Proof of Theorem radcnvrat
| Step | Hyp | Ref
| Expression |
| 1 | | radcnvrat.r |
. 2
⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*,
< ) |
| 2 | | xrltso 13183 |
. . . 4
⊢ < Or
ℝ* |
| 3 | 2 | a1i 11 |
. . 3
⊢ (𝜑 → < Or
ℝ*) |
| 4 | | radcnvrat.z |
. . . . . 6
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 5 | | radcnvrat.m |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
| 6 | 5 | nn0zd 12639 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 7 | 4 | reseq2i 5994 |
. . . . . . 7
⊢ (𝐷 ↾ 𝑍) = (𝐷 ↾ (ℤ≥‘𝑀)) |
| 8 | | radcnvrat.l |
. . . . . . . 8
⊢ (𝜑 → 𝐷 ⇝ 𝐿) |
| 9 | | radcnvrat.rat |
. . . . . . . . . 10
⊢ 𝐷 = (𝑘 ∈ ℕ0 ↦
(abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)))) |
| 10 | | nn0ex 12532 |
. . . . . . . . . . 11
⊢
ℕ0 ∈ V |
| 11 | 10 | mptex 7243 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ0
↦ (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)))) ∈ V |
| 12 | 9, 11 | eqeltri 2837 |
. . . . . . . . 9
⊢ 𝐷 ∈ V |
| 13 | | climres 15611 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℤ ∧ 𝐷 ∈ V) → ((𝐷 ↾
(ℤ≥‘𝑀)) ⇝ 𝐿 ↔ 𝐷 ⇝ 𝐿)) |
| 14 | 6, 12, 13 | sylancl 586 |
. . . . . . . 8
⊢ (𝜑 → ((𝐷 ↾ (ℤ≥‘𝑀)) ⇝ 𝐿 ↔ 𝐷 ⇝ 𝐿)) |
| 15 | 8, 14 | mpbird 257 |
. . . . . . 7
⊢ (𝜑 → (𝐷 ↾ (ℤ≥‘𝑀)) ⇝ 𝐿) |
| 16 | 7, 15 | eqbrtrid 5178 |
. . . . . 6
⊢ (𝜑 → (𝐷 ↾ 𝑍) ⇝ 𝐿) |
| 17 | 9 | reseq1i 5993 |
. . . . . . . . 9
⊢ (𝐷 ↾ 𝑍) = ((𝑘 ∈ ℕ0 ↦
(abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)))) ↾ 𝑍) |
| 18 | | eluznn0 12959 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
(ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0) |
| 19 | 5, 18 | sylan 580 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0) |
| 20 | 19 | ex 412 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈
ℕ0)) |
| 21 | 20 | ssrdv 3989 |
. . . . . . . . . . 11
⊢ (𝜑 →
(ℤ≥‘𝑀) ⊆
ℕ0) |
| 22 | 4, 21 | eqsstrid 4022 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑍 ⊆
ℕ0) |
| 23 | 22 | resmptd 6058 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦
(abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)))) ↾ 𝑍) = (𝑘 ∈ 𝑍 ↦ (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘))))) |
| 24 | 17, 23 | eqtrid 2789 |
. . . . . . . 8
⊢ (𝜑 → (𝐷 ↾ 𝑍) = (𝑘 ∈ 𝑍 ↦ (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘))))) |
| 25 | | fvexd 6921 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘))) ∈ V) |
| 26 | 24, 25 | fvmpt2d 7029 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐷 ↾ 𝑍)‘𝑘) = (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)))) |
| 27 | 4 | peano2uzs 12944 |
. . . . . . . . . 10
⊢ (𝑘 ∈ 𝑍 → (𝑘 + 1) ∈ 𝑍) |
| 28 | 22 | sselda 3983 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑘 + 1) ∈ 𝑍) → (𝑘 + 1) ∈
ℕ0) |
| 29 | | radcnvrat.a |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
| 30 | 29 | ffvelcdmda 7104 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑘 + 1) ∈ ℕ0) →
(𝐴‘(𝑘 + 1)) ∈
ℂ) |
| 31 | 28, 30 | syldan 591 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑘 + 1) ∈ 𝑍) → (𝐴‘(𝑘 + 1)) ∈ ℂ) |
| 32 | 27, 31 | sylan2 593 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴‘(𝑘 + 1)) ∈ ℂ) |
| 33 | 22 | sselda 3983 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℕ0) |
| 34 | 29 | ffvelcdmda 7104 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
| 35 | 33, 34 | syldan 591 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴‘𝑘) ∈ ℂ) |
| 36 | | radcnvrat.n0 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴‘𝑘) ≠ 0) |
| 37 | 32, 35, 36 | divcld 12043 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)) ∈ ℂ) |
| 38 | 37 | abscld 15475 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘))) ∈ ℝ) |
| 39 | 26, 38 | eqeltrd 2841 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐷 ↾ 𝑍)‘𝑘) ∈ ℝ) |
| 40 | 4, 6, 16, 39 | climrecl 15619 |
. . . . 5
⊢ (𝜑 → 𝐿 ∈ ℝ) |
| 41 | | radcnvrat.ln0 |
. . . . 5
⊢ (𝜑 → 𝐿 ≠ 0) |
| 42 | 40, 41 | rereccld 12094 |
. . . 4
⊢ (𝜑 → (1 / 𝐿) ∈ ℝ) |
| 43 | 42 | rexrd 11311 |
. . 3
⊢ (𝜑 → (1 / 𝐿) ∈
ℝ*) |
| 44 | | simpr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) → 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
| 45 | | elrabi 3687 |
. . . . 5
⊢ (𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } → 𝑥 ∈
ℝ) |
| 46 | 42 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (1 / 𝐿) ∈
ℝ) |
| 47 | | recn 11245 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℂ) |
| 48 | 47 | abscld 15475 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ →
(abs‘𝑥) ∈
ℝ) |
| 49 | 48 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (abs‘𝑥) ∈
ℝ) |
| 50 | 46, 49 | ltlend 11406 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((1 / 𝐿) < (abs‘𝑥) ↔ ((1 / 𝐿) ≤ (abs‘𝑥) ∧ (abs‘𝑥) ≠ (1 / 𝐿)))) |
| 51 | 50 | simplbda 499 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (1 / 𝐿) < (abs‘𝑥)) → (abs‘𝑥) ≠ (1 / 𝐿)) |
| 52 | 50 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) ≠ (1 / 𝐿)) → ((1 / 𝐿) < (abs‘𝑥) ↔ ((1 / 𝐿) ≤ (abs‘𝑥) ∧ (abs‘𝑥) ≠ (1 / 𝐿)))) |
