋𝑚))) ≤ (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚)))
1276pserval2 26469 . . . . . . . . 9 ((𝑋 ∈ ℂ ∧ 𝑚 ∈ ℕ0) → ((𝐺𝑋)‘𝑚) = ((𝐴𝑚) · (𝑋𝑚)))
12883, 69, 127syl2anc 584 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐺𝑋)‘𝑚) = ((𝐴𝑚) · (𝑋𝑚)))
129128fveq2d 6911 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐺𝑋)‘𝑚)) = (abs‘((𝐴𝑚) · (𝑋𝑚))))
13021recnd 11287 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑋) ∈ ℂ)
131130adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑋) ∈ ℂ)
13223recnd 11287 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑌) ∈ ℂ)
133132adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑌) ∈ ℂ)
13430ad2antrr 726 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑌) ≠ 0)
135131, 133, 134, 69expdivd 14197 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (((abs‘𝑋) / (abs‘𝑌))↑𝑚) = (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚)))
136126, 129, 1353brtr4d 5180 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐺𝑋)‘𝑚)) ≤ (((abs‘𝑋) / (abs‘𝑌))↑𝑚))
13771, 73, 74, 75, 136lemul2ad 12206 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ≤ (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
13874, 71remulcld 11289 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ ℝ)
13970absge0d 15480 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (abs‘((𝐺𝑋)‘𝑚)))
14074, 71, 75, 139mulge0d 11838 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
141138, 140absidd 15458 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑚 · (abs‘((𝐺𝑋)‘𝑚)))) = (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
14274, 73remulcld 11289 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ ℝ)
143142recnd 11287 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ ℂ)
144143mullidd 11277 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (1 · (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚))) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
145137, 141, 1443brtr4d 5180 . . . 4 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑚 · (abs‘((𝐺𝑋)‘𝑚)))) ≤ (1 · (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚))))
146 ovex 7464 . . . . . 6 (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ V
14745fvmpt2 7027 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ V) → (𝐻𝑚) = (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
14869, 146, 147sylancl 586 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝐻𝑚) = (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
149148fveq2d 6911 . . . 4 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝐻𝑚)) = (abs‘(𝑚 · (abs‘((𝐺𝑋)‘𝑚)))))
150 id 22 . . . . . . . 8 (𝑖 = 𝑚𝑖 = 𝑚)
151 oveq2 7439 . . . . . . . 8 (𝑖 = 𝑚 → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) = (((abs‘𝑋) / (abs‘𝑌))↑𝑚))
152150, 151oveq12d 7449 . . . . . . 7 (𝑖 = 𝑚 → (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
153 ovex 7464 . . . . . . 7 (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ V
154152, 36, 153fvmpt 7016 . . . . . 6 (𝑚 ∈ ℕ0 → ((𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
15569, 154syl 17 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
156155oveq2d 7447 . . . 4 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (1 · ((𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚)) = (1 · (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚))))
157145, 149, 1563brtr4d 5180 . . 3 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝐻𝑚)) ≤ (1 · ((𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚)))
1581, 16, 38, 49, 65, 66, 157cvgcmpce 15851 . 2 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → seq0( + , 𝐻) ∈ dom ⇝ )
15915, 158rexlimddv 3159 1 (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  Vcvv 3478   class class class wbr 5148  cmpt 5231  dom cdm 5689  wf 6559  cfv 6563  (class class class)co 7431  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158   < clt 11293  cle 11294   / cdiv 11918  0cn0 12524  cz 12611  cuz 12876  +crp 13032  seqcseq 14039  cexp 14099  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-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  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:  radcnvlem2  26472  radcnvlt1  26476
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