Description: Ratio test for convergence and divergence of a complex infinite series. If the ratio R of the absolute values of successive terms in an infinite sequence F converges to less than one, then the infinite sum of the terms of F converges to a complex number; and if R convergesgreater then the sum diverges. This combined form of cvgrat and dvgrat directly uses the limit of the ratio.
(It also demonstrates how to use climi2 and absltd to transform a limit to an inequality cf. https://math.stackexchange.com/q/2215191 , and how to use r19.29a in a similar fashion to Mario Carneiro's proof sketch with rexlimdva at https://groups.google.com/g/metamath/c/2RPikOiXLMo .) (Contributed by Steve Rodriguez, 28-Feb-2020)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cvgdvgrat.z | ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) | |
| cvgdvgrat.w | ⊢ 𝑊 = ( ℤ≥ ‘ 𝑁 ) | ||
| cvgdvgrat.n | ⊢ ( 𝜑 → 𝑁 ∈ 𝑍 ) | ||
| cvgdvgrat.f | ⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) | ||
| cvgdvgrat.c | ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) | ||
| cvgdvgrat.n0 | ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) | ||
| cvgdvgrat.r | ⊢ 𝑅 = ( 𝑘 ∈ 𝑊 ↦ ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ) | ||
| cvgdvgrat.cvg | ⊢ ( 𝜑 → 𝑅 ⇝ 𝐿 ) | ||
| cvgdvgrat.n1 | ⊢ ( 𝜑 → 𝐿 ≠ 1 ) | ||
| Assertion | cvgdvgrat | ⊢ ( 𝜑 → ( 𝐿 < 1 ↔ seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cvgdvgrat.z | ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) | |
| 2 | cvgdvgrat.w | ⊢ 𝑊 = ( ℤ≥ ‘ 𝑁 ) | |
| 3 | cvgdvgrat.n | ⊢ ( 𝜑 → 𝑁 ∈ 𝑍 ) | |
| 4 | cvgdvgrat.f | ⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) | |
| 5 | cvgdvgrat.c | ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) | |
| 6 | cvgdvgrat.n0 | ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) | |
| 7 | cvgdvgrat.r | ⊢ 𝑅 = ( 𝑘 ∈ 𝑊 ↦ ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ) | |
| 8 | cvgdvgrat.cvg | ⊢ ( 𝜑 → 𝑅 ⇝ 𝐿 ) | |
| 9 | cvgdvgrat.n1 | ⊢ ( 𝜑 → 𝐿 ≠ 1 ) | |
| 10 | eqid | ⊢ ( ℤ≥ ‘ 𝑛 ) = ( ℤ≥ ‘ 𝑛 ) | |
| 11 | elioore | ⊢ ( 𝑟 ∈ ( 𝐿 (,) 1 ) → 𝑟 ∈ ℝ ) | |
| 12 | 11 | ad3antlr | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → 𝑟 ∈ ℝ ) |
| 13 | 3 1 | eleqtrdi | ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
| 14 | eluzelz | ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ℤ ) | |
| 15 | 13 14 | syl | ⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
| 16 | 7 | a1i | ⊢ ( 𝜑 → 𝑅 = ( 𝑘 ∈ 𝑊 ↦ ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
| 17 | 2 | peano2uzs | ⊢ ( 𝑘 ∈ 𝑊 → ( 𝑘 + 1 ) ∈ 𝑊 ) |
| 18 | ovex | ⊢ ( 𝑘 + 1 ) ∈ V | |
| 19 | eleq1 | ⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( 𝑖 ∈ 𝑊 ↔ ( 𝑘 + 1 ) ∈ 𝑊 ) ) | |
| 20 | 19 | anbi2d | ⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( ( 𝜑 ∧ 𝑖 ∈ 𝑊 ) ↔ ( 𝜑 ∧ ( 𝑘 + 1 ) ∈ 𝑊 ) ) ) |
| 21 | fveq2 | ⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) | |
| 22 | 21 | eleq1d | ⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( ( 𝐹 ‘ 𝑖 ) ∈ ℂ ↔ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) ) |
| 23 | 20 22 | imbi12d | ⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑖 ) ∈ ℂ ) ↔ ( ( 𝜑 ∧ ( 𝑘 + 1 ) ∈ 𝑊 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) ) ) |
| 24 | eleq1 | ⊢ ( 𝑘 = 𝑖 → ( 𝑘 ∈ 𝑊 ↔ 𝑖 ∈ 𝑊 ) ) | |
| 25 | 24 | anbi2d | ⊢ ( 𝑘 = 𝑖 → ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ↔ ( 𝜑 ∧ 𝑖 ∈ 𝑊 ) ) ) |
| 26 | fveq2 | ⊢ ( 𝑘 = 𝑖 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑖 ) ) | |
| 27 | 26 | eleq1d | ⊢ ( 𝑘 = 𝑖 → ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ↔ ( 𝐹 ‘ 𝑖 ) ∈ ℂ ) ) |
| 28 | 25 27 | imbi12d | ⊢ ( 𝑘 = 𝑖 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑖 ) ∈ ℂ ) ) ) |
