| Step | Hyp | Ref
| Expression |
| 1 | | nnuz 12921 |
. . . 4
⊢ ℕ =
(ℤ≥‘1) |
| 2 | | 1zzd 12648 |
. . . 4
⊢ (⊤
→ 1 ∈ ℤ) |
| 3 | | pirp 26503 |
. . . . . . . . . 10
⊢ π
∈ ℝ+ |
| 4 | | nnrp 13046 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ+) |
| 5 | | rpdivcl 13060 |
. . . . . . . . . 10
⊢ ((π
∈ ℝ+ ∧ 𝑛 ∈ ℝ+) → (π /
𝑛) ∈
ℝ+) |
| 6 | 3, 4, 5 | sylancr 587 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ → (π /
𝑛) ∈
ℝ+) |
| 7 | 6 | rprene0d 13085 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ → ((π /
𝑛) ∈ ℝ ∧
(π / 𝑛) ≠
0)) |
| 8 | | eldifsn 4786 |
. . . . . . . 8
⊢ ((π /
𝑛) ∈ (ℝ ∖
{0}) ↔ ((π / 𝑛)
∈ ℝ ∧ (π / 𝑛) ≠ 0)) |
| 9 | 7, 8 | sylibr 234 |
. . . . . . 7
⊢ (𝑛 ∈ ℕ → (π /
𝑛) ∈ (ℝ ∖
{0})) |
| 10 | 9 | adantl 481 |
. . . . . 6
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (π / 𝑛) ∈ (ℝ ∖
{0})) |
| 11 | | eqidd 2738 |
. . . . . 6
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (π / 𝑛)) =
(𝑛 ∈ ℕ ↦
(π / 𝑛))) |
| 12 | | eqidd 2738 |
. . . . . 6
⊢ (⊤
→ (𝑦 ∈ (ℝ
∖ {0}) ↦ ((sin‘𝑦) / 𝑦)) = (𝑦 ∈ (ℝ ∖ {0}) ↦
((sin‘𝑦) / 𝑦))) |
| 13 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑦 = (π / 𝑛) → (sin‘𝑦) = (sin‘(π / 𝑛))) |
| 14 | | id 22 |
. . . . . . 7
⊢ (𝑦 = (π / 𝑛) → 𝑦 = (π / 𝑛)) |
| 15 | 13, 14 | oveq12d 7449 |
. . . . . 6
⊢ (𝑦 = (π / 𝑛) → ((sin‘𝑦) / 𝑦) = ((sin‘(π / 𝑛)) / (π / 𝑛))) |
| 16 | 10, 11, 12, 15 | fmptco 7149 |
. . . . 5
⊢ (⊤
→ ((𝑦 ∈ (ℝ
∖ {0}) ↦ ((sin‘𝑦) / 𝑦)) ∘ (𝑛 ∈ ℕ ↦ (π / 𝑛))) = (𝑛 ∈ ℕ ↦ ((sin‘(π /
𝑛)) / (π / 𝑛)))) |
| 17 | | eqid 2737 |
. . . . . . 7
⊢ (𝑛 ∈ ℕ ↦ (π /
𝑛)) = (𝑛 ∈ ℕ ↦ (π / 𝑛)) |
| 18 | 17, 9 | fmpti 7132 |
. . . . . 6
⊢ (𝑛 ∈ ℕ ↦ (π /
𝑛)):ℕ⟶(ℝ
∖ {0}) |
| 19 | | pire 26500 |
. . . . . . . 8
⊢ π
∈ ℝ |
| 20 | 19 | recni 11275 |
. . . . . . 7
⊢ π
∈ ℂ |
| 21 | | divcnv 15889 |
. . . . . . 7
⊢ (π
∈ ℂ → (𝑛
∈ ℕ ↦ (π / 𝑛)) ⇝ 0) |
| 22 | 20, 21 | mp1i 13 |
. . . . . 6
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (π / 𝑛)) ⇝
0) |
| 23 | | sinccvg 35678 |
. . . . . 6
⊢ (((𝑛 ∈ ℕ ↦ (π /
𝑛)):ℕ⟶(ℝ
∖ {0}) ∧ (𝑛
∈ ℕ ↦ (π / 𝑛)) ⇝ 0) → ((𝑦 ∈ (ℝ ∖ {0}) ↦
((sin‘𝑦) / 𝑦)) ∘ (𝑛 ∈ ℕ ↦ (π / 𝑛))) ⇝ 1) |
| 24 | 18, 22, 23 | sylancr 587 |
. . . . 5
⊢ (⊤
→ ((𝑦 ∈ (ℝ
∖ {0}) ↦ ((sin‘𝑦) / 𝑦)) ∘ (𝑛 ∈ ℕ ↦ (π / 𝑛))) ⇝ 1) |
| 25 | 16, 24 | eqbrtrrd 5167 |
. . . 4
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ ((sin‘(π / 𝑛)) / (π / 𝑛))) ⇝ 1) |
| 26 | | 2re 12340 |
. . . . . . . 8
⊢ 2 ∈
ℝ |
| 27 | 26, 19 | remulcli 11277 |
. . . . . . 7
⊢ (2
· π) ∈ ℝ |
| 28 | | circum.3 |
. . . . . . 7
⊢ 𝑅 ∈ ℝ |
| 29 | 27, 28 | remulcli 11277 |
. . . . . 6
