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

 Description: Lemma for radcnvlt1 23976, radcnvle 23978. 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))))
psergf.x (𝜑𝑋 ∈ 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)

Dummy variables 𝑖 𝑘 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0uz 11598 . . 3 0 = (ℤ‘0)
2 0zd 11266 . . 3 (𝜑 → 0 ∈ ℤ)
3 1rp 11712 . . . 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 23969 . . . 4 ((𝑌 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) = ((𝐴𝑘) · (𝑌𝑘)))
85, 7sylan 487 . . 3 ((𝜑𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) = ((𝐴𝑘) · (𝑌𝑘)))
9 fvex 6113 . . . . 5 (𝐺𝑌) ∈ V
109a1i 11 . . . 4 (𝜑 → (𝐺𝑌) ∈ V)
11 radcnvlem2.c . . . 4 (𝜑 → seq0( + , (𝐺𝑌)) ∈ dom ⇝ )
12 radcnv.a . . . . . 6 (𝜑𝐴:ℕ0⟶ℂ)
136, 12, 5psergf 23970 . . . . 5 (𝜑 → (𝐺𝑌):ℕ0⟶ℂ)
1413ffvelrnda 6267 . . . 4 ((𝜑𝑘 ∈ ℕ0) → ((𝐺𝑌)‘𝑘) ∈ ℂ)
151, 2, 10, 11, 14serf0 14259 . . 3 (𝜑 → (𝐺𝑌) ⇝ 0)
161, 2, 4, 8, 15climi0 14091 . 2 (𝜑 → ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)
17 simprl 790 . . 3 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝑗 ∈ ℕ0)
18 nn0re 11178 . . . . . . 7 (𝑖 ∈ ℕ0𝑖 ∈ ℝ)
1918adantl 481 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℝ)
20 psergf.x . . . . . . . . . 10 (𝜑𝑋 ∈ ℂ)
2120adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝑋 ∈ ℂ)
2221abscld 14023 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑋) ∈ ℝ)
235adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝑌 ∈ ℂ)
2423abscld 14023 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑌) ∈ ℝ)
25 0red 9920 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℝ)
2620abscld 14023 . . . . . . . . . . 11 (𝜑 → (abs‘𝑋) ∈ ℝ)
275abscld 14023 . . . . . . . . . . 11 (𝜑 → (abs‘𝑌) ∈ ℝ)
2820absge0d 14031 . . . . . . . . . . 11 (𝜑 → 0 ≤ (abs‘𝑋))
29 radcnvlem2.a . . . . . . . . . . 11 (𝜑 → (abs‘𝑋) < (abs‘𝑌))
3025, 26, 27, 28, 29lelttrd 10074 . . . . . . . . . 10 (𝜑 → 0 < (abs‘𝑌))
3130gt0ne0d 10471 . . . . . . . . 9 (𝜑 → (abs‘𝑌) ≠ 0)
3231adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑌) ≠ 0)
3322, 24, 32redivcld 10732 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
34 reexpcl 12739 . . . . . . 7 ((((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ ∧ 𝑖 ∈ ℕ0) → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) ∈ ℝ)
3533, 34sylan 487 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑖 ∈ ℕ0) → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) ∈ ℝ)
3619, 35remulcld 9949 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑖 ∈ ℕ0) → (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)) ∈ ℝ)
37 eqid 2610 . . . . 5 (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖))) = (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))
3836, 37fmptd 6292 . . . 4 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖))):ℕ0⟶ℝ)
3938ffvelrnda 6267 . . 3 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → ((𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚) ∈ ℝ)
40 nn0re 11178 . . . . . . . . 9 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
4140adantl 481 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ0) → 𝑚 ∈ ℝ)
426, 12, 20psergf 23970 . . . . . . . . . 10 (𝜑 → (𝐺𝑋):ℕ0⟶ℂ)
4342ffvelrnda 6267 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ0) → ((𝐺𝑋)‘𝑚) ∈ ℂ)
4443abscld 14023 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ0) → (abs‘((𝐺𝑋)‘𝑚)) ∈ ℝ)
4541, 44remulcld 9949 . . . . . . 7 ((𝜑𝑚 ∈ ℕ0) → (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ ℝ)
46 radcnvlem1.h . . . . . . 7 𝐻 = (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
4745, 46fmptd 6292 . . . . . 6 (𝜑𝐻:ℕ0⟶ℝ)
4847adantr 480 . . . . 5 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 𝐻:ℕ0⟶ℝ)
4948ffvelrnda 6267 . . . 4 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → (𝐻𝑚) ∈ ℝ)
