| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | nn0uz 12920 | . . 3
⊢
ℕ0 = (ℤ≥‘0) | 
| 2 |  | 0zd 12625 | . . 3
⊢ (𝜑 → 0 ∈
ℤ) | 
| 3 |  | 1rp 13038 | . . . 4
⊢ 1 ∈
ℝ+ | 
| 4 | 3 | a1i 11 | . . 3
⊢ (𝜑 → 1 ∈
ℝ+) | 
| 5 |  | radcnvlem2.y | . . . 4
⊢ (𝜑 → 𝑌 ∈ ℂ) | 
| 6 |  | pser.g | . . . . 5
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) | 
| 7 | 6 | pserval2 26454 | . . . 4
⊢ ((𝑌 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ ((𝐺‘𝑌)‘𝑘) = ((𝐴‘𝑘) · (𝑌↑𝑘))) | 
| 8 | 5, 7 | sylan 580 | . . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘𝑌)‘𝑘) = ((𝐴‘𝑘) · (𝑌↑𝑘))) | 
| 9 |  | fvexd 6921 | . . . 4
⊢ (𝜑 → (𝐺‘𝑌) ∈ V) | 
| 10 |  | radcnvlem2.c | . . . 4
⊢ (𝜑 → seq0( + , (𝐺‘𝑌)) ∈ dom ⇝ ) | 
| 11 |  | radcnv.a | . . . . . 6
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | 
| 12 | 6, 11, 5 | psergf 26455 | . . . . 5
⊢ (𝜑 → (𝐺‘𝑌):ℕ0⟶ℂ) | 
| 13 | 12 | ffvelcdmda 7104 | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘𝑌)‘𝑘) ∈ ℂ) | 
| 14 | 1, 2, 9, 10, 13 | serf0 15717 | . . 3
⊢ (𝜑 → (𝐺‘𝑌) ⇝ 0) | 
| 15 | 1, 2, 4, 8, 14 | climi0 15548 | . 2
⊢ (𝜑 → ∃𝑗 ∈ ℕ0 ∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1) | 
| 16 |  | simprl 771 | . . 3
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → 𝑗 ∈ ℕ0) | 
| 17 |  | nn0re 12535 | . . . . . . 7
⊢ (𝑖 ∈ ℕ0
→ 𝑖 ∈
ℝ) | 
| 18 | 17 | adantl 481 | . . . . . 6
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈
ℝ) | 
| 19 |  | psergf.x | . . . . . . . . . 10
⊢ (𝜑 → 𝑋 ∈ ℂ) | 
| 20 | 19 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → 𝑋 ∈ ℂ) | 
| 21 | 20 | abscld 15475 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → (abs‘𝑋) ∈
ℝ) | 
| 22 | 5 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → 𝑌 ∈ ℂ) | 
| 23 | 22 | abscld 15475 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → (abs‘𝑌) ∈
ℝ) | 
| 24 |  | 0red 11264 | . . . . . . . . . . 11
⊢ (𝜑 → 0 ∈
ℝ) | 
| 25 | 19 | abscld 15475 | . . . . . . . . . . 11
⊢ (𝜑 → (abs‘𝑋) ∈
ℝ) | 
| 26 | 5 | abscld 15475 | . . . . . . . . . . 11
⊢ (𝜑 → (abs‘𝑌) ∈
ℝ) | 
| 27 | 19 | absge0d 15483 | . . . . . . . . . . 11
⊢ (𝜑 → 0 ≤ (abs‘𝑋)) | 
| 28 |  | radcnvlem2.a | . . . . . . . . . . 11
⊢ (𝜑 → (abs‘𝑋) < (abs‘𝑌)) | 
| 29 | 24, 25, 26, 27, 28 | lelttrd 11419 | . . . . . . . . . 10
⊢ (𝜑 → 0 < (abs‘𝑌)) | 
| 30 | 29 | gt0ne0d 11827 | . . . . . . . . 9
⊢ (𝜑 → (abs‘𝑌) ≠ 0) | 
| 31 | 30 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → (abs‘𝑌) ≠ 0) | 
| 32 | 21, 23, 31 | redivcld 12095 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ) | 
| 33 |  | reexpcl 14119 | . . . . . . 7
⊢
((((abs‘𝑋) /
(abs‘𝑌)) ∈
ℝ ∧ 𝑖 ∈
ℕ0) → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) ∈ ℝ) | 
| 34 | 32, 33 | sylan 580 | . . . . . 6
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑖 ∈ ℕ0) →
(((abs‘𝑋) /
(abs‘𝑌))↑𝑖) ∈
ℝ) | 
| 35 | 18, 34 | remulcld 11291 | . . . . 5
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 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))) | 