| 29 | 2 | eleq2i | ⊢ ( 𝑘 ∈ 𝑊 ↔ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
| 30 | 1 | uztrn2 | ⊢ ( ( 𝑁 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑘 ∈ 𝑍 ) |
| 31 | 3 30 | sylan | ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑘 ∈ 𝑍 ) |
| 32 | 29 31 | sylan2b | ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → 𝑘 ∈ 𝑍 ) |
| 33 | 32 5 | syldan | ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
| 34 | 28 33 | chvarvv | ⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑖 ) ∈ ℂ ) |
| 35 | 18 23 34 | vtocl | ⊢ ( ( 𝜑 ∧ ( 𝑘 + 1 ) ∈ 𝑊 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
| 36 | 17 35 | sylan2 | ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
| 37 | 36 33 6 | divcld | ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ) |
| 38 | 37 | abscld | ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ∈ ℝ ) |
| 39 | 16 38 | fvmpt2d | ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝑅 ‘ 𝑘 ) = ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ) |
| 40 | 39 38 | eqeltrd | ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝑅 ‘ 𝑘 ) ∈ ℝ ) |
| 41 | 2 15 8 40 | climrecl | ⊢ ( 𝜑 → 𝐿 ∈ ℝ ) |
| 42 | 41 | rexrd | ⊢ ( 𝜑 → 𝐿 ∈ ℝ* ) |
| 43 | 1xr | ⊢ 1 ∈ ℝ* | |
| 44 | elioo2 | ⊢ ( ( 𝐿 ∈ ℝ* ∧ 1 ∈ ℝ* ) → ( 𝑟 ∈ ( 𝐿 (,) 1 ) ↔ ( 𝑟 ∈ ℝ ∧ 𝐿 < 𝑟 ∧ 𝑟 < 1 ) ) ) | |
| 45 | 42 43 44 | sylancl | ⊢ ( 𝜑 → ( 𝑟 ∈ ( 𝐿 (,) 1 ) ↔ ( 𝑟 ∈ ℝ ∧ 𝐿 < 𝑟 ∧ 𝑟 < 1 ) ) ) |
| 46 | 45 | biimpa | ⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) → ( 𝑟 ∈ ℝ ∧ 𝐿 < 𝑟 ∧ 𝑟 < 1 ) ) |
| 47 | 46 | simp3d | ⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) → 𝑟 < 1 ) |
| 48 | 47 | ad2antrr | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → 𝑟 < 1 ) |
| 49 | simplr | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → 𝑛 ∈ 𝑊 ) | |
| 50 | 34 | ex | ⊢ ( 𝜑 → ( 𝑖 ∈ 𝑊 → ( 𝐹 ‘ 𝑖 ) ∈ ℂ ) ) |
| 51 | 50 | ad3antrrr | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( 𝑖 ∈ 𝑊 → ( 𝐹 ‘ 𝑖 ) ∈ ℂ ) ) |
| 52 | 51 | imp | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ∧ 𝑖 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑖 ) ∈ ℂ ) |
| 53 | fvoveq1 | ⊢ ( 𝑘 = 𝑖 → ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) | |
| 54 | 53 | fveq2d | ⊢ ( 𝑘 = 𝑖 → ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) = ( abs ‘ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ) |
| 55 | 26 | fveq2d | ⊢ ( 𝑘 = 𝑖 → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) = ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) |
| 56 | 55 | oveq2d | ⊢ ( 𝑘 = 𝑖 → ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) = ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) |
| 57 | 54 56 | breq12d | ⊢ ( 𝑘 = 𝑖 → ( ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ ( abs ‘ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) |
| 58 | 57 | rspccva | ⊢ ( ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( abs ‘ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) |
| 59 | 58 | adantll | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( abs ‘ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) |
| 60 | 2 10 12 48 49 52 59 | cvgrat | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → seq 𝑁 ( + , 𝐹 ) ∈ dom ⇝ ) |
| 61 | 15 | adantr | ⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) → 𝑁 ∈ ℤ ) |
| 62 | 46 | simp2d | ⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) → 𝐿 < 𝑟 ) |
| 63 | difrp | ⊢ ( ( 𝐿 ∈ ℝ ∧ 𝑟 ∈ ℝ ) → ( 𝐿 < 𝑟 ↔ ( 𝑟 − 𝐿 ) ∈ ℝ+ ) ) | |
| 64 | 41 11 63 | syl2an | ⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) → ( 𝐿 < 𝑟 ↔ ( 𝑟 − 𝐿 ) ∈ ℝ+ ) ) |
| 65 | 62 64 | mpbid | ⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) → ( 𝑟 − 𝐿 ) ∈ ℝ+ ) |