| 53 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) ≠ (1 / 𝐿)) → (abs‘𝑥) ≠ (1 / 𝐿)) |
| 54 | 53 | biantrud 531 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) ≠ (1 / 𝐿)) → ((1 / 𝐿) ≤ (abs‘𝑥) ↔ ((1 / 𝐿) ≤ (abs‘𝑥) ∧ (abs‘𝑥) ≠ (1 / 𝐿)))) |
| 55 | 46, 49 | lenltd 11407 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((1 / 𝐿) ≤ (abs‘𝑥) ↔ ¬ (abs‘𝑥) < (1 / 𝐿))) |
| 56 | 55 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) ≠ (1 / 𝐿)) → ((1 / 𝐿) ≤ (abs‘𝑥) ↔ ¬ (abs‘𝑥) < (1 / 𝐿))) |
| 57 | 52, 54, 56 | 3bitr2d 307 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) ≠ (1 / 𝐿)) → ((1 / 𝐿) < (abs‘𝑥) ↔ ¬ (abs‘𝑥) < (1 / 𝐿))) |
| 58 | | 1cnd 11256 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈
ℂ) |
| 59 | 49 | recnd 11289 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (abs‘𝑥) ∈
ℂ) |
| 60 | 40 | recnd 11289 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐿 ∈ ℂ) |
| 61 | 60 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐿 ∈ ℂ) |
| 62 | 41 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐿 ≠ 0) |
| 63 | 58, 59, 61, 62 | divmul3d 12077 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((1 / 𝐿) = (abs‘𝑥) ↔ 1 = ((abs‘𝑥) · 𝐿))) |
| 64 | | eqcom 2744 |
. . . . . . . . . . . . . . . . 17
⊢ ((1 /
𝐿) = (abs‘𝑥) ↔ (abs‘𝑥) = (1 / 𝐿)) |
| 65 | | eqcom 2744 |
. . . . . . . . . . . . . . . . 17
⊢ (1 =
((abs‘𝑥) ·
𝐿) ↔ ((abs‘𝑥) · 𝐿) = 1) |
| 66 | 63, 64, 65 | 3bitr3g 313 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((abs‘𝑥) = (1 / 𝐿) ↔ ((abs‘𝑥) · 𝐿) = 1)) |
| 67 | 66 | necon3bid 2985 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((abs‘𝑥) ≠ (1 / 𝐿) ↔ ((abs‘𝑥) · 𝐿) ≠ 1)) |
| 68 | 67 | biimpa 476 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) ≠ (1 / 𝐿)) → ((abs‘𝑥) · 𝐿) ≠ 1) |
| 69 | | 1red 11262 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈
ℝ) |
| 70 | | fvres 6925 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ 𝑍 → ((𝐷 ↾ 𝑍)‘𝑘) = (𝐷‘𝑘)) |
| 71 | 70 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐷 ↾ 𝑍)‘𝑘) = (𝐷‘𝑘)) |
| 72 | 71, 39 | eqeltrrd 2842 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐷‘𝑘) ∈ ℝ) |
| 73 | 37 | absge0d 15483 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)))) |
| 74 | 73, 26 | breqtrrd 5171 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ ((𝐷 ↾ 𝑍)‘𝑘)) |
| 75 | 74, 71 | breqtrd 5169 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐷‘𝑘)) |
| 76 | 4, 6, 8, 72, 75 | climge0 15620 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 0 ≤ 𝐿) |
| 77 | 40, 76, 41 | ne0gt0d 11398 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 0 < 𝐿) |
| 78 | 40, 77 | elrpd 13074 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐿 ∈
ℝ+) |
| 79 | 78 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐿 ∈
ℝ+) |
| 80 | 49, 69, 79 | ltmuldivd 13124 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (((abs‘𝑥) · 𝐿) < 1 ↔ (abs‘𝑥) < (1 / 𝐿))) |
| 81 | 80 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ((abs‘𝑥) · 𝐿) ≠ 1) → (((abs‘𝑥) · 𝐿) < 1 ↔ (abs‘𝑥) < (1 / 𝐿))) |
| 82 | | elun 4153 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ((ℝ ∩ {0})
∪ (ℝ ∖ {0})) ↔ (𝑥 ∈ (ℝ ∩ {0}) ∨ 𝑥 ∈ (ℝ ∖
{0}))) |
| 83 | | inundif 4479 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((ℝ
∩ {0}) ∪ (ℝ ∖ {0})) = ℝ |
| 84 | 83 | eleq2i 2833 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ((ℝ ∩ {0})
∪ (ℝ ∖ {0})) ↔ 𝑥 ∈ ℝ) |
| 85 | 82, 84 | bitr3i 277 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ (ℝ ∩ {0}) ∨
𝑥 ∈ (ℝ ∖
{0})) ↔ 𝑥 ∈
ℝ) |
| 86 | | elin 3967 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ (ℝ ∩ {0})
↔ (𝑥 ∈ ℝ
∧ 𝑥 ∈
{0})) |
| 87 | 86 | simprbi 496 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (ℝ ∩ {0})
→ 𝑥 ∈
{0}) |
| 88 | | elsni 4643 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ {0} → 𝑥 = 0) |
| 89 | 87, 88 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (ℝ ∩ {0})
→ 𝑥 =
0) |
| 90 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = 0 → (abs‘𝑥) =
(abs‘0)) |
| 91 | | abs0 15324 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(abs‘0) = 0 |
| 92 | 90, 91 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 0 → (abs‘𝑥) = 0) |
| 93 | 92 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 0 → ((abs‘𝑥) · 𝐿) = (0 · 𝐿)) |
| 94 | 60 | mul02d 11459 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (0 · 𝐿) = 0) |
| 95 | 93, 94 | sylan9eqr 2799 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 = 0) → ((abs‘𝑥) · 𝐿) = 0) |
| 96 | | 0lt1 11785 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 0 <
1 |
| 97 | 95, 96 | eqbrtrdi 5182 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 = 0) → ((abs‘𝑥) · 𝐿) < 1) |
| 98 | | radcnvrat.g |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) |
| 99 | 98, 29 | radcnv0 26459 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 0 ∈ {𝑟 ∈ ℝ ∣ seq0( +
, (𝐺‘𝑟)) ∈ dom ⇝
}) |