⊢ ((2
· π) · 𝑅)
∈ ℝ |
| 30 | 29 | recni 11275 |
. . . . 5
⊢ ((2
· π) · 𝑅)
∈ ℂ |
| 31 | 30 | a1i 11 |
. . . 4
⊢ (⊤
→ ((2 · π) · 𝑅) ∈ ℂ) |
| 32 | | circum.2 |
. . . . . 6
⊢ 𝑃 = (𝑛 ∈ ℕ ↦ ((2 · 𝑛) · (𝑅 · (sin‘(𝐴 / 2))))) |
| 33 | | nnex 12272 |
. . . . . . 7
⊢ ℕ
∈ V |
| 34 | 33 | mptex 7243 |
. . . . . 6
⊢ (𝑛 ∈ ℕ ↦ ((2
· 𝑛) · (𝑅 · (sin‘(𝐴 / 2))))) ∈
V |
| 35 | 32, 34 | eqeltri 2837 |
. . . . 5
⊢ 𝑃 ∈ V |
| 36 | 35 | a1i 11 |
. . . 4
⊢ (⊤
→ 𝑃 ∈
V) |
| 37 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (ℝ ∖ {0})
↦ ((sin‘𝑦) /
𝑦)) = (𝑦 ∈ (ℝ ∖ {0}) ↦
((sin‘𝑦) / 𝑦)) |
| 38 | | eldifi 4131 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ (ℝ ∖ {0})
→ 𝑦 ∈
ℝ) |
| 39 | 38 | resincld 16179 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ (ℝ ∖ {0})
→ (sin‘𝑦) ∈
ℝ) |
| 40 | | eldifsni 4790 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ (ℝ ∖ {0})
→ 𝑦 ≠
0) |
| 41 | 39, 38, 40 | redivcld 12095 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (ℝ ∖ {0})
→ ((sin‘𝑦) /
𝑦) ∈
ℝ) |
| 42 | 37, 41 | fmpti 7132 |
. . . . . . . . 9
⊢ (𝑦 ∈ (ℝ ∖ {0})
↦ ((sin‘𝑦) /
𝑦)):(ℝ ∖
{0})⟶ℝ |
| 43 | | fco 6760 |
. . . . . . . . 9
⊢ (((𝑦 ∈ (ℝ ∖ {0})
↦ ((sin‘𝑦) /
𝑦)):(ℝ ∖
{0})⟶ℝ ∧ (𝑛 ∈ ℕ ↦ (π / 𝑛)):ℕ⟶(ℝ
∖ {0})) → ((𝑦
∈ (ℝ ∖ {0}) ↦ ((sin‘𝑦) / 𝑦)) ∘ (𝑛 ∈ ℕ ↦ (π / 𝑛))):ℕ⟶ℝ) |
| 44 | 42, 18, 43 | mp2an 692 |
. . . . . . . 8
⊢ ((𝑦 ∈ (ℝ ∖ {0})
↦ ((sin‘𝑦) /
𝑦)) ∘ (𝑛 ∈ ℕ ↦ (π /
𝑛))):ℕ⟶ℝ |
| 45 | 16 | mptru 1547 |
. . . . . . . . 9
⊢ ((𝑦 ∈ (ℝ ∖ {0})
↦ ((sin‘𝑦) /
𝑦)) ∘ (𝑛 ∈ ℕ ↦ (π /
𝑛))) = (𝑛 ∈ ℕ ↦ ((sin‘(π /
𝑛)) / (π / 𝑛))) |
| 46 | 45 | feq1i 6727 |
. . . . . . . 8
⊢ (((𝑦 ∈ (ℝ ∖ {0})
↦ ((sin‘𝑦) /
𝑦)) ∘ (𝑛 ∈ ℕ ↦ (π /
𝑛))):ℕ⟶ℝ
↔ (𝑛 ∈ ℕ
↦ ((sin‘(π / 𝑛)) / (π / 𝑛))):ℕ⟶ℝ) |
| 47 | 44, 46 | mpbi 230 |
. . . . . . 7
⊢ (𝑛 ∈ ℕ ↦
((sin‘(π / 𝑛)) /
(π / 𝑛))):ℕ⟶ℝ |
| 48 | 47 | ffvelcdmi 7103 |
. . . . . 6
⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦
((sin‘(π / 𝑛)) /
(π / 𝑛)))‘𝑘) ∈
ℝ) |
| 49 | 48 | adantl 481 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ ↦ ((sin‘(π / 𝑛)) / (π / 𝑛)))‘𝑘) ∈ ℝ) |
| 50 | 49 | recnd 11289 |
. . . 4
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ ↦ ((sin‘(π / 𝑛)) / (π / 𝑛)))‘𝑘) ∈ ℂ) |
| 51 | 26 | recni 11275 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℂ |
| 52 | 51 | a1i 11 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 2 ∈ ℂ) |
| 53 | 20 | a1i 11 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → π ∈ ℂ) |
| 54 | | nncn 12274 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℂ) |
| 55 | 54 | adantl 481 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