5049recnd 9947 . . 3 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → (𝐻𝑚) ∈ ℂ)
5126, 27, 31redivcld 10732 . . . . . 6 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
5251recnd 9947 . . . . 5 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℂ)
53 divge0 10771 . . . . . . . 8 ((((abs‘𝑋) ∈ ℝ ∧ 0 ≤ (abs‘𝑋)) ∧ ((abs‘𝑌) ∈ ℝ ∧ 0 < (abs‘𝑌))) → 0 ≤ ((abs‘𝑋) / (abs‘𝑌)))
5426, 28, 27, 30, 53syl22anc 1319 . . . . . . 7 (𝜑 → 0 ≤ ((abs‘𝑋) / (abs‘𝑌)))
5551, 54absidd 14009 . . . . . 6 (𝜑 → (abs‘((abs‘𝑋) / (abs‘𝑌))) = ((abs‘𝑋) / (abs‘𝑌)))
5627recnd 9947 . . . . . . . . 9 (𝜑 → (abs‘𝑌) ∈ ℂ)
5756mulid1d 9936 . . . . . . . 8 (𝜑 → ((abs‘𝑌) · 1) = (abs‘𝑌))
5829, 57breqtrrd 4611 . . . . . . 7 (𝜑 → (abs‘𝑋) < ((abs‘𝑌) · 1))
59 1red 9934 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
60 ltdivmul 10777 . . . . . . . 8 (((abs‘𝑋) ∈ ℝ ∧ 1 ∈ ℝ ∧ ((abs‘𝑌) ∈ ℝ ∧ 0 < (abs‘𝑌))) → (((abs‘𝑋) / (abs‘𝑌)) < 1 ↔ (abs‘𝑋) < ((abs‘𝑌) · 1)))
6126, 59, 27, 30, 60syl112anc 1322 . . . . . . 7 (𝜑 → (((abs‘𝑋) / (abs‘𝑌)) < 1 ↔ (abs‘𝑋) < ((abs‘𝑌) · 1)))
6258, 61mpbird 246 . . . . . 6 (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) < 1)
6355, 62eqbrtrd 4605 . . . . 5 (𝜑 → (abs‘((abs‘𝑋) / (abs‘𝑌))) < 1)
6437geomulcvg 14446 . . . . 5 ((((abs‘𝑋) / (abs‘𝑌)) ∈ ℂ ∧ (abs‘((abs‘𝑋) / (abs‘𝑌))) < 1) → seq0( + , (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))) ∈ dom ⇝ )
6552, 63, 64syl2anc 691 . . . 4 (𝜑 → seq0( + , (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))) ∈ dom ⇝ )
6665adantr 480 . . 3 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → seq0( + , (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))) ∈ dom ⇝ )
67 1red 9934 . . 3 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → 1 ∈ ℝ)
6842ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝐺𝑋):ℕ0⟶ℂ)
69 eluznn0 11633 . . . . . . . . 9 ((𝑗 ∈ ℕ0𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℕ0)
7017, 69sylan 487 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℕ0)
7168, 70ffvelrnd 6268 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐺𝑋)‘𝑚) ∈ ℂ)
7271abscld 14023 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐺𝑋)‘𝑚)) ∈ ℝ)
7333adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ)
7473, 70reexpcld 12887 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (((abs‘𝑋) / (abs‘𝑌))↑𝑚) ∈ ℝ)
7570nn0red 11229 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℝ)
7670nn0ge0d 11231 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ 𝑚)
7712ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝐴:ℕ0⟶ℂ)
7877, 70ffvelrnd 6268 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝐴𝑚) ∈ ℂ)
795ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑌 ∈ ℂ)
8079, 70expcld 12870 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑌𝑚) ∈ ℂ)
8178, 80mulcld 9939 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐴𝑚) · (𝑌𝑚)) ∈ ℂ)
8281abscld 14023 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑌𝑚))) ∈ ℝ)
83 1red 9934 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 1 ∈ ℝ)
8420ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑋 ∈ ℂ)
8584abscld 14023 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑋) ∈ ℝ)
8685, 70reexpcld 12887 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑋)↑𝑚) ∈ ℝ)
8784absge0d 14031 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (abs‘𝑋))
8885, 70, 87expge0d 12888 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ ((abs‘𝑋)↑𝑚))
89 simprr 792 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)
90 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝐴𝑘) = (𝐴𝑚))
91 oveq2 6557 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑌𝑘) = (𝑌𝑚))
9290, 91oveq12d 6567 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((𝐴𝑘) · (𝑌𝑘)) = ((𝐴𝑚) · (𝑌𝑚)))
9392fveq2d 6107 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → (abs‘((𝐴𝑘) · (𝑌𝑘))) = (abs‘((𝐴𝑚) · (𝑌𝑚))))