| 66 | 39 | adantlr | ⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑘 ∈ 𝑊 ) → ( 𝑅 ‘ 𝑘 ) = ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ) |
| 67 | 8 | adantr | ⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) → 𝑅 ⇝ 𝐿 ) |
| 68 | 2 61 65 66 67 | climi2 | ⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) → ∃ 𝑛 ∈ 𝑊 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) |
| 69 | 2 | uztrn2 | ⊢ ( ( 𝑛 ∈ 𝑊 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑘 ∈ 𝑊 ) |
| 70 | 69 36 | sylan2 | ⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ 𝑊 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
| 71 | 70 | anassrs | ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
| 72 | 71 | adantllr | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
| 73 | 72 | adantr | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
| 74 | 73 | abscld | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ∈ ℝ ) |
| 75 | 11 | ad4antlr | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → 𝑟 ∈ ℝ ) |
| 76 | 69 33 | sylan2 | ⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ 𝑊 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
| 77 | 76 | anassrs | ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
| 78 | 77 | adantllr | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
| 79 | 78 | adantr | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
| 80 | 79 | abscld | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
| 81 | 75 80 | remulcld | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ∈ ℝ ) |
| 82 | 69 6 | sylan2 | ⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ 𝑊 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ) → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) |
| 83 | 82 | anassrs | ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) |
| 84 | 83 | adantllr | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) |
| 85 | 84 | adantr | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) |
| 86 | 73 79 85 | absdivd | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) = ( ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) / ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
| 87 | 72 78 84 | divcld | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ) |
| 88 | 87 | abscld | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ∈ ℝ ) |
| 89 | 41 | ad3antrrr | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝐿 ∈ ℝ ) |
| 90 | 88 89 | resubcld | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ∈ ℝ ) |
| 91 | 11 | ad3antlr | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑟 ∈ ℝ ) |
| 92 | 91 89 | resubcld | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝑟 − 𝐿 ) ∈ ℝ ) |
| 93 | 90 92 | absltd | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ↔ ( - ( 𝑟 − 𝐿 ) < ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ∧ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) < ( 𝑟 − 𝐿 ) ) ) ) |
| 94 | 93 | simplbda | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) < ( 𝑟 − 𝐿 ) ) |
| 95 | 73 79 85 | divcld | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ) |
| 96 | 95 | abscld | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ∈ ℝ ) |
| 97 | 41 | ad4antr | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → 𝐿 ∈ ℝ ) |
| 98 | 96 75 97 | ltsub1d | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) < 𝑟 ↔ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) < ( 𝑟 − 𝐿 ) ) ) |
| 99 | 94 98 | mpbird | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) < 𝑟 ) |
| 100 | 86 99 | eqbrtrrd | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) / ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) < 𝑟 ) |