| 100 | | eleq1 2829 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 0 → (𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } ↔ 0 ∈
{𝑟 ∈ ℝ ∣
seq0( + , (𝐺‘𝑟)) ∈ dom ⇝
})) |
| 101 | 99, 100 | syl5ibrcom 247 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑥 = 0 → 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 102 | 101 | imp 406 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 = 0) → 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
| 103 | 97, 102 | 2thd 265 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 = 0) → (((abs‘𝑥) · 𝐿) < 1 ↔ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 104 | 89, 103 | sylan2 593 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∩ {0})) →
(((abs‘𝑥) ·
𝐿) < 1 ↔ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 105 | 104 | adantlr 715 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ((abs‘𝑥) · 𝐿) ≠ 1) ∧ 𝑥 ∈ (ℝ ∩ {0})) →
(((abs‘𝑥) ·
𝐿) < 1 ↔ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 106 | | ax-resscn 11212 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ℝ
⊆ ℂ |
| 107 | | ssdif 4144 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (ℝ
⊆ ℂ → (ℝ ∖ {0}) ⊆ (ℂ ∖
{0})) |
| 108 | 106, 107 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (ℝ
∖ {0}) ⊆ (ℂ ∖ {0}) |
| 109 | 108 | sseli 3979 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (ℝ ∖ {0})
→ 𝑥 ∈ (ℂ
∖ {0})) |
| 110 | | nn0uz 12920 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
ℕ0 = (ℤ≥‘0) |
| 111 | 5 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧
((abs‘𝑥) ·
𝐿) ≠ 1) → 𝑀 ∈
ℕ0) |
| 112 | | fvexd 6921 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧
((abs‘𝑥) ·
𝐿) ≠ 1) → (𝐺‘𝑥) ∈ V) |
| 113 | | eldifi 4131 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 ∈ (ℂ ∖ {0})
→ 𝑥 ∈
ℂ) |
| 114 | 98 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))) |
| 115 | 10 | mptex 7243 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑛 ∈ ℕ0
↦ ((𝐴‘𝑛) · (𝑥↑𝑛))) ∈ V |
| 116 | 115 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))) ∈ V) |
| 117 | 114, 116 | fvmpt2d 7029 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) = (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) |
| 118 | 117 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑥) = (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) |
| 119 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 = 𝑘 → (𝐴‘𝑛) = (𝐴‘𝑘)) |
| 120 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 = 𝑘 → (𝑥↑𝑛) = (𝑥↑𝑘)) |
| 121 | 119, 120 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 = 𝑘 → ((𝐴‘𝑛) · (𝑥↑𝑛)) = ((𝐴‘𝑘) · (𝑥↑𝑘))) |
| 122 | 121 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) ∧ 𝑛 = 𝑘) → ((𝐴‘𝑛) · (𝑥↑𝑛)) = ((𝐴‘𝑘) · (𝑥↑𝑘))) |
| 123 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℕ0) |
| 124 | | ovexd 7466 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ V) |
| 125 | 118, 122,
123, 124 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐺‘𝑥)‘𝑘) = ((𝐴‘𝑘) · (𝑥↑𝑘))) |
| 126 | 34 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
| 127 | | simplr 769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → 𝑥 ∈
ℂ) |
| 128 | 127, 123 | expcld 14186 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑥↑𝑘) ∈ ℂ) |
| 129 | 126, 128 | mulcld 11281 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ) |
| 130 | 125, 129 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐺‘𝑥)‘𝑘) ∈ ℂ) |
| 131 | 113, 130 | sylanl2 681 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ ℕ0)
→ ((𝐺‘𝑥)‘𝑘) ∈ ℂ) |
| 132 | 131 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧
((abs‘𝑥) ·
𝐿) ≠ 1) ∧ 𝑘 ∈ ℕ0)
→ ((𝐺‘𝑥)‘𝑘) ∈ ℂ) |
| 133 | 33 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℕ0) |
| 134 | 133, 125 | syldan 591 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑥)‘𝑘) = ((𝐴‘𝑘) · (𝑥↑𝑘))) |
| 135 | 113, 134 | sylanl2 681 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑥)‘𝑘) = ((𝐴‘𝑘) · (𝑥↑𝑘))) |
| 136 | 35 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (𝐴‘𝑘) ∈ ℂ) |
| 137 | 113 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈
ℂ) |
| 138 | 137 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → 𝑥 ∈ ℂ) |
| 139 | 33 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℕ0) |
| 140 | 138, 139 | expcld 14186 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (𝑥↑𝑘) ∈ ℂ) |
| 141 | 36 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (𝐴‘𝑘) ≠ 0) |
| 142 | | eldifsni 4790 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 ∈ (ℂ ∖ {0})
→ 𝑥 ≠
0) |
| 143 | 142 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → 𝑥 ≠ 0) |
| 144 | 139 | nn0zd 12639 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℤ) |
| 145 | 138, 143,
144 | expne0d 14192 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (𝑥↑𝑘) ≠ 0) |
| 146 | 136, 140,
141, 145 | mulne0d 11915 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ≠ 0) |
| 147 | 135, 146 | eqnetrd 3008 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑥)‘𝑘) ≠ 0) |
