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

Theorem radcnvlem1 24928
Description: Lemma for radcnvlt1 24933, radcnvle 24935. 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 12268 . . 3 0 = (ℤ‘0)
2 0zd 11981 . . 3 (𝜑 → 0 ∈ ℤ)
3 1rp 12381 . . . 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 24926 . . . 4 ((𝑌 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) = ((𝐴𝑘) · (𝑌𝑘)))
85, 7sylan 580 . . 3 ((𝜑𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) = ((𝐴𝑘) · (𝑌𝑘)))
9 fvexd 6678 . . . 4 (𝜑 → (𝐺𝑌) ∈ V)
10 radcnvlem2.c . . . 4 (𝜑 → seq0( + , (𝐺𝑌)) ∈ dom ⇝ )
11 radcnv.a . . . . . 6 (𝜑𝐴:ℕ0⟶ℂ)
126, 11, 5psergf 24927 . . . . 5 (𝜑 → (𝐺𝑌):ℕ0⟶ℂ)
1312ffvelrnda 6843 . . . 4 ((𝜑𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) ∈ ℂ)
141, 2, 9, 10, 13serf0 15025 . . 3 (𝜑 → (𝐺𝑌) ⇝ 0)
151, 2, 4, 8, 14climi0 14857 . 2 (𝜑 → ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)
16 simprl 767 . . 3 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝑗 ∈ ℕ0)
17 nn0re 11894 . . . . . . 7 (𝑖 ∈ ℕ0𝑖 ∈ ℝ)
1817adantl 482 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℝ)
19 psergf.x . . . . . . . . . 10 (𝜑𝑋 ∈ ℂ)
2019adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝑋 ∈ ℂ)
2120abscld 14784 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑋) ∈ ℝ)
225adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝑌 ∈ ℂ)
2322abscld 14784 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑌) ∈ ℝ)
24 0red 10632 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℝ)
2519abscld 14784 . . . . . . . . . . 11 (𝜑 → (abs‘𝑋) ∈ ℝ)
265abscld 14784 . . . . . . . . . . 11 (𝜑 → (abs‘𝑌) ∈ ℝ)
2719absge0d 14792 . . . . . . . . . . 11 (𝜑 → 0 ≤ (abs‘𝑋))
28 radcnvlem2.a . . . . . . . . . . 11 (𝜑 → (abs‘𝑋) < (abs‘𝑌))
2924, 25, 26, 27, 28lelttrd 10786 . . . . . . . . . 10 (𝜑 → 0 < (abs‘𝑌))
3029gt0ne0d 11192 . . . . . . . . 9 (𝜑 → (abs‘𝑌) ≠ 0)
3130adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑌) ≠ 0)
3221, 23, 31redivcld 11456 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
33 reexpcl 13434 . . . . . . 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 10659 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑖 ∈ ℕ0) → (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)) ∈ ℝ)
36 eqid 2818 . . . . 5 (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖))) = (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))
3735, 36fmptd 6870 . . . 4 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖))):ℕ0⟶ℝ)
3837ffvelrnda 6843 . . 3 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → ((𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚) ∈ ℝ)
39 nn0re 11894 . . . . . . . . 9 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
4039adantl 482 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ0) → 𝑚 ∈ ℝ)
416, 11, 19psergf 24927 . . . . . . . . . 10 (𝜑 → (𝐺𝑋):ℕ0⟶ℂ)
4241ffvelrnda 6843 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ0) → ((𝐺𝑋)‘𝑚) ∈ ℂ)
4342abscld 14784 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ0) → (abs‘((𝐺𝑋)‘𝑚)) ∈ ℝ)
4440, 43remulcld 10659 . . . . . . 7 ((𝜑𝑚 ∈ ℕ0) → (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ ℝ)
45 radcnvlem1.h . . . . . . 7 𝐻 = (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
4644, 45fmptd 6870 . . . . . 6 (𝜑𝐻:ℕ0⟶ℝ)
4746adantr 481 . . . . 5 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝐻:ℕ0⟶ℝ)
4847ffvelrnda 6843 . . . 4 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → (𝐻𝑚) ∈ ℝ)
4948recnd 10657 . . 3 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → (𝐻𝑚) ∈ ℂ)
5025, 26, 30redivcld 11456 . . . . . 6 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
5150recnd 10657 . . . . 5 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℂ)
52 divge0 11497 . . . . . . . 8 ((((abs‘𝑋) ∈ ℝ ∧ 0 ≤ (abs‘𝑋)) ∧ ((abs‘𝑌) ∈ ℝ ∧ 0 < (abs‘𝑌))) → 0 ≤ ((abs‘𝑋) / (abs‘𝑌)))
5325, 27, 26, 29, 52syl22anc 834 . . . . . . 7 (𝜑 → 0 ≤ ((abs‘𝑋) / (abs‘𝑌)))
5450, 53absidd 14770 . . . . . 6 (𝜑 → (abs‘((abs‘𝑋) / (abs‘𝑌))) = ((abs‘𝑋) / (abs‘𝑌)))
5526recnd 10657 . . . . . . . . 9 (𝜑 → (abs‘𝑌) ∈ ℂ)
5655mulid1d 10646 . . . . . . . 8 (𝜑 → ((abs‘𝑌) · 1) = (abs‘𝑌))
5728, 56breqtrrd 5085 . . . . . . 7 (𝜑 → (abs‘𝑋) < ((abs‘𝑌) · 1))
58 1red 10630 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
59 ltdivmul 11503 . . . . . . . 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 1366 . . . . . . 7 (𝜑 → (((abs‘𝑋) / (abs‘𝑌)) < 1 ↔ (abs‘𝑋) < ((abs‘𝑌) · 1)))