| 37 | 35, 36 | fmptd 7134 | . . . 4
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → (𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖))):ℕ0⟶ℝ) | 
| 38 | 37 | ffvelcdmda 7104 | . . 3
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → ((𝑖 ∈ ℕ0
↦ (𝑖 ·
(((abs‘𝑋) /
(abs‘𝑌))↑𝑖)))‘𝑚) ∈ ℝ) | 
| 39 |  | nn0re 12535 | . . . . . . . . 9
⊢ (𝑚 ∈ ℕ0
→ 𝑚 ∈
ℝ) | 
| 40 | 39 | adantl 481 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈
ℝ) | 
| 41 | 6, 11, 19 | psergf 26455 | . . . . . . . . . 10
⊢ (𝜑 → (𝐺‘𝑋):ℕ0⟶ℂ) | 
| 42 | 41 | ffvelcdmda 7104 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝐺‘𝑋)‘𝑚) ∈ ℂ) | 
| 43 | 42 | abscld 15475 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
(abs‘((𝐺‘𝑋)‘𝑚)) ∈ ℝ) | 
| 44 | 40, 43 | remulcld 11291 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))) ∈ ℝ) | 
| 45 |  | radcnvlem1.h | . . . . . . 7
⊢ 𝐻 = (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))) | 
| 46 | 44, 45 | fmptd 7134 | . . . . . 6
⊢ (𝜑 → 𝐻:ℕ0⟶ℝ) | 
| 47 | 46 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → 𝐻:ℕ0⟶ℝ) | 
| 48 | 47 | ffvelcdmda 7104 | . . . 4
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → (𝐻‘𝑚) ∈ ℝ) | 
| 49 | 48 | recnd 11289 | . . 3
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ ℕ0) → (𝐻‘𝑚) ∈ ℂ) | 
| 50 | 25, 26, 30 | redivcld 12095 | . . . . . 6
⊢ (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ) | 
| 51 | 50 | recnd 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))) | 
| 53 | 25, 27, 26, 29, 52 | syl22anc 839 | . . . . . . 7
⊢ (𝜑 → 0 ≤ ((abs‘𝑋) / (abs‘𝑌))) | 
| 54 | 50, 53 | absidd 15461 | . . . . . 6
⊢ (𝜑 →
(abs‘((abs‘𝑋) /
(abs‘𝑌))) =
((abs‘𝑋) /
(abs‘𝑌))) | 
| 55 | 26 | recnd 11289 | . . . . . . . . 9
⊢ (𝜑 → (abs‘𝑌) ∈
ℂ) | 
| 56 | 55 | mulridd 11278 | . . . . . . . 8
⊢ (𝜑 → ((abs‘𝑌) · 1) = (abs‘𝑌)) | 
| 57 | 28, 56 | breqtrrd 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))) | 
| 60 | 25, 58, 26, 29, 59 | syl112anc 1376 | . . . . . . 7
⊢ (𝜑 → (((abs‘𝑋) / (abs‘𝑌)) < 1 ↔ (abs‘𝑋) < ((abs‘𝑌) · 1))) | 
| 61 | 57, 60 | mpbird 257 | . . . . . 6
⊢ (𝜑 → ((abs‘𝑋) / (abs‘𝑌)) < 1) | 
| 62 | 54, 61 | eqbrtrd 5165 | . . . . 5
⊢ (𝜑 →
(abs‘((abs‘𝑋) /
(abs‘𝑌))) <
1) | 
| 63 | 36 | geomulcvg 15912 | . . . . 5
⊢
((((abs‘𝑋) /
(abs‘𝑌)) ∈
ℂ ∧ (abs‘((abs‘𝑋) / (abs‘𝑌))) < 1) → seq0( + , (𝑖 ∈ ℕ0
↦ (𝑖 ·
(((abs‘𝑋) /
(abs‘𝑌))↑𝑖)))) ∈ dom ⇝
) | 
| 64 | 51, 62, 63 | syl2anc 584 | . . . 4
⊢ (𝜑 → seq0( + , (𝑖 ∈ ℕ0
↦ (𝑖 ·
(((abs‘𝑋) /
(abs‘𝑌))↑𝑖)))) ∈ dom ⇝
) | 
| 65 | 64 | adantr 480 | . . 3
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → seq0( + , (𝑖 ∈ ℕ0
↦ (𝑖 ·
(((abs‘𝑋) /
(abs‘𝑌))↑𝑖)))) ∈ dom ⇝
) | 
| 66 |  | 1red 11262 | . . 3
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → 1 ∈
ℝ) | 
| 67 | 41 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝐺‘𝑋):ℕ0⟶ℂ) | 
| 68 |  | eluznn0 12959 | . . . . . . . . 9
⊢ ((𝑗 ∈ ℕ0
∧ 𝑚 ∈
(ℤ≥‘𝑗)) → 𝑚 ∈ ℕ0) | 