| 101 | 79 85 | absrpcld | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ+ ) |
| 102 | 74 75 101 | ltdivmuld | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( ( ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) / ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) < 𝑟 ↔ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) < ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) · 𝑟 ) ) ) |
| 103 | 100 102 | mpbid | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) < ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) · 𝑟 ) ) |
| 104 | 101 | rpcnd | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ) |
| 105 | 75 | recnd | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → 𝑟 ∈ ℂ ) |
| 106 | 104 105 | mulcomd | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) · 𝑟 ) = ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
| 107 | 103 106 | breqtrd | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) < ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
| 108 | 74 81 107 | ltled | ⊢ ( ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) ) → ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
| 109 | 108 | ex | ⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) → ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
| 110 | 109 | ralimdva | ⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) ∧ 𝑛 ∈ 𝑊 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
| 111 | 110 | reximdva | ⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) → ( ∃ 𝑛 ∈ 𝑊 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝑟 − 𝐿 ) → ∃ 𝑛 ∈ 𝑊 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
| 112 | 68 111 | mpd | ⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) → ∃ 𝑛 ∈ 𝑊 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝑟 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
| 113 | 60 112 | r19.29a | ⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( 𝐿 (,) 1 ) ) → seq 𝑁 ( + , 𝐹 ) ∈ dom ⇝ ) |
| 114 | 113 | ralrimiva | ⊢ ( 𝜑 → ∀ 𝑟 ∈ ( 𝐿 (,) 1 ) seq 𝑁 ( + , 𝐹 ) ∈ dom ⇝ ) |
| 115 | 114 | adantr | ⊢ ( ( 𝜑 ∧ 𝐿 < 1 ) → ∀ 𝑟 ∈ ( 𝐿 (,) 1 ) seq 𝑁 ( + , 𝐹 ) ∈ dom ⇝ ) |
| 116 | ioon0 | ⊢ ( ( 𝐿 ∈ ℝ* ∧ 1 ∈ ℝ* ) → ( ( 𝐿 (,) 1 ) ≠ ∅ ↔ 𝐿 < 1 ) ) | |
| 117 | 42 43 116 | sylancl | ⊢ ( 𝜑 → ( ( 𝐿 (,) 1 ) ≠ ∅ ↔ 𝐿 < 1 ) ) |
| 118 | 117 | biimpar | ⊢ ( ( 𝜑 ∧ 𝐿 < 1 ) → ( 𝐿 (,) 1 ) ≠ ∅ ) |
| 119 | r19.3rzv | ⊢ ( ( 𝐿 (,) 1 ) ≠ ∅ → ( seq 𝑁 ( + , 𝐹 ) ∈ dom ⇝ ↔ ∀ 𝑟 ∈ ( 𝐿 (,) 1 ) seq 𝑁 ( + , 𝐹 ) ∈ dom ⇝ ) ) | |
| 120 | 118 119 | syl | ⊢ ( ( 𝜑 ∧ 𝐿 < 1 ) → ( seq 𝑁 ( + , 𝐹 ) ∈ dom ⇝ ↔ ∀ 𝑟 ∈ ( 𝐿 (,) 1 ) seq 𝑁 ( + , 𝐹 ) ∈ dom ⇝ ) ) |
| 121 | 115 120 | mpbird | ⊢ ( ( 𝜑 ∧ 𝐿 < 1 ) → seq 𝑁 ( + , 𝐹 ) ∈ dom ⇝ ) |
| 122 | 1 3 5 | iserex | ⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ↔ seq 𝑁 ( + , 𝐹 ) ∈ dom ⇝ ) ) |
| 123 | 122 | adantr | ⊢ ( ( 𝜑 ∧ 𝐿 < 1 ) → ( seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ↔ seq 𝑁 ( + , 𝐹 ) ∈ dom ⇝ ) ) |
| 124 | 121 123 | mpbird | ⊢ ( ( 𝜑 ∧ 𝐿 < 1 ) → seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) |
| 125 | 124 | ex | ⊢ ( 𝜑 → ( 𝐿 < 1 → seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) ) |
| 126 | 1red | ⊢ ( 𝜑 → 1 ∈ ℝ ) | |
| 127 | 41 126 | lttri2d | ⊢ ( 𝜑 → ( 𝐿 ≠ 1 ↔ ( 𝐿 < 1 ∨ 1 < 𝐿 ) ) ) |
| 128 | 9 127 | mpbid | ⊢ ( 𝜑 → ( 𝐿 < 1 ∨ 1 < 𝐿 ) ) |
| 129 | 128 | orcanai | ⊢ ( ( 𝜑 ∧ ¬ 𝐿 < 1 ) → 1 < 𝐿 ) |
| 130 | simplr | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝑛 ∈ 𝑊 ) | |
| 131 | 4 | ad3antrrr | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝐹 ∈ 𝑉 ) |
| 132 | 50 | ad3antrrr | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝑖 ∈ 𝑊 → ( 𝐹 ‘ 𝑖 ) ∈ ℂ ) ) |
| 133 | 132 | imp | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ∧ 𝑖 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑖 ) ∈ ℂ ) |