9493breq1d 4593 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → ((abs‘((𝐴𝑘) · (𝑌𝑘))) < 1 ↔ (abs‘((𝐴𝑚) · (𝑌𝑚))) < 1))
9594rspccva 3281 . . . . . . . . . . . 12 ((∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1 ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑌𝑚))) < 1)
9689, 95sylan 487 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑌𝑚))) < 1)
97 1re 9918 . . . . . . . . . . . 12 1 ∈ ℝ
98 ltle 10005 . . . . . . . . . . . 12 (((abs‘((𝐴𝑚) · (𝑌𝑚))) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘((𝐴𝑚) · (𝑌𝑚))) < 1 → (abs‘((𝐴𝑚) · (𝑌𝑚))) ≤ 1))
9982, 97, 98sylancl 693 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑌𝑚))) < 1 → (abs‘((𝐴𝑚) · (𝑌𝑚))) ≤ 1))
10096, 99mpd 15 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑌𝑚))) ≤ 1)
10182, 83, 86, 88, 100lemul1ad 10842 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑌𝑚))) · ((abs‘𝑋)↑𝑚)) ≤ (1 · ((abs‘𝑋)↑𝑚)))
10284, 70expcld 12870 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑋𝑚) ∈ ℂ)
10378, 102mulcld 9939 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐴𝑚) · (𝑋𝑚)) ∈ ℂ)
104103, 80absmuld 14041 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))) = ((abs‘((𝐴𝑚) · (𝑋𝑚))) · (abs‘(𝑌𝑚))))
10581, 102absmuld 14041 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚))) = ((abs‘((𝐴𝑚) · (𝑌𝑚))) · (abs‘(𝑋𝑚))))
10678, 80, 102mul32d 10125 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚)) = (((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚)))
107106fveq2d 6107 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑌𝑚)) · (𝑋𝑚))) = (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))))
10884, 70absexpd 14039 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑋𝑚)) = ((abs‘𝑋)↑𝑚))
109108oveq2d 6565 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑌𝑚))) · (abs‘(𝑋𝑚))) = ((abs‘((𝐴𝑚) · (𝑌𝑚))) · ((abs‘𝑋)↑𝑚)))
110105, 107, 1093eqtr3d 2652 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(((𝐴𝑚) · (𝑋𝑚)) · (𝑌𝑚))) = ((abs‘((𝐴𝑚) · (𝑌𝑚))) · ((abs‘𝑋)↑𝑚)))
11179, 70absexpd 14039 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑌𝑚)) = ((abs‘𝑌)↑𝑚))
112111oveq2d 6565 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑋𝑚))) · (abs‘(𝑌𝑚))) = ((abs‘((𝐴𝑚) · (𝑋𝑚))) · ((abs‘𝑌)↑𝑚)))
113104, 110, 1123eqtr3d 2652 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑌𝑚))) · ((abs‘𝑋)↑𝑚)) = ((abs‘((𝐴𝑚) · (𝑋𝑚))) · ((abs‘𝑌)↑𝑚)))
11486recnd 9947 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑋)↑𝑚) ∈ ℂ)
115114mulid2d 9937 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (1 · ((abs‘𝑋)↑𝑚)) = ((abs‘𝑋)↑𝑚))
116101, 113, 1153brtr3d 4614 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘((𝐴𝑚) · (𝑋𝑚))) · ((abs‘𝑌)↑𝑚)) ≤ ((abs‘𝑋)↑𝑚))
117103abscld 14023 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑋𝑚))) ∈ ℝ)
11824adantr 480 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑌) ∈ ℝ)
119118, 70reexpcld 12887 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((abs‘𝑌)↑𝑚) ∈ ℝ)
120 eluzelz 11573 . . . . . . . . . . 11 (𝑚 ∈ (ℤ𝑗) → 𝑚 ∈ ℤ)
121120adantl 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℤ)
12230ad2antrr 758 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 < (abs‘𝑌))
123 expgt0 12755 . . . . . . . . . 10 (((abs‘𝑌) ∈ ℝ ∧ 𝑚 ∈ ℤ ∧ 0 < (abs‘𝑌)) → 0 < ((abs‘𝑌)↑𝑚))
124118, 121, 122, 123syl3anc 1318 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 < ((abs‘𝑌)↑𝑚))
125 lemuldiv 10782 . . . . . . . . 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))))
126117, 86, 119, 124, 125syl112anc 1322 . . . . . . . 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))))
127116, 126mpbid 221 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐴𝑚) · (𝑋𝑚))) ≤ (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚)))
1286pserval2 23969 . . . . . . . . 9 ((𝑋 ∈ ℂ ∧ 𝑚 ∈ ℕ0) → ((𝐺𝑋)‘𝑚) = ((𝐴𝑚) · (𝑋𝑚)))