| 69 | 16, 68 | sylan 580 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑚 ∈ ℕ0) | 
| 70 | 67, 69 | ffvelcdmd 7105 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((𝐺‘𝑋)‘𝑚) ∈ ℂ) | 
| 71 | 70 | abscld 15475 | . . . . . 6
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘((𝐺‘𝑋)‘𝑚)) ∈ ℝ) | 
| 72 | 32 | adantr 480 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((abs‘𝑋) / (abs‘𝑌)) ∈ ℝ) | 
| 73 | 72, 69 | reexpcld 14203 | . . . . . 6
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (((abs‘𝑋) / (abs‘𝑌))↑𝑚) ∈ ℝ) | 
| 74 | 69 | nn0red 12588 | . . . . . 6
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑚 ∈ ℝ) | 
| 75 | 69 | nn0ge0d 12590 | . . . . . 6
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 0 ≤ 𝑚) | 
| 76 | 11 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝐴:ℕ0⟶ℂ) | 
| 77 | 76, 69 | ffvelcdmd 7105 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝐴‘𝑚) ∈ ℂ) | 
| 78 | 5 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑌 ∈ ℂ) | 
| 79 | 78, 69 | expcld 14186 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝑌↑𝑚) ∈ ℂ) | 
| 80 | 77, 79 | mulcld 11281 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((𝐴‘𝑚) · (𝑌↑𝑚)) ∈ ℂ) | 
| 81 | 80 | abscld 15475 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) ∈ ℝ) | 
| 82 |  | 1red 11262 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 1 ∈
ℝ) | 
| 83 | 19 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑋 ∈ ℂ) | 
| 84 | 83 | abscld 15475 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘𝑋) ∈
ℝ) | 
| 85 | 84, 69 | reexpcld 14203 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((abs‘𝑋)↑𝑚) ∈ ℝ) | 
| 86 | 83 | absge0d 15483 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 0 ≤
(abs‘𝑋)) | 
| 87 | 84, 69, 86 | expge0d 14204 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 0 ≤
((abs‘𝑋)↑𝑚)) | 
| 88 |  | simprr 773 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → ∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1) | 
| 89 |  | fveq2 6906 | . . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑚 → (𝐴‘𝑘) = (𝐴‘𝑚)) | 
| 90 |  | oveq2 7439 | . . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑚 → (𝑌↑𝑘) = (𝑌↑𝑚)) | 
| 91 | 89, 90 | oveq12d 7449 | . . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑚 → ((𝐴‘𝑘) · (𝑌↑𝑘)) = ((𝐴‘𝑚) · (𝑌↑𝑚))) | 
| 92 | 91 | fveq2d 6910 | . . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑚 → (abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) = (abs‘((𝐴‘𝑚) · (𝑌↑𝑚)))) | 
| 93 | 92 | breq1d 5153 | . . . . . . . . . . . . 13
⊢ (𝑘 = 𝑚 → ((abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1 ↔ (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) < 1)) | 
| 94 | 93 | rspccva 3621 | . . . . . . . . . . . 12
⊢
((∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1 ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) < 1) | 
| 95 | 88, 94 | sylan 580 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) < 1) | 