6157, 60mpbird 258 . . . . . 6 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) < 1)
6254, 61eqbrtrd 5079 . . . . 5 (𝜑 → (abs‘((abs‘𝑋) / (abs‘𝑌))) < 1)
6336geomulcvg 15220 . . . . 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 481 . . 3 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → seq0( + , (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))) ∈ dom ⇝ )
66 1red 10630 . . 3 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 1 ∈ ℝ)
6741ad2antrr 722 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝐺𝑋):ℕ0⟶ℂ)
68 eluznn0 12305 . . . . . . . . 9 ((𝑗 ∈ ℕ0𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℕ0)
6916, 68sylan 580 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℕ0)
7067, 69ffvelrnd 6844 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐺𝑋)‘𝑚) ∈ ℂ)
7170abscld 14784 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐺𝑋)‘𝑚)) ∈ ℝ)
7232adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
7372, 69reexpcld 13515 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (((abs‘𝑋) / (abs‘𝑌))↑𝑚) ∈ ℝ)
7469nn0red 11944 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℝ)
7569nn0ge0d 11946 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ 𝑚)
7611ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝐴:ℕ0⟶ℂ)
7776, 69ffvelrnd 6844 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝐴𝑚) ∈ ℂ)
785ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑌 ∈ ℂ)
7978, 69expcld 13498 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑌𝑚) ∈ ℂ)
8077, 79mulcld 10649 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐴𝑚) · (𝑌𝑚)) ∈ ℂ)
8180abscld 14784 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑌𝑚))) ∈ ℝ)
82 1red 10630 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 1 ∈ ℝ)
8319ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑋 ∈ ℂ)
8483abscld 14784 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑋) ∈ ℝ)
8584, 69reexpcld 13515 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑋)↑𝑚) ∈ ℝ)
8683absge0d 14792 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (abs‘𝑋))
8784, 69, 86expge0d 13516 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ ((abs‘𝑋)↑𝑚))
88 simprr 769 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)
89 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝐴𝑘) = (𝐴𝑚))
90 oveq2 7153 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑌𝑘) = (𝑌𝑚))
9189, 90oveq12d 7163 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((𝐴𝑘) · (𝑌𝑘)) = ((𝐴𝑚) · (𝑌𝑚)))
9291fveq2d 6667 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → (abs‘((𝐴𝑘) · (𝑌𝑘))) = (abs‘((𝐴𝑚) · (𝑌𝑚))))
9392breq1d 5067 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → ((abs‘((𝐴𝑘) · (𝑌𝑘))) < 1 ↔ (abs‘((𝐴𝑚) · (𝑌𝑚))) < 1))
9493rspccva 3619 . . . . . . . . . . . 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 10629 . . . . . . . . . . . 12 1 ∈ ℝ
97 ltle 10717 . . . . . . . . . . . 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 11567 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑌𝑚))) · ((abs‘𝑋)↑𝑚)) ≤ (1 · ((abs‘𝑋)↑𝑚)))
10183, 69expcld 13498 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑋𝑚) ∈ ℂ)
10277, 101mulcld 10649 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐴𝑚) · (𝑋𝑚)) ∈ ℂ)
103102, 79absmuld 14802 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))) = ((abs‘((𝐴𝑚) · (𝑋𝑚))) · (abs‘(𝑌𝑚))))
10480, 101absmuld 14802 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚))) = ((abs‘((𝐴𝑚) · (𝑌𝑚))) · (abs‘(𝑋𝑚))))
10577, 79, 101mul32d 10838 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚)) = (((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚)))
106105fveq2d 6667 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚))) = (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))))
10783, 69absexpd 14800 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑋𝑚)) = ((abs‘𝑋)↑𝑚))
108107oveq2d 7161 . . . . . . . . . . 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 2861 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))) = ((abs‘((𝐴𝑚) · (𝑌𝑚))) · ((abs‘𝑋)↑𝑚)))
11078, 69absexpd 14800 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑌𝑚)) = ((abs‘𝑌)↑𝑚))
111110oveq2d 7161 . . . . . . . . . 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 2861 . . . . . . . . 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 10657 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑋)↑𝑚) ∈ ℂ)