| 148 | 147 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧
((abs‘𝑥) ·
𝐿) ≠ 1) ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑥)‘𝑘) ≠ 0) |
| 149 | | fvoveq1 7454 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 = 𝑘 → ((𝐺‘𝑥)‘(𝑛 + 1)) = ((𝐺‘𝑥)‘(𝑘 + 1))) |
| 150 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 = 𝑘 → ((𝐺‘𝑥)‘𝑛) = ((𝐺‘𝑥)‘𝑘)) |
| 151 | 149, 150 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝑘 → (((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)) = (((𝐺‘𝑥)‘(𝑘 + 1)) / ((𝐺‘𝑥)‘𝑘))) |
| 152 | 151 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑘 → (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛))) = (abs‘(((𝐺‘𝑥)‘(𝑘 + 1)) / ((𝐺‘𝑥)‘𝑘)))) |
| 153 | 152 | cbvmptv 5255 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 ∈ 𝑍 ↦ (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) = (𝑘 ∈ 𝑍 ↦ (abs‘(((𝐺‘𝑥)‘(𝑘 + 1)) / ((𝐺‘𝑥)‘𝑘)))) |
| 154 | 4 | reseq2i 5994 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑛 ∈ ℕ0
↦ (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ↾ 𝑍) = ((𝑛 ∈ ℕ0 ↦
(abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ↾
(ℤ≥‘𝑀)) |
| 155 | 22 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑍 ⊆
ℕ0) |
| 156 | 155 | resmptd 6058 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((𝑛 ∈ ℕ0
↦ (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ↾ 𝑍) = (𝑛 ∈ 𝑍 ↦ (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛))))) |
| 157 | 154, 156 | eqtr3id 2791 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((𝑛 ∈ ℕ0
↦ (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ↾
(ℤ≥‘𝑀)) = (𝑛 ∈ 𝑍 ↦ (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛))))) |
| 158 | 6 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑀 ∈
ℤ) |
| 159 | 8 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝐷 ⇝ 𝐿) |
| 160 | 137 | abscld 15475 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) →
(abs‘𝑥) ∈
ℝ) |
| 161 | 160 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) →
(abs‘𝑥) ∈
ℂ) |
| 162 | 10 | mptex 7243 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 ∈ ℕ0
↦ (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ∈ V |
| 163 | 162 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑛 ∈ ℕ0
↦ (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ∈ V) |
| 164 | 72 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐷‘𝑘) ∈ ℂ) |
| 165 | 164 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (𝐷‘𝑘) ∈ ℂ) |
| 166 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (𝑛 ∈ ℕ0 ↦
(abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) = (𝑛 ∈ ℕ0 ↦
(abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛))))) |
| 167 | 152 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) ∧ 𝑛 = 𝑘) → (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛))) = (abs‘(((𝐺‘𝑥)‘(𝑘 + 1)) / ((𝐺‘𝑥)‘𝑘)))) |
| 168 | | fvexd 6921 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (abs‘(((𝐺‘𝑥)‘(𝑘 + 1)) / ((𝐺‘𝑥)‘𝑘))) ∈ V) |
| 169 | 166, 167,
139, 168 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → ((𝑛 ∈ ℕ0 ↦
(abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛))))‘𝑘) = (abs‘(((𝐺‘𝑥)‘(𝑘 + 1)) / ((𝐺‘𝑥)‘𝑘)))) |
| 170 | 117 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑥) = (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) |
| 171 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑛 = (𝑘 + 1)) → 𝑛 = (𝑘 + 1)) |
| 172 | 171 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑛 = (𝑘 + 1)) → (𝐴‘𝑛) = (𝐴‘(𝑘 + 1))) |
| 173 | 171 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑛 = (𝑘 + 1)) → (𝑥↑𝑛) = (𝑥↑(𝑘 + 1))) |
| 174 | 172, 173 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑛 = (𝑘 + 1)) → ((𝐴‘𝑛) · (𝑥↑𝑛)) = ((𝐴‘(𝑘 + 1)) · (𝑥↑(𝑘 + 1)))) |
| 175 | | 1nn0 12542 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ 1 ∈
ℕ0 |
| 176 | 175 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) → 1 ∈
ℕ0) |
| 177 | 133, 176 | nn0addcld 12591 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) → (𝑘 + 1) ∈
ℕ0) |
| 178 | | ovexd 7466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) → ((𝐴‘(𝑘 + 1)) · (𝑥↑(𝑘 + 1))) ∈ V) |
| 179 | 170, 174,
177, 178 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑥)‘(𝑘 + 1)) = ((𝐴‘(𝑘 + 1)) · (𝑥↑(𝑘 + 1)))) |
| 180 | 121 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑛 = 𝑘) → ((𝐴‘𝑛) · (𝑥↑𝑛)) = ((𝐴‘𝑘) · (𝑥↑𝑘))) |
| 181 | | ovexd 7466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ V) |
| 182 | 170, 180,
133, 181 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑥)‘𝑘) = ((𝐴‘𝑘) · (𝑥↑𝑘))) |
| 183 | 179, 182 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝑍) → (((𝐺‘𝑥)‘(𝑘 + 1)) / ((𝐺‘𝑥)‘𝑘)) = (((𝐴‘(𝑘 + 1)) · (𝑥↑(𝑘 + 1))) / ((𝐴‘𝑘) · (𝑥↑𝑘)))) |
| 184 | 113, 183 | sylanl2 681 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (((𝐺‘𝑥)‘(𝑘 + 1)) / ((𝐺‘𝑥)‘𝑘)) = (((𝐴‘(𝑘 + 1)) · (𝑥↑(𝑘 + 1))) / ((𝐴‘𝑘) · (𝑥↑𝑘)))) |