| 96 |  | 1re 11261 | . . . . . . . . . . . 12
⊢ 1 ∈
ℝ | 
| 97 |  | ltle 11349 | . . . . . . . . . . . 12
⊢
(((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) ∈ ℝ ∧ 1 ∈ ℝ)
→ ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) < 1 → (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) ≤ 1)) | 
| 98 | 81, 96, 97 | sylancl 586 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) < 1 → (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) ≤ 1)) | 
| 99 | 95, 98 | mpd 15 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) ≤ 1) | 
| 100 | 81, 82, 85, 87, 99 | lemul1ad 12207 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) · ((abs‘𝑋)↑𝑚)) ≤ (1 · ((abs‘𝑋)↑𝑚))) | 
| 101 | 83, 69 | expcld 14186 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝑋↑𝑚) ∈ ℂ) | 
| 102 | 77, 101 | mulcld 11281 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((𝐴‘𝑚) · (𝑋↑𝑚)) ∈ ℂ) | 
| 103 | 102, 79 | absmuld 15493 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘(((𝐴‘𝑚) · (𝑋↑𝑚)) · (𝑌↑𝑚))) = ((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) · (abs‘(𝑌↑𝑚)))) | 
| 104 | 80, 101 | absmuld 15493 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘(((𝐴‘𝑚) · (𝑌↑𝑚)) · (𝑋↑𝑚))) = ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) · (abs‘(𝑋↑𝑚)))) | 
| 105 | 77, 79, 101 | mul32d 11471 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (((𝐴‘𝑚) · (𝑌↑𝑚)) · (𝑋↑𝑚)) = (((𝐴‘𝑚) · (𝑋↑𝑚)) · (𝑌↑𝑚))) | 
| 106 | 105 | fveq2d 6910 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘(((𝐴‘𝑚) · (𝑌↑𝑚)) · (𝑋↑𝑚))) = (abs‘(((𝐴‘𝑚) · (𝑋↑𝑚)) · (𝑌↑𝑚)))) | 
| 107 | 83, 69 | absexpd 15491 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘(𝑋↑𝑚)) = ((abs‘𝑋)↑𝑚)) | 
| 108 | 107 | oveq2d 7447 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) · (abs‘(𝑋↑𝑚))) = ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) · ((abs‘𝑋)↑𝑚))) | 
| 109 | 104, 106,
108 | 3eqtr3d 2785 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘(((𝐴‘𝑚) · (𝑋↑𝑚)) · (𝑌↑𝑚))) = ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) · ((abs‘𝑋)↑𝑚))) | 
| 110 | 78, 69 | absexpd 15491 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘(𝑌↑𝑚)) = ((abs‘𝑌)↑𝑚)) | 
| 111 | 110 | oveq2d 7447 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) · (abs‘(𝑌↑𝑚))) = ((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) · ((abs‘𝑌)↑𝑚))) | 
| 112 | 103, 109,
111 | 3eqtr3d 2785 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((abs‘((𝐴‘𝑚) · (𝑌↑𝑚))) · ((abs‘𝑋)↑𝑚)) = ((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) · ((abs‘𝑌)↑𝑚))) | 
| 113 | 85 | recnd 11289 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((abs‘𝑋)↑𝑚) ∈ ℂ) | 
| 114 | 113 | mullidd 11279 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (1 ·
((abs‘𝑋)↑𝑚)) = ((abs‘𝑋)↑𝑚)) | 
| 115 | 100, 112,
114 | 3brtr3d 5174 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) · ((abs‘𝑌)↑𝑚)) ≤ ((abs‘𝑋)↑𝑚)) | 
| 116 | 102 | abscld 15475 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) ∈ ℝ) | 
