MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  radcnvlem1 Structured version   Visualization version   GIF version

Theorem radcnvlem1 24366
Description: Lemma for radcnvlt1 24371, radcnvle 24373. 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 11915 . . 3 0 = (ℤ‘0)
2 0zd 11581 . . 3 (𝜑 → 0 ∈ ℤ)
3 1rp 12029 . . . 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 24364 . . . 4 ((𝑌 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) = ((𝐴𝑘) · (𝑌𝑘)))
85, 7sylan 489 . . 3 ((𝜑𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) = ((𝐴𝑘) · (𝑌𝑘)))
9 fvexd 6364 . . . 4 (𝜑 → (𝐺𝑌) ∈ V)
10 radcnvlem2.c . . . 4 (𝜑 → seq0( + , (𝐺𝑌)) ∈ dom ⇝ )
11 radcnv.a . . . . . 6 (𝜑𝐴:ℕ0⟶ℂ)
126, 11, 5psergf 24365 . . . . 5 (𝜑 → (𝐺𝑌):ℕ0⟶ℂ)
1312ffvelrnda 6522 . . . 4 ((𝜑𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) ∈ ℂ)
141, 2, 9, 10, 13serf0 14610 . . 3 (𝜑 → (𝐺𝑌) ⇝ 0)
151, 2, 4, 8, 14climi0 14442 . 2 (𝜑 → ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)
16 simprl 811 . . 3 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝑗 ∈ ℕ0)
17 nn0re 11493 . . . . . . 7 (𝑖 ∈ ℕ0𝑖 ∈ ℝ)
1817adantl 473 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℝ)
19 psergf.x . . . . . . . . . 10 (𝜑𝑋 ∈ ℂ)
2019adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝑋 ∈ ℂ)
2120abscld 14374 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑋) ∈ ℝ)
225adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝑌 ∈ ℂ)
2322abscld 14374 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑌) ∈ ℝ)
24 0red 10233 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℝ)
2519abscld 14374 . . . . . . . . . . 11 (𝜑 → (abs‘𝑋) ∈ ℝ)
265abscld 14374 . . . . . . . . . . 11 (𝜑 → (abs‘𝑌) ∈ ℝ)
2719absge0d 14382 . . . . . . . . . . 11 (𝜑 → 0 ≤ (abs‘𝑋))
28 radcnvlem2.a . . . . . . . . . . 11 (𝜑 → (abs‘𝑋) < (abs‘𝑌))
2924, 25, 26, 27, 28lelttrd 10387 . . . . . . . . . 10 (𝜑 → 0 < (abs‘𝑌))
3029gt0ne0d 10784 . . . . . . . . 9 (𝜑 → (abs‘𝑌) ≠ 0)
3130adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑌) ≠ 0)
3221, 23, 31redivcld 11045 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
33 reexpcl 13071 . . . . . . 7 ((((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ ∧ 𝑖 ∈ ℕ0) → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) ∈ ℝ)
3432, 33sylan 489 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑖 ∈ ℕ0) → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) ∈ ℝ)
3518, 34remulcld 10262 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑖 ∈ ℕ0) → (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)) ∈ ℝ)
36 eqid 2760 . . . . 5 (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖))) = (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))
3735, 36fmptd 6548 . . . 4 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖))):ℕ0⟶ℝ)
3837ffvelrnda 6522 . . 3 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → ((𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚) ∈ ℝ)
39 nn0re 11493 . . . . . . . . 9 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
4039adantl 473 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ0) → 𝑚 ∈ ℝ)
416, 11, 19psergf 24365 . . . . . . . . . 10 (𝜑 → (𝐺𝑋):ℕ0⟶ℂ)
4241ffvelrnda 6522 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ0) → ((𝐺𝑋)‘𝑚) ∈ ℂ)
4342abscld 14374 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ0) → (abs‘((𝐺𝑋)‘𝑚)) ∈ ℝ)
4440, 43remulcld 10262 . . . . . . 7 ((𝜑𝑚 ∈ ℕ0) → (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ ℝ)
45 radcnvlem1.h . . . . . . 7 𝐻 = (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
4644, 45fmptd 6548 . . . . . 6 (𝜑𝐻:ℕ0⟶ℝ)
4746adantr 472 . . . . 5 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝐻:ℕ0⟶ℝ)
4847ffvelrnda 6522 . . . 4 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → (𝐻𝑚) ∈ ℝ)
4948recnd 10260 . . 3 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → (𝐻𝑚) ∈ ℂ)
5025, 26, 30redivcld 11045 . . . . . 6 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
5150recnd 10260 . . . . 5 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℂ)
52 divge0 11084 . . . . . . . 8 ((((abs‘𝑋) ∈ ℝ ∧ 0 ≤ (abs‘𝑋)) ∧ ((abs‘𝑌) ∈ ℝ ∧ 0 < (abs‘𝑌))) → 0 ≤ ((abs‘𝑋) / (abs‘𝑌)))
5325, 27, 26, 29, 52syl22anc 1478 . . . . . . 7 (𝜑 → 0 ≤ ((abs‘𝑋) / (abs‘𝑌)))
5450, 53absidd 14360 . . . . . 6 (𝜑 → (abs‘((abs‘𝑋) / (abs‘𝑌))) = ((abs‘𝑋) / (abs‘𝑌)))
5526recnd 10260 . . . . . . . . 9 (𝜑 → (abs‘𝑌) ∈ ℂ)
5655mulid1d 10249 . . . . . . . 8 (𝜑 → ((abs‘𝑌) · 1) = (abs‘𝑌))
5728, 56breqtrrd 4832 . . . . . . 7 (𝜑 → (abs‘𝑋) < ((abs‘𝑌) · 1))
58 1red 10247 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
59 ltdivmul 11090 . . . . . . . 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 1481 . . . . . . 7 (𝜑 → (((abs‘𝑋) / (abs‘𝑌)) < 1 ↔ (abs‘𝑋) < ((abs‘𝑌) · 1)))
6157, 60mpbird 247 . . . . . 6 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) < 1)