12984, 70, 128syl2anc 691 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝐺𝑋)‘𝑚) = ((𝐴𝑚) · (𝑋𝑚)))
130129fveq2d 6107 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐺𝑋)‘𝑚)) = (abs‘((𝐴𝑚) · (𝑋𝑚))))
13122recnd 9947 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑋) ∈ ℂ)
132131adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑋) ∈ ℂ)
13324recnd 9947 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → (abs‘𝑌) ∈ ℂ)
134133adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑌) ∈ ℂ)
13531ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘𝑌) ≠ 0)
136132, 134, 135, 70expdivd 12884 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (((abs‘𝑋) / (abs‘𝑌))↑𝑚) = (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚)))
137127, 130, 1363brtr4d 4615 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘((𝐺𝑋)‘𝑚)) ≤ (((abs‘𝑋) / (abs‘𝑌))↑𝑚))
13872, 74, 75, 76, 137lemul2ad 10843 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ≤ (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
13975, 72remulcld 9949 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ ℝ)
14071absge0d 14031 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (abs‘((𝐺𝑋)‘𝑚)))
14175, 72, 76, 140mulge0d 10483 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → 0 ≤ (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
142139, 141absidd 14009 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑚 · (abs‘((𝐺𝑋)‘𝑚)))) = (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
14375, 74remulcld 9949 . . . . . . 7 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ ℝ)
144143recnd 9947 . . . . . 6 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ ℂ)
145144mulid2d 9937 . . . . 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)))
146138, 142, 1453brtr4d 4615 . . . 4 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝑚 · (abs‘((𝐺𝑋)‘𝑚)))) ≤ (1 · (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚))))
147 ovex 6577 . . . . . 6 (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ V
14846fvmpt2 6200 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ (𝑚 · (abs‘((𝐺𝑋)‘𝑚))) ∈ V) → (𝐻𝑚) = (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
14970, 147, 148sylancl 693 . . . . 5 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝐻𝑚) = (𝑚 · (abs‘((𝐺𝑋)‘𝑚))))
150149fveq2d 6107 . . . 4 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝐻𝑚)) = (abs‘(𝑚 · (abs‘((𝐺𝑋)‘𝑚)))))
151 id 22 . . . . . . . 8 (𝑖 = 𝑚𝑖 = 𝑚)
152 oveq2 6557 . . . . . . . 8 (𝑖 = 𝑚 → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) = (((abs‘𝑋) / (abs‘𝑌))↑𝑚))
153151, 152oveq12d 6567 . . . . . . 7 (𝑖 = 𝑚 → (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
154 ovex 6577 . . . . . . 7 (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ V
155153, 37, 154fvmpt 6191 . . . . . 6 (𝑚 ∈ ℕ0 → ((𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))
15670, 155syl 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)))
157156oveq2d 6565 . . . 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))))
158146, 150, 1573brtr4d 4615 . . 3 (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ𝑗)) → (abs‘(𝐻𝑚)) ≤ (1 · ((𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚)))
1591, 17, 39, 50, 66, 67, 158cvgcmpce 14391 . 2 ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐴𝑘) · (𝑌𝑘))) < 1)) → seq0( + , 𝐻) ∈ dom ⇝ )
16016, 159rexlimddv 3017 1 (𝜑 → seq0( + , 𝐻) ∈ dom ⇝ )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953   ≤ cle 9954   / cdiv 10563  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ℝ+crp 11708  seqcseq 12663  ↑cexp 12722  abscabs 13822   ⇝ cli 14063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265 This theorem is referenced by:  radcnvlem2  23972  radcnvlt1  23976
 Copyright terms: Public domain W3C validator