| 134 | 2 | uztrn2 | ⊢ ( ( 𝑛 ∈ 𝑊 ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑖 ∈ 𝑊 ) |
| 135 | 26 | neeq1d | ⊢ ( 𝑘 = 𝑖 → ( ( 𝐹 ‘ 𝑘 ) ≠ 0 ↔ ( 𝐹 ‘ 𝑖 ) ≠ 0 ) ) |
| 136 | 25 135 | imbi12d | ⊢ ( 𝑘 = 𝑖 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑖 ) ≠ 0 ) ) ) |
| 137 | 136 6 | chvarvv | ⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑖 ) ≠ 0 ) |
| 138 | 134 137 | sylan2 | ⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ 𝑊 ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) ) → ( 𝐹 ‘ 𝑖 ) ≠ 0 ) |
| 139 | 138 | anassrs | ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑖 ) ≠ 0 ) |
| 140 | 139 | adantllr | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑖 ) ≠ 0 ) |
| 141 | 140 | adantlr | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑖 ) ≠ 0 ) |
| 142 | 55 54 | breq12d | ⊢ ( 𝑘 = 𝑖 → ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ↔ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ) ) |
| 143 | 142 | rspccva | ⊢ ( ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ) |
| 144 | 143 | adantll | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ) |
| 145 | 2 10 130 131 133 141 144 | dvgrat | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → seq 𝑁 ( + , 𝐹 ) ∉ dom ⇝ ) |
| 146 | 15 | adantr | ⊢ ( ( 𝜑 ∧ 1 < 𝐿 ) → 𝑁 ∈ ℤ ) |
| 147 | 1re | ⊢ 1 ∈ ℝ | |
| 148 | difrp | ⊢ ( ( 1 ∈ ℝ ∧ 𝐿 ∈ ℝ ) → ( 1 < 𝐿 ↔ ( 𝐿 − 1 ) ∈ ℝ+ ) ) | |
| 149 | 147 41 148 | sylancr | ⊢ ( 𝜑 → ( 1 < 𝐿 ↔ ( 𝐿 − 1 ) ∈ ℝ+ ) ) |
| 150 | 149 | biimpa | ⊢ ( ( 𝜑 ∧ 1 < 𝐿 ) → ( 𝐿 − 1 ) ∈ ℝ+ ) |
| 151 | 39 | adantlr | ⊢ ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑘 ∈ 𝑊 ) → ( 𝑅 ‘ 𝑘 ) = ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ) |
| 152 | 8 | adantr | ⊢ ( ( 𝜑 ∧ 1 < 𝐿 ) → 𝑅 ⇝ 𝐿 ) |
| 153 | 2 146 150 151 152 | climi2 | ⊢ ( ( 𝜑 ∧ 1 < 𝐿 ) → ∃ 𝑛 ∈ 𝑊 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) |
| 154 | 77 | adantllr | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
| 155 | 154 | adantr | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
| 156 | 155 | abscld | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
| 157 | 71 | adantllr | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
| 158 | 157 | adantr | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
| 159 | 158 | abscld | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ∈ ℝ ) |
| 160 | 83 | adantllr | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) |
| 161 | 160 | adantr | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) |
| 162 | 155 161 | absrpcld | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ+ ) |
| 163 | 162 | rpcnd | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ) |
| 164 | 163 | mullidd | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( 1 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) = ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
| 165 | 41 | ad4antr | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → 𝐿 ∈ ℝ ) |
| 166 | 165 | recnd | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → 𝐿 ∈ ℂ ) |
| 167 | 1cnd | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → 1 ∈ ℂ ) | |
| 168 | 166 167 | negsubdi2d | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → - ( 𝐿 − 1 ) = ( 1 − 𝐿 ) ) |
| 169 | 157 154 160 | divcld | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ) |
| 170 | 169 | abscld | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ∈ ℝ ) |
| 171 | 41 | ad3antrrr | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝐿 ∈ ℝ ) |
| 172 | 170 171 | resubcld | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ∈ ℝ ) |
| 173 | 1red | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 1 ∈ ℝ ) | |