6254, 61eqbrtrd 4826 . . . . 5 (𝜑 → (abs‘((abs‘𝑋) / (abs‘𝑌))) < 1)
6336geomulcvg 14806 . . . . 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 696 . . . 4 (𝜑 → seq0( + , (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))) ∈ dom ⇝ )
6564adantr 472 . . 3 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → seq0( + , (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))) ∈ dom ⇝ )
66 1red 10247 . . 3 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 1 ∈ ℝ)
6741ad2antrr 764 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝐺𝑋):ℕ0⟶ℂ)
68 eluznn0 11950 . . . . . . . . 9 ((𝑗 ∈ ℕ0𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℕ0)
6916, 68sylan 489 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℕ0)
7067, 69ffvelrnd 6523 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐺𝑋)‘𝑚) ∈ ℂ)
7170abscld 14374 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐺𝑋)‘𝑚)) ∈ ℝ)
7232adantr 472 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
7372, 69reexpcld 13219 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (((abs‘𝑋) / (abs‘𝑌))↑𝑚) ∈ ℝ)
7469nn0red 11544 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℝ)
7569nn0ge0d 11546 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ 𝑚)
7611ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝐴:ℕ0⟶ℂ)
7776, 69ffvelrnd 6523 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝐴𝑚) ∈ ℂ)
785ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑌 ∈ ℂ)
7978, 69expcld 13202 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑌𝑚) ∈ ℂ)
8077, 79mulcld 10252 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐴𝑚) · (𝑌𝑚)) ∈ ℂ)
8180abscld 14374 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑌𝑚))) ∈ ℝ)
82 1red 10247 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 1 ∈ ℝ)
8319ad2antrr 764 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑋 ∈ ℂ)
8483abscld 14374 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑋) ∈ ℝ)
8584, 69reexpcld 13219 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑋)↑𝑚) ∈ ℝ)
8683absge0d 14382 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (abs‘𝑋))
8784, 69, 86expge0d 13220 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ ((abs‘𝑋)↑𝑚))
88 simprr 813 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)
89 fveq2 6352 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝐴𝑘) = (𝐴𝑚))
90 oveq2 6821 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑌𝑘) = (𝑌𝑚))
9189, 90oveq12d 6831 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((𝐴𝑘) · (𝑌𝑘)) = ((𝐴𝑚) · (𝑌𝑚)))
9291fveq2d 6356 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → (abs‘((𝐴𝑘) · (𝑌𝑘))) = (abs‘((𝐴𝑚) · (𝑌𝑚))))
9392breq1d 4814 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → ((abs‘((𝐴𝑘) · (𝑌𝑘))) < 1 ↔ (abs‘((𝐴𝑚) · (𝑌𝑚))) < 1))
9493rspccva 3448 . . . . . . . . . . . 12 ((∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1 ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑌𝑚))) < 1)
9588, 94sylan 489 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑌𝑚))) < 1)
96 1re 10231 . . . . . . . . . . . 12 1 ∈ ℝ
97 ltle 10318 . . . . . . . . . . . 12 (((abs‘((𝐴𝑚) · (𝑌𝑚))) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘((𝐴𝑚) · (𝑌𝑚))) < 1 → (abs‘((𝐴𝑚) · (𝑌𝑚))) ≤ 1))
9881, 96, 97sylancl 697 . . . . . . . . . . 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 11155 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑌𝑚))) · ((abs‘𝑋)↑𝑚)) ≤ (1 · ((abs‘𝑋)↑𝑚)))
10183, 69expcld 13202 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑋𝑚) ∈ ℂ)
10277, 101mulcld 10252 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐴𝑚) · (𝑋𝑚)) ∈ ℂ)
103102, 79absmuld 14392 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))) = ((abs‘((𝐴𝑚) · (𝑋𝑚))) · (abs‘(𝑌𝑚))))
10480, 101absmuld 14392 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚))) = ((abs‘((𝐴𝑚) · (𝑌𝑚))) · (abs‘(𝑋𝑚))))
10577, 79, 101mul32d 10438 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚)) = (((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚)))
106105fveq2d 6356 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚))) = (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))))
10783, 69absexpd 14390 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑋𝑚)) = ((abs‘𝑋)↑𝑚))
108107oveq2d 6829 . . . . . . . . . . 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 2802 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))) = ((abs‘((𝐴𝑚) · (𝑌𝑚))) · ((abs‘𝑋)↑𝑚)))
11078, 69absexpd 14390 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑌𝑚)) = ((abs‘𝑌)↑𝑚))
111110oveq2d 6829 . . . . . . . . . 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 2802 . . . . . . . . 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 10260 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑋)↑𝑚) ∈ ℂ)