∈ ℂ) |
| 56 | | nnne0 12300 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0) |
| 57 | 56 | adantl 481 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
≠ 0) |
| 58 | 52, 53, 55, 57 | divassd 12078 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · π) / 𝑘) = (2 · (π / 𝑘))) |
| 59 | 58 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (((2 · π) / 𝑘) / 2) = ((2 · (π / 𝑘)) / 2)) |
| 60 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
∈ ℕ) |
| 61 | | nndivre 12307 |
. . . . . . . . . . . . . . 15
⊢ ((π
∈ ℝ ∧ 𝑘
∈ ℕ) → (π / 𝑘) ∈ ℝ) |
| 62 | 19, 60, 61 | sylancr 587 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (π / 𝑘) ∈ ℝ) |
| 63 | 62 | recnd 11289 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (π / 𝑘) ∈ ℂ) |
| 64 | | 2ne0 12370 |
. . . . . . . . . . . . . 14
⊢ 2 ≠
0 |
| 65 | 64 | a1i 11 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 2 ≠ 0) |
| 66 | 63, 52, 65 | divcan3d 12048 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · (π / 𝑘)) / 2) = (π / 𝑘)) |
| 67 | 59, 66 | eqtrd 2777 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (((2 · π) / 𝑘) / 2) = (π / 𝑘)) |
| 68 | 67 | fveq2d 6910 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (sin‘(((2 · π) / 𝑘) / 2)) = (sin‘(π / 𝑘))) |
| 69 | 62 | resincld 16179 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (sin‘(π / 𝑘)) ∈ ℝ) |
| 70 | 69 | recnd 11289 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (sin‘(π / 𝑘)) ∈ ℂ) |
| 71 | | nnrp 13046 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℝ+) |
| 72 | 71 | adantl 481 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
∈ ℝ+) |
| 73 | | rpdivcl 13060 |
. . . . . . . . . . . . 13
⊢ ((π
∈ ℝ+ ∧ 𝑘 ∈ ℝ+) → (π /
𝑘) ∈
ℝ+) |
| 74 | 3, 72, 73 | sylancr 587 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (π / 𝑘) ∈
ℝ+) |
| 75 | 74 | rpne0d 13082 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (π / 𝑘) ≠ 0) |
| 76 | 70, 63, 75 | divcan2d 12045 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((π / 𝑘) · ((sin‘(π / 𝑘)) / (π / 𝑘))) = (sin‘(π / 𝑘))) |
| 77 | 68, 76 | eqtr4d 2780 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (sin‘(((2 · π) / 𝑘) / 2)) = ((π / 𝑘) · ((sin‘(π / 𝑘)) / (π / 𝑘)))) |
| 78 | 77 | oveq2d 7447 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝑅
· (sin‘(((2 · π) / 𝑘) / 2))) = (𝑅 · ((π / 𝑘) · ((sin‘(π / 𝑘)) / (π / 𝑘))))) |
| 79 | 28 | recni 11275 |
. . . . . . . . . 10
⊢ 𝑅 ∈ ℂ |
| 80 | 79 | a1i 11 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑅
