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Theorem radcnvlem1 26456
Description: Lemma for radcnvlt1 26461, radcnvle 26463. 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 12920 . . 3 0 = (ℤ‘0)
2 0zd 12625 . . 3 (𝜑 → 0 ∈ ℤ)
3 1rp 13038 . . . 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 26454 . . . 4 ((𝑌 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) = ((𝐴𝑘) · (𝑌𝑘)))
85, 7sylan 580 . . 3 ((𝜑𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) = ((𝐴𝑘) · (𝑌𝑘)))
9 fvexd 6921 . . . 4 (𝜑 → (𝐺𝑌) ∈ V)
10 radcnvlem2.c . . . 4 (𝜑 → seq0( + , (𝐺𝑌)) ∈ dom ⇝ )
11 radcnv.a . . . . . 6 (𝜑𝐴:ℕ0⟶ℂ)
126, 11, 5psergf 26455 . . . . 5 (𝜑 → (𝐺𝑌):ℕ0⟶ℂ)
1312ffvelcdmda 7104 . . . 4 ((𝜑𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) ∈ ℂ)
141, 2, 9, 10, 13serf0 15717 . . 3 (𝜑 → (𝐺𝑌) ⇝ 0)
151, 2, 4, 8, 14climi0 15548 . 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 12535 . . . . . . 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 15475 . . . . . . . 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 15475 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑌) ∈ ℝ)
24 0red 11264 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℝ)
2519abscld 15475 . . . . . . . . . . 11 (𝜑 → (abs‘𝑋) ∈ ℝ)
265abscld 15475 . . . . . . . . . . 11 (𝜑 → (abs‘𝑌) ∈ ℝ)
2719absge0d 15483 . . . . . . . . . . 11 (𝜑 → 0 ≤ (abs‘𝑋))
28 radcnvlem2.a . . . . . . . . . . 11 (𝜑 → (abs‘𝑋) < (abs‘𝑌))
2924, 25, 26, 27, 28lelttrd 11419 . . . . . . . . . 10 (𝜑 → 0 < (abs‘𝑌))
3029gt0ne0d 11827 . . . . . . . . 9 (𝜑 → (abs‘𝑌) ≠ 0)
3130adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑌) ≠ 0)
3221, 23, 31redivcld 12095 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
33 reexpcl 14119 . . . . . . 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 11291 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑖 ∈ ℕ0) → (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)) ∈ ℝ)
36 eqid 2737 . . . . 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 12535 . . . . . . . . 9 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
4039adantl 481 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ0) → 𝑚 ∈ ℝ)
416, 11, 19psergf 26455 . . . . . . . . . 10 (𝜑 → (𝐺𝑋):ℕ0⟶ℂ)
4241ffvelcdmda 7104 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ0) → ((𝐺𝑋)‘𝑚) ∈ ℂ)
4342abscld 15475 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ0) → (abs‘((𝐺𝑋)‘𝑚)) ∈ ℝ)
4440, 43remulcld 11291 . . . . . . 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 11289 . . 3 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → (𝐻𝑚) ∈ ℂ)
5025, 26, 30redivcld 12095 . . . . . 6 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
5150recnd 11289 . . . . 5 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℂ)
52 divge0 12137 . . . . . . . 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 15461 . . . . . 6 (𝜑 → (abs‘((abs‘𝑋) / (abs‘𝑌))) = ((abs‘𝑋) / (abs‘𝑌)))
5526recnd 11289 . . . . . . . . 9 (𝜑 → (abs‘𝑌) ∈ ℂ)
5655mulridd 11278 . . . . . . . 8 (𝜑 → ((abs‘𝑌) · 1) = (abs‘𝑌))
5728, 56breqtrrd 5171 . . . . . . 7 (𝜑 → (abs‘𝑋) < ((abs‘𝑌) · 1))
58 1red 11262 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
59 ltdivmul 12143 . . . . . . . 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 1376 . . . . . . 7 (𝜑 → (((abs‘𝑋) / (abs‘𝑌)) < 1 ↔ (abs‘𝑋) < ((abs‘𝑌) · 1)))
6157, 60mpbird 257 . . . . . 6 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) < 1)
6254, 61eqbrtrd 5165 . . . . 5 (𝜑 → (abs‘((abs‘𝑋) / (abs‘𝑌))) < 1)
6336geomulcvg 15912 . . . . 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 11262 . . 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 12959 . . . . . . . . 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 15475 . . . . . 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 14203 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (((abs‘𝑋) / (abs‘𝑌))↑𝑚) ∈ ℝ)
7469nn0red 12588 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℝ)
7569nn0ge0d 12590 . . . . . 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 14186 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑌𝑚) ∈ ℂ)
8077, 79mulcld 11281 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐴𝑚) · (𝑌𝑚)) ∈ ℂ)
8180abscld 15475 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑌𝑚))) ∈ ℝ)
82 1red 11262 . . . . . . . . . 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 15475 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑋) ∈ ℝ)
8584, 69reexpcld 14203 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑋)↑𝑚) ∈ ℝ)
8683absge0d 15483 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (abs‘𝑋))
8784, 69, 86expge0d 14204 . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝐴𝑘) = (𝐴𝑚))