114113mulid2d 10647 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (1 · ((abs‘𝑋)↑𝑚)) = ((abs‘𝑋)↑𝑚))
115100, 112, 1143brtr3d 5088 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑋𝑚))) · ((abs‘𝑌)↑𝑚)) ≤ ((abs‘𝑋)↑𝑚))
116102abscld 14784 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑋𝑚))) ∈ ℝ)
11723adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑌) ∈ ℝ)
118117, 69reexpcld 13515 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑌)↑𝑚) ∈ ℝ)
119 eluzelz 12241 . . . . . . . . . . 11 (𝑚 ∈ (ℤ𝑗) → 𝑚 ∈ ℤ)
120119adantl 482 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℤ)
12129ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 < (abs‘𝑌))
122 expgt0 13450 . . . . . . . . . 10 (((abs‘𝑌) ∈ ℝ ∧ 𝑚 ∈ ℤ ∧ 0 < (abs‘𝑌)) → 0 < ((abs‘𝑌)↑𝑚))
123117, 120, 121, 122syl3anc 1363 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 < ((abs‘𝑌)↑𝑚))
124 lemuldiv 11508 . . . . . . . . 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 1366 . . . . . . . 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 233 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑋𝑚))) ≤ (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚)))
1276pserval2 24926 . . . . . . . . 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 6667 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐺𝑋)‘𝑚)) = (abs‘((𝐴𝑚) · (𝑋𝑚))))
13021recnd 10657 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑋) ∈ ℂ)
131130adantr 481 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑋) ∈ ℂ)
13223recnd 10657 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑌) ∈ ℂ)
133132adantr 481 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑌) ∈ ℂ)
13430ad2antrr 722 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑌) ≠ 0)
135131, 133, 134, 69expdivd 13512 . . . . . . 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 5089 . . . . . 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 11568 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ≤ (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
13874, 71remulcld 10659 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ ℝ)
13970absge0d 14792 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (abs‘((𝐺𝑋)‘𝑚)))
14074, 71, 75, 139mulge0d 11205 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
141138, 140absidd 14770 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑚 · (abs‘((𝐺𝑋)‘𝑚)))) = (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
14274, 73remulcld 10659 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ ℝ)
143142recnd 10657 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ ℂ)
144143mulid2d 10647 . . . . 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 5089 . . . 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 7178 . . . . . 6 (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ V
14745fvmpt2 6771 . . . . . 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 6667 . . . 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 7153 . . . . . . . 8 (𝑖 = 𝑚 → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) = (((abs‘𝑋) / (abs‘𝑌))↑𝑚))
152150, 151oveq12d 7163 . . . . . . 7 (𝑖 = 𝑚 → (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
153 ovex 7178 . . . . . . 7 (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ V
154152, 36, 153fvmpt 6761 . . . . . 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 7161 . . . 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 5089 . . 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 15161 . 2 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → seq0( + , 𝐻) ∈ dom ⇝ )
15915, 158rexlimddv 3288 1 (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  Vcvv 3492   class class class wbr 5057  cmpt 5137  dom cdm 5548  wf 6344  cfv 6348  (class class class)co 7145  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530   < clt 10663  cle 10664   / cdiv 11285  0cn0 11885  cz 11969  cuz 12231  +crp 12377  seqcseq 13357  cexp 13417  abscabs 14581  cli 14829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-ico 12732  df-fz 12881  df-fzo 13022  df-fl 13150  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-limsup 14816  df-clim 14833  df-rlim 14834  df-sum 15031
This theorem is referenced by:  radcnvlem2  24929  radcnvlt1  24933
  Copyright terms: Public domain W3C validator