∈ ℂ) |
| 81 | | oveq2 7439 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑘 → (π / 𝑛) = (π / 𝑘)) |
| 82 | 81 | fveq2d 6910 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑘 → (sin‘(π / 𝑛)) = (sin‘(π / 𝑘))) |
| 83 | 82, 81 | oveq12d 7449 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → ((sin‘(π / 𝑛)) / (π / 𝑛)) = ((sin‘(π / 𝑘)) / (π / 𝑘))) |
| 84 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ ↦
((sin‘(π / 𝑛)) /
(π / 𝑛))) = (𝑛 ∈ ℕ ↦
((sin‘(π / 𝑛)) /
(π / 𝑛))) |
| 85 | | ovex 7464 |
. . . . . . . . . . . 12
⊢
((sin‘(π / 𝑘)) / (π / 𝑘)) ∈ V |
| 86 | 83, 84, 85 | fvmpt 7016 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦
((sin‘(π / 𝑛)) /
(π / 𝑛)))‘𝑘) = ((sin‘(π / 𝑘)) / (π / 𝑘))) |
| 87 | 86 | adantl 481 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ ↦ ((sin‘(π / 𝑛)) / (π / 𝑛)))‘𝑘) = ((sin‘(π / 𝑘)) / (π / 𝑘))) |
| 88 | 87, 50 | eqeltrrd 2842 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((sin‘(π / 𝑘)) / (π / 𝑘)) ∈ ℂ) |
| 89 | 80, 63, 88 | mulassd 11284 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑅
· (π / 𝑘))
· ((sin‘(π / 𝑘)) / (π / 𝑘))) = (𝑅 · ((π / 𝑘) · ((sin‘(π / 𝑘)) / (π / 𝑘))))) |
| 90 | 78, 89 | eqtr4d 2780 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝑅
· (sin‘(((2 · π) / 𝑘) / 2))) = ((𝑅 · (π / 𝑘)) · ((sin‘(π / 𝑘)) / (π / 𝑘)))) |
| 91 | 90 | oveq2d 7447 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑘) · (𝑅 · (sin‘(((2 · π) /
𝑘) / 2)))) = ((2 ·
𝑘) · ((𝑅 · (π / 𝑘)) · ((sin‘(π /
𝑘)) / (π / 𝑘))))) |
| 92 | | mulcl 11239 |
. . . . . . . 8
⊢ ((2
∈ ℂ ∧ 𝑘
∈ ℂ) → (2 · 𝑘) ∈ ℂ) |
| 93 | 51, 55, 92 | sylancr 587 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (2 · 𝑘) ∈ ℂ) |
| 94 | | mulcl 11239 |
. . . . . . . 8
⊢ ((𝑅 ∈ ℂ ∧ (π /
𝑘) ∈ ℂ) →
(𝑅 · (π / 𝑘)) ∈
ℂ) |
| 95 | 79, 63, 94 | sylancr 587 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝑅
· (π / 𝑘)) ∈
ℂ) |
| 96 | 93, 95, 88 | mulassd 11284 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (((2 · 𝑘) · (𝑅 · (π / 𝑘))) · ((sin‘(π / 𝑘)) / (π / 𝑘))) = ((2 · 𝑘) · ((𝑅 · (π / 𝑘)) · ((sin‘(π / 𝑘)) / (π / 𝑘))))) |
| 97 | 52, 55, 80, 63 | mul4d 11473 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑘) · (𝑅 · (π / 𝑘))) = ((2 · 𝑅) · (𝑘 · (π / 𝑘)))) |
| 98 | 53, 55, 57 | divcan2d 12045 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝑘
· (π / 𝑘)) =
π) |
| 99 | 98 | oveq2d 7447 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑅) · (𝑘 · (π / 𝑘))) = ((2 · 𝑅) · π)) |
| 100 | 52, 80, 53 | mul32d 11471 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑅) · π) = ((2 · π)