| 185 | 32 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (𝐴‘(𝑘 + 1)) ∈ ℂ) |
| 186 | 113, 177 | sylanl2 681 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (𝑘 + 1) ∈
ℕ0) |
| 187 | 138, 186 | expcld 14186 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (𝑥↑(𝑘 + 1)) ∈ ℂ) |
| 188 | 185, 136,
187, 140, 141, 145 | divmuldivd 12084 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)) · ((𝑥↑(𝑘 + 1)) / (𝑥↑𝑘))) = (((𝐴‘(𝑘 + 1)) · (𝑥↑(𝑘 + 1))) / ((𝐴‘𝑘) · (𝑥↑𝑘)))) |
| 189 | 139 | nn0cnd 12589 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℂ) |
| 190 | | 1cnd 11256 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → 1 ∈ ℂ) |
| 191 | 189, 190 | pncan2d 11622 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → ((𝑘 + 1) − 𝑘) = 1) |
| 192 | 191 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (𝑥↑((𝑘 + 1) − 𝑘)) = (𝑥↑1)) |
| 193 | 186 | nn0zd 12639 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (𝑘 + 1) ∈ ℤ) |
| 194 | 138, 143,
144, 193 | expsubd 14197 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (𝑥↑((𝑘 + 1) − 𝑘)) = ((𝑥↑(𝑘 + 1)) / (𝑥↑𝑘))) |
| 195 | 138 | exp1d 14181 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (𝑥↑1) = 𝑥) |
| 196 | 192, 194,
195 | 3eqtr3d 2785 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → ((𝑥↑(𝑘 + 1)) / (𝑥↑𝑘)) = 𝑥) |
| 197 | 196 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)) · ((𝑥↑(𝑘 + 1)) / (𝑥↑𝑘))) = (((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)) · 𝑥)) |
| 198 | 184, 188,
197 | 3eqtr2d 2783 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (((𝐺‘𝑥)‘(𝑘 + 1)) / ((𝐺‘𝑥)‘𝑘)) = (((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)) · 𝑥)) |
| 199 | 198 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (abs‘(((𝐺‘𝑥)‘(𝑘 + 1)) / ((𝐺‘𝑥)‘𝑘))) = (abs‘(((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)) · 𝑥))) |
| 200 | 37 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → ((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)) ∈ ℂ) |
| 201 | 200, 138 | absmuld 15493 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (abs‘(((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)) · 𝑥)) = ((abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘))) · (abs‘𝑥))) |
| 202 | 169, 199,
201 | 3eqtrd 2781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → ((𝑛 ∈ ℕ0 ↦
(abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛))))‘𝑘) = ((abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘))) · (abs‘𝑥))) |
| 203 | 71, 26 | eqtr3d 2779 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐷‘𝑘) = (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)))) |
| 204 | 203 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (𝐷‘𝑘) = (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘)))) |
| 205 | 204 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘))) = (𝐷‘𝑘)) |
| 206 | 205 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → ((abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘))) · (abs‘𝑥)) = ((𝐷‘𝑘) · (abs‘𝑥))) |
| 207 | 161 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → (abs‘𝑥) ∈ ℂ) |
| 208 | 165, 207 | mulcomd 11282 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → ((𝐷‘𝑘) · (abs‘𝑥)) = ((abs‘𝑥) · (𝐷‘𝑘))) |
| 209 | 202, 206,
208 | 3eqtrd 2781 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧ 𝑘 ∈ 𝑍) → ((𝑛 ∈ ℕ0 ↦
(abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛))))‘𝑘) = ((abs‘𝑥) · (𝐷‘𝑘))) |
| 210 | 4, 158, 159, 161, 163, 165, 209 | climmulc2 15673 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑛 ∈ ℕ0
↦ (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ⇝ ((abs‘𝑥) · 𝐿)) |
| 211 | | climres 15611 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℕ0
↦ (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ∈ V) → (((𝑛 ∈ ℕ0 ↦
(abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ↾
(ℤ≥‘𝑀)) ⇝ ((abs‘𝑥) · 𝐿) ↔ (𝑛 ∈ ℕ0 ↦
(abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ⇝ ((abs‘𝑥) · 𝐿))) |
| 212 | 158, 162,
211 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) →
(((𝑛 ∈
ℕ0 ↦ (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ↾
(ℤ≥‘𝑀)) ⇝ ((abs‘𝑥) · 𝐿) ↔ (𝑛 ∈ ℕ0 ↦
(abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ⇝ ((abs‘𝑥) · 𝐿))) |
| 213 | 210, 212 | mpbird 257 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((𝑛 ∈ ℕ0
↦ (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ↾
(ℤ≥‘𝑀)) ⇝ ((abs‘𝑥) · 𝐿)) |
| 214 | 157, 213 | eqbrtrrd 5167 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑛 ∈ 𝑍 ↦ (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ⇝ ((abs‘𝑥) · 𝐿)) |
| 215 | 214 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧
((abs‘𝑥) ·
𝐿) ≠ 1) → (𝑛 ∈ 𝑍 ↦ (abs‘(((𝐺‘𝑥)‘(𝑛 + 1)) / ((𝐺‘𝑥)‘𝑛)))) ⇝ ((abs‘𝑥) · 𝐿)) |
| 216 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧
((abs‘𝑥) ·
𝐿) ≠ 1) →
((abs‘𝑥) ·
𝐿) ≠ 1) |
| 217 | 110, 4, 111, 112, 132, 148, 153, 215, 216 | cvgdvgrat 44332 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) ∧
((abs‘𝑥) ·
𝐿) ≠ 1) →
(((abs‘𝑥) ·