90 oveq2 7439 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑌𝑘) = (𝑌𝑚))
9189, 90oveq12d 7449 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((𝐴𝑘) · (𝑌𝑘)) = ((𝐴𝑚) · (𝑌𝑚)))
9291fveq2d 6910 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → (abs‘((𝐴𝑘) · (𝑌𝑘))) = (abs‘((𝐴𝑚) · (𝑌𝑚))))
9392breq1d 5153 . . . . . . . . . . . . 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 11261 . . . . . . . . . . . 12 1 ∈ ℝ
97 ltle 11349 . . . . . . . . . . . 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 12207 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑌𝑚))) · ((abs‘𝑋)↑𝑚)) ≤ (1 · ((abs‘𝑋)↑𝑚)))
10183, 69expcld 14186 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑋𝑚) ∈ ℂ)
10277, 101mulcld 11281 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐴𝑚) · (𝑋𝑚)) ∈ ℂ)
103102, 79absmuld 15493 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))) = ((abs‘((𝐴𝑚) · (𝑋𝑚))) · (abs‘(𝑌𝑚))))
10480, 101absmuld 15493 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚))) = ((abs‘((𝐴𝑚) · (𝑌𝑚))) · (abs‘(𝑋𝑚))))
10577, 79, 101mul32d 11471 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚)) = (((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚)))
106105fveq2d 6910 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚))) = (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))))
10783, 69absexpd 15491 . . . . . . . . . . . 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 2785 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))) = ((abs‘((𝐴𝑚) · (𝑌𝑚))) · ((abs‘𝑋)↑𝑚)))
11078, 69absexpd 15491 . . . . . . . . . . 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 2785 . . . . . . . . 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 11289 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑋)↑𝑚) ∈ ℂ)
114113mullidd 11279 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (1 · ((abs‘𝑋)↑𝑚)) = ((abs‘𝑋)↑𝑚))
115100, 112, 1143brtr3d 5174 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑋𝑚))) · ((abs‘𝑌)↑𝑚)) ≤ ((abs‘𝑋)↑𝑚))
116102abscld 15475 . . . . . . . . 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 14203 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑌)↑𝑚) ∈ ℝ)
119 eluzelz 12888 . . . . . . . . . . 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 14136 . . . . . . . . . 10 (((abs‘𝑌) ∈ ℝ ∧ 𝑚 ∈ ℤ ∧ 0 < (abs‘𝑌)) → 0 < ((abs‘𝑌)↑𝑚))
123117, 120, 121, 122syl3anc 1373 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 < ((abs‘𝑌)↑𝑚))
124 lemuldiv 12148 . . . . . . . . 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 1376 . . . . . . . 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 26454 . . . . . . . . 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 6910 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐺𝑋)‘𝑚)) = (abs‘((𝐴𝑚) · (𝑋𝑚))))
13021recnd 11289 . . . . . . . . 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 11289 . . . . . . . . 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 14200 . . . . . . 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 5175 . . . . . 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 12208 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ≤ (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
13874, 71remulcld 11291 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ ℝ)
13970absge0d 15483 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (abs‘((𝐺𝑋)‘𝑚)))
14074, 71, 75, 139mulge0d 11840 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
141138, 140absidd 15461 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑚 · (abs‘((𝐺𝑋)‘𝑚)))) = (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
14274, 73remulcld 11291 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ ℝ)
143142recnd 11289 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ ℂ)
144143mullidd 11279 . . . . 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 5175 . . . 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 6910 . . . 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 5175 . . 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 15854 . 2 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → seq0( + , 𝐻) ∈ dom ⇝ )
15915, 158rexlimddv 3161 1 (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480   class class class wbr 5143  cmpt 5225  dom cdm 5685  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295  cle 11296   / cdiv 11920  0cn0 12526  cz 12613  cuz 12878  +crp 13034  seqcseq 14042  cexp 14102  abscabs 15273  cli 15520
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-ico 13393  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-sum 15723
This theorem is referenced by:  radcnvlem2  26457  radcnvlt1  26461
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