· 𝑅)) |
| 101 | 99, 100 | eqtrd 2777 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑅) · (𝑘 · (π / 𝑘))) = ((2 · π) · 𝑅)) |
| 102 | 97, 101 | eqtrd 2777 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑘) · (𝑅 · (π / 𝑘))) = ((2 · π) · 𝑅)) |
| 103 | 102 | oveq1d 7446 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (((2 · 𝑘) · (𝑅 · (π / 𝑘))) · ((sin‘(π / 𝑘)) / (π / 𝑘))) = (((2 · π) · 𝑅) · ((sin‘(π /
𝑘)) / (π / 𝑘)))) |
| 104 | 91, 96, 103 | 3eqtr2d 2783 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑘) · (𝑅 · (sin‘(((2 · π) /
𝑘) / 2)))) = (((2 ·
π) · 𝑅) ·
((sin‘(π / 𝑘)) /
(π / 𝑘)))) |
| 105 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑛 = 𝑘 → (2 · 𝑛) = (2 · 𝑘)) |
| 106 | | circum.1 |
. . . . . . . . . . . 12
⊢ 𝐴 = ((2 · π) / 𝑛) |
| 107 | | oveq2 7439 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → ((2 · π) / 𝑛) = ((2 · π) / 𝑘)) |
| 108 | 106, 107 | eqtrid 2789 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑘 → 𝐴 = ((2 · π) / 𝑘)) |
| 109 | 108 | oveq1d 7446 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑘 → (𝐴 / 2) = (((2 · π) / 𝑘) / 2)) |
| 110 | 109 | fveq2d 6910 |
. . . . . . . . 9
⊢ (𝑛 = 𝑘 → (sin‘(𝐴 / 2)) = (sin‘(((2 · π) /
𝑘) / 2))) |
| 111 | 110 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑛 = 𝑘 → (𝑅 · (sin‘(𝐴 / 2))) = (𝑅 · (sin‘(((2 · π) /
𝑘) / 2)))) |
| 112 | 105, 111 | oveq12d 7449 |
. . . . . . 7
⊢ (𝑛 = 𝑘 → ((2 · 𝑛) · (𝑅 · (sin‘(𝐴 / 2)))) = ((2 · 𝑘) · (𝑅 · (sin‘(((2 · π) /
𝑘) /
2))))) |
| 113 | | ovex 7464 |
. . . . . . 7
⊢ ((2
· 𝑘) · (𝑅 · (sin‘(((2
· π) / 𝑘) / 2))))
∈ V |
| 114 | 112, 32, 113 | fvmpt 7016 |
. . . . . 6
⊢ (𝑘 ∈ ℕ → (𝑃‘𝑘) = ((2 · 𝑘) · (𝑅 · (sin‘(((2 · π) /
𝑘) /
2))))) |
| 115 | 114 | adantl 481 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝑃‘𝑘) = ((2 · 𝑘) · (𝑅 · (sin‘(((2 · π) /
𝑘) /
2))))) |
| 116 | 87 | oveq2d 7447 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (((2 · π) · 𝑅) · ((𝑛 ∈ ℕ ↦ ((sin‘(π /
𝑛)) / (π / 𝑛)))‘𝑘)) = (((2 · π) · 𝑅) · ((sin‘(π /
𝑘)) / (π / 𝑘)))) |
| 117 | 104, 115,
116 | 3eqtr4d 2787 |
. . . 4
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝑃‘𝑘) = (((2 · π) · 𝑅) · ((𝑛 ∈ ℕ ↦ ((sin‘(π /
𝑛)) / (π / 𝑛)))‘𝑘))) |
| 118 | 1, 2, 25, 31, 36, 50, 117 | climmulc2 15673 |
. . 3
⊢ (⊤
→ 𝑃 ⇝ (((2
· π) · 𝑅)
· 1)) |
| 119 | 118 | mptru 1547 |
. 2
⊢ 𝑃 ⇝ (((2 · π)
· 𝑅) ·
1) |
| 120 | 30 | mulridi 11265 |
. 2
⊢ (((2
· π) · 𝑅)
· 1) = ((2 · π) · 𝑅) |
| 121 | 119, 120 | breqtri 5168 |
1
⊢ 𝑃 ⇝ ((2 · π)
· 𝑅) |