𝐿) < 1 ↔ seq0( + ,
(𝐺‘𝑥)) ∈ dom ⇝ )) |
| 218 | 109, 217 | sylanl2 681 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ {0})) ∧
((abs‘𝑥) ·
𝐿) ≠ 1) →
(((abs‘𝑥) ·
𝐿) < 1 ↔ seq0( + ,
(𝐺‘𝑥)) ∈ dom ⇝ )) |
| 219 | | eldifi 4131 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ (ℝ ∖ {0})
→ 𝑥 ∈
ℝ) |
| 220 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑟 = 𝑥 → (𝐺‘𝑟) = (𝐺‘𝑥)) |
| 221 | 220 | seqeq3d 14050 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑟 = 𝑥 → seq0( + , (𝐺‘𝑟)) = seq0( + , (𝐺‘𝑥))) |
| 222 | 221 | eleq1d 2826 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑟 = 𝑥 → (seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ ↔ seq0( + , (𝐺‘𝑥)) ∈ dom ⇝ )) |
| 223 | 222 | elrab3 3693 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ ℝ → (𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } ↔ seq0( + ,
(𝐺‘𝑥)) ∈ dom ⇝ )) |
| 224 | 219, 223 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (ℝ ∖ {0})
→ (𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( +
, (𝐺‘𝑟)) ∈ dom ⇝ } ↔
seq0( + , (𝐺‘𝑥)) ∈ dom ⇝
)) |
| 225 | 224 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ {0})) ∧
((abs‘𝑥) ·
𝐿) ≠ 1) → (𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } ↔ seq0( + ,
(𝐺‘𝑥)) ∈ dom ⇝ )) |
| 226 | 218, 225 | bitr4d 282 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ {0})) ∧
((abs‘𝑥) ·
𝐿) ≠ 1) →
(((abs‘𝑥) ·
𝐿) < 1 ↔ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 227 | 226 | an32s 652 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ((abs‘𝑥) · 𝐿) ≠ 1) ∧ 𝑥 ∈ (ℝ ∖ {0})) →
(((abs‘𝑥) ·
𝐿) < 1 ↔ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 228 | 105, 227 | jaodan 960 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ((abs‘𝑥) · 𝐿) ≠ 1) ∧ (𝑥 ∈ (ℝ ∩ {0}) ∨ 𝑥 ∈ (ℝ ∖ {0})))
→ (((abs‘𝑥)
· 𝐿) < 1 ↔
𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( +
, (𝐺‘𝑟)) ∈ dom ⇝
})) |
| 229 | 85, 228 | sylan2br 595 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ((abs‘𝑥) · 𝐿) ≠ 1) ∧ 𝑥 ∈ ℝ) → (((abs‘𝑥) · 𝐿) < 1 ↔ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 230 | 229 | an32s 652 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ((abs‘𝑥) · 𝐿) ≠ 1) → (((abs‘𝑥) · 𝐿) < 1 ↔ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 231 | 81, 230 | bitr3d 281 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ((abs‘𝑥) · 𝐿) ≠ 1) → ((abs‘𝑥) < (1 / 𝐿) ↔ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 232 | 68, 231 | syldan 591 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) ≠ (1 / 𝐿)) → ((abs‘𝑥) < (1 / 𝐿) ↔ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 233 | 232 | notbid 318 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) ≠ (1 / 𝐿)) → (¬ (abs‘𝑥) < (1 / 𝐿) ↔ ¬ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 234 | 57, 233 | bitrd 279 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) ≠ (1 / 𝐿)) → ((1 / 𝐿) < (abs‘𝑥) ↔ ¬ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 235 | 234 | biimpd 229 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) ≠ (1 / 𝐿)) → ((1 / 𝐿) < (abs‘𝑥) → ¬ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 236 | 235 | impancom 451 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (1 / 𝐿) < (abs‘𝑥)) → ((abs‘𝑥) ≠ (1 / 𝐿) → ¬ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 237 | 51, 236 | mpd 15 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (1 / 𝐿) < (abs‘𝑥)) → ¬ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
| 238 | 237 | ex 412 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((1 / 𝐿) < (abs‘𝑥) → ¬ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 239 | 238 | con2d 134 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } → ¬ (1 /
𝐿) < (abs‘𝑥))) |
| 240 | 46 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (1 / 𝐿) < 𝑥) → (1 / 𝐿) ∈ ℝ) |
| 241 | | simplr 769 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (1 / 𝐿) < 𝑥) → 𝑥 ∈ ℝ) |
| 242 | 49 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (1 / 𝐿) < 𝑥) → (abs‘𝑥) ∈ ℝ) |
| 243 | | simpr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (1 / 𝐿) < 𝑥) → (1 / 𝐿) < 𝑥) |
| 244 | 241 | leabsd 15453 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (1 / 𝐿) < 𝑥) → 𝑥 ≤ (abs‘𝑥)) |
| 245 | 240, 241,
242, 243, 244 | ltletrd 11421 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (1 / 𝐿) < 𝑥) → (1 / 𝐿) < (abs‘𝑥)) |
| 246 | 245 | ex 412 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((1 / 𝐿) < 𝑥 → (1 / 𝐿) < (abs‘𝑥))) |
| 247 | 239, 246 | nsyld 156 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } → ¬ (1 /
𝐿) < 𝑥)) |
| 248 | 45, 247 | sylan2 593 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) → (𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } → ¬ (1 /
𝐿) < 𝑥)) |
| 249 | 44, 248 | mpd 15 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) → ¬ (1 /
𝐿) < 𝑥) |
| 250 | 42 | renegcld 11690 |
. . . . . . . . 9
⊢ (𝜑 → -(1 / 𝐿) ∈ ℝ) |
| 251 | 250 | rexrd 11311 |
. . . . . . . 8
⊢ (𝜑 → -(1 / 𝐿) ∈