| 174 | 171 173 | resubcld | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐿 − 1 ) ∈ ℝ ) |
| 175 | 172 174 | absltd | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ↔ ( - ( 𝐿 − 1 ) < ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ∧ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) < ( 𝐿 − 1 ) ) ) ) |
| 176 | 175 | simprbda | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → - ( 𝐿 − 1 ) < ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) |
| 177 | 168 176 | eqbrtrrd | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( 1 − 𝐿 ) < ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) |
| 178 | 1red | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → 1 ∈ ℝ ) | |
| 179 | 158 155 161 | divcld | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ) |
| 180 | 179 | abscld | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ∈ ℝ ) |
| 181 | 178 180 165 | ltsub1d | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( 1 < ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ↔ ( 1 − 𝐿 ) < ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) ) |
| 182 | 177 181 | mpbird | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → 1 < ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ) |
| 183 | 158 155 161 | absdivd | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) = ( ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) / ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
| 184 | 182 183 | breqtrd | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → 1 < ( ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) / ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
| 185 | 178 159 162 | ltmuldivd | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( ( 1 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) < ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ↔ 1 < ( ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) / ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
| 186 | 184 185 | mpbird | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( 1 · ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) < ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
| 187 | 164 186 | eqbrtrrd | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
| 188 | 156 159 187 | ltled | ⊢ ( ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) ∧ ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
| 189 | 188 | ex | ⊢ ( ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) |
| 190 | 189 | ralimdva | ⊢ ( ( ( 𝜑 ∧ 1 < 𝐿 ) ∧ 𝑛 ∈ 𝑊 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) |
| 191 | 190 | reximdva | ⊢ ( ( 𝜑 ∧ 1 < 𝐿 ) → ( ∃ 𝑛 ∈ 𝑊 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) − 𝐿 ) ) < ( 𝐿 − 1 ) → ∃ 𝑛 ∈ 𝑊 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) |
| 192 | 153 191 | mpd | ⊢ ( ( 𝜑 ∧ 1 < 𝐿 ) → ∃ 𝑛 ∈ 𝑊 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
| 193 | 145 192 | r19.29a | ⊢ ( ( 𝜑 ∧ 1 < 𝐿 ) → seq 𝑁 ( + , 𝐹 ) ∉ dom ⇝ ) |
| 194 | df-nel | ⊢ ( seq 𝑁 ( + , 𝐹 ) ∉ dom ⇝ ↔ ¬ seq 𝑁 ( + , 𝐹 ) ∈ dom ⇝ ) | |
| 195 | 193 194 | sylib | ⊢ ( ( 𝜑 ∧ 1 < 𝐿 ) → ¬ seq 𝑁 ( + , 𝐹 ) ∈ dom ⇝ ) |
| 196 | 122 | adantr | ⊢ ( ( 𝜑 ∧ 1 < 𝐿 ) → ( seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ↔ seq 𝑁 ( + , 𝐹 ) ∈ dom ⇝ ) ) |
| 197 | 195 196 | mtbird | ⊢ ( ( 𝜑 ∧ 1 < 𝐿 ) → ¬ seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) |
| 198 | 129 197 | syldan | ⊢ ( ( 𝜑 ∧ ¬ 𝐿 < 1 ) → ¬ seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) |
| 199 | 198 | ex | ⊢ ( 𝜑 → ( ¬ 𝐿 < 1 → ¬ seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) ) |
| 200 | 125 199 | impcon4bid | ⊢ ( 𝜑 → ( 𝐿 < 1 ↔ seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) ) |