| 117 | 23 | adantr 480 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘𝑌) ∈
ℝ) | 
| 118 | 117, 69 | reexpcld 14203 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((abs‘𝑌)↑𝑚) ∈ ℝ) | 
| 119 |  | eluzelz 12888 | . . . . . . . . . . 11
⊢ (𝑚 ∈
(ℤ≥‘𝑗) → 𝑚 ∈ ℤ) | 
| 120 | 119 | adantl 481 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑚 ∈ ℤ) | 
| 121 | 29 | ad2antrr 726 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 0 <
(abs‘𝑌)) | 
| 122 |  | expgt0 14136 | . . . . . . . . . 10
⊢
(((abs‘𝑌)
∈ ℝ ∧ 𝑚
∈ ℤ ∧ 0 < (abs‘𝑌)) → 0 < ((abs‘𝑌)↑𝑚)) | 
| 123 | 117, 120,
121, 122 | syl3anc 1373 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 0 <
((abs‘𝑌)↑𝑚)) | 
| 124 |  | lemuldiv 12148 | . . . . . . . . 9
⊢
(((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) ∈ ℝ ∧ ((abs‘𝑋)↑𝑚) ∈ ℝ ∧ (((abs‘𝑌)↑𝑚) ∈ ℝ ∧ 0 <
((abs‘𝑌)↑𝑚))) → (((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) · ((abs‘𝑌)↑𝑚)) ≤ ((abs‘𝑋)↑𝑚) ↔ (abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) ≤ (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚)))) | 
| 125 | 116, 85, 118, 123, 124 | syl112anc 1376 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (((abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) · ((abs‘𝑌)↑𝑚)) ≤ ((abs‘𝑋)↑𝑚) ↔ (abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) ≤ (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚)))) | 
| 126 | 115, 125 | mpbid 232 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘((𝐴‘𝑚) · (𝑋↑𝑚))) ≤ (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚))) | 
| 127 | 6 | pserval2 26454 | . . . . . . . . 9
⊢ ((𝑋 ∈ ℂ ∧ 𝑚 ∈ ℕ0)
→ ((𝐺‘𝑋)‘𝑚) = ((𝐴‘𝑚) · (𝑋↑𝑚))) | 
| 128 | 83, 69, 127 | syl2anc 584 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((𝐺‘𝑋)‘𝑚) = ((𝐴‘𝑚) · (𝑋↑𝑚))) | 
| 129 | 128 | fveq2d 6910 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘((𝐺‘𝑋)‘𝑚)) = (abs‘((𝐴‘𝑚) · (𝑋↑𝑚)))) | 
| 130 | 21 | recnd 11289 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → (abs‘𝑋) ∈
ℂ) | 
| 131 | 130 | adantr 480 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘𝑋) ∈
ℂ) | 
| 132 | 23 | recnd 11289 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → (abs‘𝑌) ∈
ℂ) | 
| 133 | 132 | adantr 480 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘𝑌) ∈
ℂ) | 
| 134 | 30 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘𝑌) ≠ 0) | 
| 135 | 131, 133,
134, 69 | expdivd 14200 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (((abs‘𝑋) / (abs‘𝑌))↑𝑚) = (((abs‘𝑋)↑𝑚) / ((abs‘𝑌)↑𝑚))) | 
| 136 | 126, 129,
135 | 3brtr4d 5175 | . . . . . 6
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘((𝐺‘𝑋)‘𝑚)) ≤ (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) | 
| 137 | 71, 73, 74, 75, 136 | lemul2ad 12208 | . . . . 5
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))) ≤ (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚))) | 
| 138 | 74, 71 | remulcld 11291 | . . . . . 6