ℝ*) |
| 252 | | iooss1 13422 |
. . . . . . . 8
⊢ ((-(1 /
𝐿) ∈
ℝ* ∧ -(1 / 𝐿) ≤ 𝑥) → (𝑥(,)(1 / 𝐿)) ⊆ (-(1 / 𝐿)(,)(1 / 𝐿))) |
| 253 | 251, 252 | sylan 580 |
. . . . . . 7
⊢ ((𝜑 ∧ -(1 / 𝐿) ≤ 𝑥) → (𝑥(,)(1 / 𝐿)) ⊆ (-(1 / 𝐿)(,)(1 / 𝐿))) |
| 254 | 253 | adantlr 715 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < (1 / 𝐿))) ∧ -(1 / 𝐿) ≤ 𝑥) → (𝑥(,)(1 / 𝐿)) ⊆ (-(1 / 𝐿)(,)(1 / 𝐿))) |
| 255 | | eliooord 13446 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (𝑥(,)(1 / 𝐿)) → (𝑥 < 𝑘 ∧ 𝑘 < (1 / 𝐿))) |
| 256 | 255 | simpld 494 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (𝑥(,)(1 / 𝐿)) → 𝑥 < 𝑘) |
| 257 | 256 | rgen 3063 |
. . . . . . . . 9
⊢
∀𝑘 ∈
(𝑥(,)(1 / 𝐿))𝑥 < 𝑘 |
| 258 | | ioon0 13413 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ*
∧ (1 / 𝐿) ∈
ℝ*) → ((𝑥(,)(1 / 𝐿)) ≠ ∅ ↔ 𝑥 < (1 / 𝐿))) |
| 259 | 43, 258 | sylan2 593 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ*
∧ 𝜑) → ((𝑥(,)(1 / 𝐿)) ≠ ∅ ↔ 𝑥 < (1 / 𝐿))) |
| 260 | 259 | ancoms 458 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ*) → ((𝑥(,)(1 / 𝐿)) ≠ ∅ ↔ 𝑥 < (1 / 𝐿))) |
| 261 | 260 | biimpar 477 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ*) ∧ 𝑥 < (1 / 𝐿)) → (𝑥(,)(1 / 𝐿)) ≠ ∅) |
| 262 | | r19.2zb 4496 |
. . . . . . . . . 10
⊢ ((𝑥(,)(1 / 𝐿)) ≠ ∅ ↔ (∀𝑘 ∈ (𝑥(,)(1 / 𝐿))𝑥 < 𝑘 → ∃𝑘 ∈ (𝑥(,)(1 / 𝐿))𝑥 < 𝑘)) |
| 263 | 261, 262 | sylib 218 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ*) ∧ 𝑥 < (1 / 𝐿)) → (∀𝑘 ∈ (𝑥(,)(1 / 𝐿))𝑥 < 𝑘 → ∃𝑘 ∈ (𝑥(,)(1 / 𝐿))𝑥 < 𝑘)) |
| 264 | 257, 263 | mpi 20 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ*) ∧ 𝑥 < (1 / 𝐿)) → ∃𝑘 ∈ (𝑥(,)(1 / 𝐿))𝑥 < 𝑘) |
| 265 | 264 | anasss 466 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < (1 / 𝐿))) → ∃𝑘 ∈ (𝑥(,)(1 / 𝐿))𝑥 < 𝑘) |
| 266 | 265 | adantr 480 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < (1 / 𝐿))) ∧ -(1 / 𝐿) ≤ 𝑥) → ∃𝑘 ∈ (𝑥(,)(1 / 𝐿))𝑥 < 𝑘) |
| 267 | | ssrexv 4053 |
. . . . . 6
⊢ ((𝑥(,)(1 / 𝐿)) ⊆ (-(1 / 𝐿)(,)(1 / 𝐿)) → (∃𝑘 ∈ (𝑥(,)(1 / 𝐿))𝑥 < 𝑘 → ∃𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘)) |
| 268 | 254, 266,
267 | sylc 65 |
. . . . 5
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < (1 / 𝐿))) ∧ -(1 / 𝐿) ≤ 𝑥) → ∃𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘) |
| 269 | | simplr 769 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ*) ∧ ¬ -(1
/ 𝐿) ≤ 𝑥) → 𝑥 ∈ ℝ*) |
| 270 | | xrltnle 11328 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ*
∧ -(1 / 𝐿) ∈
ℝ*) → (𝑥 < -(1 / 𝐿) ↔ ¬ -(1 / 𝐿) ≤ 𝑥)) |
| 271 | | xrltle 13191 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ*
∧ -(1 / 𝐿) ∈
ℝ*) → (𝑥 < -(1 / 𝐿) → 𝑥 ≤ -(1 / 𝐿))) |
| 272 | 270, 271 | sylbird 260 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ*
∧ -(1 / 𝐿) ∈
ℝ*) → (¬ -(1 / 𝐿) ≤ 𝑥 → 𝑥 ≤ -(1 / 𝐿))) |
| 273 | 251, 272 | sylan2 593 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ*
∧ 𝜑) → (¬ -(1 /
𝐿) ≤ 𝑥 → 𝑥 ≤ -(1 / 𝐿))) |
| 274 | 273 | ancoms 458 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ*) → (¬
-(1 / 𝐿) ≤ 𝑥 → 𝑥 ≤ -(1 / 𝐿))) |
| 275 | 274 | imp 406 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ*) ∧ ¬ -(1
/ 𝐿) ≤ 𝑥) → 𝑥 ≤ -(1 / 𝐿)) |
| 276 | | iooss1 13422 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ*
∧ 𝑥 ≤ -(1 / 𝐿)) → (-(1 / 𝐿)(,)(1 / 𝐿)) ⊆ (𝑥(,)(1 / 𝐿))) |
| 277 | 269, 275,
276 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ*) ∧ ¬ -(1
/ 𝐿) ≤ 𝑥) → (-(1 / 𝐿)(,)(1 / 𝐿)) ⊆ (𝑥(,)(1 / 𝐿))) |
| 278 | 277 | sselda 3983 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ*) ∧ ¬ -(1
/ 𝐿) ≤ 𝑥) ∧ 𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))) → 𝑘 ∈ (𝑥(,)(1 / 𝐿))) |
| 279 | 278, 256 | syl 17 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ*) ∧ ¬ -(1
/ 𝐿) ≤ 𝑥) ∧ 𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))) → 𝑥 < 𝑘) |
| 280 | 279 | ralrimiva 3146 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ*) ∧ ¬ -(1
/ 𝐿) ≤ 𝑥) → ∀𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘) |
| 281 | 40, 77 | recgt0d 12202 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 < (1 / 𝐿)) |
| 282 | 42, 42, 281, 281 | addgt0d 11838 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 < ((1 / 𝐿) + (1 / 𝐿))) |
| 283 | 42 | recnd 11289 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1 / 𝐿) ∈ ℂ) |
| 284 | 283, 283 | subnegd 11627 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((1 / 𝐿) − -(1 / 𝐿)) = ((1 / 𝐿) + (1 / 𝐿))) |
| 285 | 282, 284 | breqtrrd 5171 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 < ((1 / 𝐿) − -(1 / 𝐿))) |
| 286 | 250, 42 | posdifd 11850 |
. . . . . . . . . . 11
⊢ (𝜑 → (-(1 / 𝐿) < (1 / 𝐿) ↔ 0 < ((1 / 𝐿) − -(1 / 𝐿)))) |
| 287 | 285, 286 | mpbird 257 |
. . . . . . . . . 10
⊢ (𝜑 → -(1 / 𝐿) < (1 / 𝐿)) |
| 288 | | ioon0 13413 |
. . . . . . . . . . 11
⊢ ((-(1 /
𝐿) ∈
ℝ* ∧ (1 / 𝐿) ∈ ℝ*) → ((-(1 /
𝐿)(,)(1 / 𝐿)) ≠ ∅ ↔ -(1 / 𝐿) < (1 / 𝐿))) |
| 289 | 251, 43, 288 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → ((-(1 / 𝐿)(,)(1 / 𝐿)) ≠ ∅ ↔ -(1 / 𝐿) < (1 / 𝐿))) |
| 290 | 287, 289 | mpbird 257 |