114113mulid2d 10250 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (1 · ((abs‘𝑋)↑𝑚)) = ((abs‘𝑋)↑𝑚))
115100, 112, 1143brtr3d 4835 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑋𝑚))) · ((abs‘𝑌)↑𝑚)) ≤ ((abs‘𝑋)↑𝑚))
116102abscld 14374 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑋𝑚))) ∈ ℝ)
11723adantr 472 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑌) ∈ ℝ)
118117, 69reexpcld 13219 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑌)↑𝑚) ∈ ℝ)
119 eluzelz 11889 . . . . . . . . . . 11 (𝑚 ∈ (ℤ𝑗) → 𝑚 ∈ ℤ)
120119adantl 473 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℤ)
12129ad2antrr 764 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 < (abs‘𝑌))
122 expgt0 13087 . . . . . . . . . 10 (((abs‘𝑌) ∈ ℝ ∧ 𝑚 ∈ ℤ ∧ 0 < (abs‘𝑌)) → 0 < ((abs‘𝑌)↑𝑚))
123117, 120, 121, 122syl3anc 1477 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 < ((abs‘𝑌)↑𝑚))
124 lemuldiv 11095 . . . . . . . . 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 1481 . . . . . . . 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 222 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑋𝑚))) ≤ (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚)))
1276pserval2 24364 . . . . . . . . 9 ((𝑋 ∈ ℂ ∧ 𝑚 ∈ ℕ0) → ((𝐺𝑋)‘𝑚) = ((𝐴𝑚) · (𝑋𝑚)))
12883, 69, 127syl2anc 696 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐺𝑋)‘𝑚) = ((𝐴𝑚) · (𝑋𝑚)))
129128fveq2d 6356 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐺𝑋)‘𝑚)) = (abs‘((𝐴𝑚) · (𝑋𝑚))))
13021recnd 10260 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑋) ∈ ℂ)
131130adantr 472 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑋) ∈ ℂ)
13223recnd 10260 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑌) ∈ ℂ)
133132adantr 472 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑌) ∈ ℂ)
13430ad2antrr 764 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑌) ≠ 0)
135131, 133, 134, 69expdivd 13216 . . . . . . 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 4836 . . . . . 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 11156 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ≤ (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
13874, 71remulcld 10262 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ ℝ)
13970absge0d 14382 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (abs‘((𝐺𝑋)‘𝑚)))
14074, 71, 75, 139mulge0d 10796 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
141138, 140absidd 14360 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑚 · (abs‘((𝐺𝑋)‘𝑚)))) = (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
14274, 73remulcld 10262 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ ℝ)
143142recnd 10260 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ ℂ)
144143mulid2d 10250 . . . . 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 4836 . . . 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 6841 . . . . . 6 (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ V
14745fvmpt2 6453 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ V) → (𝐻𝑚) = (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
14869, 146, 147sylancl 697 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝐻𝑚) = (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
149148fveq2d 6356 . . . 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 6821 . . . . . . . 8 (𝑖 = 𝑚 → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) = (((abs‘𝑋) / (abs‘𝑌))↑𝑚))
152150, 151oveq12d 6831 . . . . . . 7 (𝑖 = 𝑚 → (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
153 ovex 6841 . . . . . . 7 (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ V
154152, 36, 153fvmpt 6444 . . . . . 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 6829 . . . 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 4836 . . 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 14749 . 2 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → seq0( + , 𝐻) ∈ dom ⇝ )
15915, 158rexlimddv 3173 1 (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  Vcvv 3340   class class class wbr 4804  cmpt 4881  dom cdm 5266  wf 6045  cfv 6049  (class class class)co 6813  cc 10126  cr 10127  0cc0 10128  1c1 10129   + caddc 10131   · cmul 10133   < clt 10266  cle 10267   / cdiv 10876  0cn0 11484  cz 11569  cuz 11879  +crp 12025  seqcseq 12995  cexp 13054  abscabs 14173  cli 14414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206  ax-addf 10207  ax-mulf 10208
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-rp 12026  df-ico 12374  df-fz 12520  df-fzo 12660  df-fl 12787  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-limsup 14401  df-clim 14418  df-rlim 14419  df-sum 14616
This theorem is referenced by:  radcnvlem2  24367  radcnvlt1  24371
  Copyright terms: Public domain W3C validator