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))) ∈ ℝ) | 
| 139 | 70 | absge0d 15483 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 0 ≤
(abs‘((𝐺‘𝑋)‘𝑚))) | 
| 140 | 74, 71, 75, 139 | mulge0d 11840 | . . . . . 6
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 0 ≤ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))) | 
| 141 | 138, 140 | absidd 15461 | . . . . 5
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘(𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))) = (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))) | 
| 142 | 74, 73 | remulcld 11291 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ ℝ) | 
| 143 | 142 | recnd 11289 | . . . . . 6
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) ∈ ℂ) | 
| 144 | 143 | mullidd 11279 | . . . . 5
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (1 · (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚))) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚))) | 
| 145 | 137, 141,
144 | 3brtr4d 5175 | . . . 4
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘(𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))) ≤ (1 · (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))) | 
| 146 |  | ovex 7464 | . . . . . 6
⊢ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))) ∈ V | 
| 147 | 45 | fvmpt2 7027 | . . . . . 6
⊢ ((𝑚 ∈ ℕ0
∧ (𝑚 ·
(abs‘((𝐺‘𝑋)‘𝑚))) ∈ V) → (𝐻‘𝑚) = (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))) | 
| 148 | 69, 146, 147 | sylancl 586 | . . . . 5
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝐻‘𝑚) = (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))) | 
| 149 | 148 | fveq2d 6910 | . . . 4
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘(𝐻‘𝑚)) = (abs‘(𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))))) | 
| 150 |  | id 22 | . . . . . . . 8
⊢ (𝑖 = 𝑚 → 𝑖 = 𝑚) | 
| 151 |  | oveq2 7439 | . . . . . . . 8
⊢ (𝑖 = 𝑚 → (((abs‘𝑋) / (abs‘𝑌))↑𝑖) = (((abs‘𝑋) / (abs‘𝑌))↑𝑚)) | 
| 152 | 150, 151 | oveq12d 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 | 
| 154 | 152, 36, 153 | fvmpt 7016 | . . . . . 6
⊢ (𝑚 ∈ ℕ0
→ ((𝑖 ∈
ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚))) | 
| 155 | 69, 154 | syl 17 | . . . . 5
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚) = (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚))) | 
| 156 | 155 | oveq2d 7447 | . . . 4
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (1 · ((𝑖 ∈ ℕ0
↦ (𝑖 ·
(((abs‘𝑋) /
(abs‘𝑌))↑𝑖)))‘𝑚)) = (1 · (𝑚 · (((abs‘𝑋) / (abs‘𝑌))↑𝑚)))) | 
| 157 | 145, 149,
156 | 3brtr4d 5175 | . . 3
⊢ (((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (abs‘(𝐻‘𝑚)) ≤ (1 · ((𝑖 ∈ ℕ0 ↦ (𝑖 · (((abs‘𝑋) / (abs‘𝑌))↑𝑖)))‘𝑚))) | 
| 158 | 1, 16, 38, 49, 65, 66, 157 | cvgcmpce 15854 | . 2
⊢ ((𝜑 ∧ (𝑗 ∈ ℕ0 ∧
∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐴‘𝑘) · (𝑌↑𝑘))) < 1)) → seq0( + , 𝐻) ∈ dom ⇝
) | 
| 159 | 15, 158 | rexlimddv 3161 | 1
⊢ (𝜑 → seq0( + , 𝐻) ∈ dom ⇝
) |