. . . . . . . . 9
⊢ (𝜑 → (-(1 / 𝐿)(,)(1 / 𝐿)) ≠ ∅) |
| 291 | | r19.2zb 4496 |
. . . . . . . . 9
⊢ ((-(1 /
𝐿)(,)(1 / 𝐿)) ≠ ∅ ↔ (∀𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘 → ∃𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘)) |
| 292 | 290, 291 | sylib 218 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘 → ∃𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘)) |
| 293 | 292 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ*) ∧ ¬ -(1
/ 𝐿) ≤ 𝑥) → (∀𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘 → ∃𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘)) |
| 294 | 280, 293 | mpd 15 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ*) ∧ ¬ -(1
/ 𝐿) ≤ 𝑥) → ∃𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘) |
| 295 | 294 | adantlrr 721 |
. . . . 5
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < (1 / 𝐿))) ∧ ¬ -(1 / 𝐿) ≤ 𝑥) → ∃𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘) |
| 296 | 268, 295 | pm2.61dan 813 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < (1 / 𝐿))) → ∃𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘) |
| 297 | | elioo2 13428 |
. . . . . . . . . . 11
⊢ ((-(1 /
𝐿) ∈
ℝ* ∧ (1 / 𝐿) ∈ ℝ*) → (𝑥 ∈ (-(1 / 𝐿)(,)(1 / 𝐿)) ↔ (𝑥 ∈ ℝ ∧ -(1 / 𝐿) < 𝑥 ∧ 𝑥 < (1 / 𝐿)))) |
| 298 | 251, 43, 297 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ (-(1 / 𝐿)(,)(1 / 𝐿)) ↔ (𝑥 ∈ ℝ ∧ -(1 / 𝐿) < 𝑥 ∧ 𝑥 < (1 / 𝐿)))) |
| 299 | 298 | biimpa 476 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))) → (𝑥 ∈ ℝ ∧ -(1 / 𝐿) < 𝑥 ∧ 𝑥 < (1 / 𝐿))) |
| 300 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) |
| 301 | 300, 46 | absltd 15468 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((abs‘𝑥) < (1 / 𝐿) ↔ (-(1 / 𝐿) < 𝑥 ∧ 𝑥 < (1 / 𝐿)))) |
| 302 | 49 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) < (1 / 𝐿)) → (abs‘𝑥) ∈ ℝ) |
| 303 | | simpr 484 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) < (1 / 𝐿)) → (abs‘𝑥) < (1 / 𝐿)) |
| 304 | 302, 303 | ltned 11397 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) < (1 / 𝐿)) → (abs‘𝑥) ≠ (1 / 𝐿)) |
| 305 | 232 | biimpd 229 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) ≠ (1 / 𝐿)) → ((abs‘𝑥) < (1 / 𝐿) → 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 306 | 305 | impancom 451 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) < (1 / 𝐿)) → ((abs‘𝑥) ≠ (1 / 𝐿) → 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 307 | 304, 306 | mpd 15 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (abs‘𝑥) < (1 / 𝐿)) → 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
| 308 | 307 | ex 412 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((abs‘𝑥) < (1 / 𝐿) → 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 309 | 301, 308 | sylbird 260 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((-(1 / 𝐿) < 𝑥 ∧ 𝑥 < (1 / 𝐿)) → 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 310 | 309 | impr 454 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (-(1 / 𝐿) < 𝑥 ∧ 𝑥 < (1 / 𝐿)))) → 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
| 311 | 310 | expcom 413 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ (-(1 /
𝐿) < 𝑥 ∧ 𝑥 < (1 / 𝐿))) → (𝜑 → 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 312 | 311 | 3impb 1115 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ -(1 /
𝐿) < 𝑥 ∧ 𝑥 < (1 / 𝐿)) → (𝜑 → 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 313 | 312 | impcom 407 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ -(1 / 𝐿) < 𝑥 ∧ 𝑥 < (1 / 𝐿))) → 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
| 314 | 299, 313 | syldan 591 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))) → 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
| 315 | 314 | ex 412 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (-(1 / 𝐿)(,)(1 / 𝐿)) → 𝑥 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ })) |
| 316 | 315 | ssrdv 3989 |
. . . . . 6
⊢ (𝜑 → (-(1 / 𝐿)(,)(1 / 𝐿)) ⊆ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
| 317 | | ssrexv 4053 |
. . . . . 6
⊢ ((-(1 /
𝐿)(,)(1 / 𝐿)) ⊆ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } → (∃𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘 → ∃𝑘 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }𝑥 < 𝑘)) |
| 318 | 316, 317 | syl 17 |
. . . . 5
⊢ (𝜑 → (∃𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘 → ∃𝑘 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }𝑥 < 𝑘)) |
| 319 | 318 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < (1 / 𝐿))) → (∃𝑘 ∈ (-(1 / 𝐿)(,)(1 / 𝐿))𝑥 < 𝑘 → ∃𝑘 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }𝑥 < 𝑘)) |
| 320 | 296, 319 | mpd 15 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < (1 / 𝐿))) → ∃𝑘 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }𝑥 < 𝑘) |
| 321 | 3, 43, 249, 320 | eqsupd 9497 |
. 2
⊢ (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*,
< ) = (1 / 𝐿)) |
| 322 | 1, 321 | eqtrid 2789 |
1
⊢ (𝜑 → 𝑅 = (1 / 𝐿)) |