Step |
Hyp |
Ref |
Expression |
1 |
|
circum.1 |
⊢ 𝐴 = ( ( 2 · π ) / 𝑛 ) |
2 |
|
circum.2 |
⊢ 𝑃 = ( 𝑛 ∈ ℕ ↦ ( ( 2 · 𝑛 ) · ( 𝑅 · ( sin ‘ ( 𝐴 / 2 ) ) ) ) ) |
3 |
|
circum.3 |
⊢ 𝑅 ∈ ℝ |
4 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
5 |
|
1zzd |
⊢ ( ⊤ → 1 ∈ ℤ ) |
6 |
|
pirp |
⊢ π ∈ ℝ+ |
7 |
|
nnrp |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℝ+ ) |
8 |
|
rpdivcl |
⊢ ( ( π ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) → ( π / 𝑛 ) ∈ ℝ+ ) |
9 |
6 7 8
|
sylancr |
⊢ ( 𝑛 ∈ ℕ → ( π / 𝑛 ) ∈ ℝ+ ) |
10 |
9
|
rprene0d |
⊢ ( 𝑛 ∈ ℕ → ( ( π / 𝑛 ) ∈ ℝ ∧ ( π / 𝑛 ) ≠ 0 ) ) |
11 |
|
eldifsn |
⊢ ( ( π / 𝑛 ) ∈ ( ℝ ∖ { 0 } ) ↔ ( ( π / 𝑛 ) ∈ ℝ ∧ ( π / 𝑛 ) ≠ 0 ) ) |
12 |
10 11
|
sylibr |
⊢ ( 𝑛 ∈ ℕ → ( π / 𝑛 ) ∈ ( ℝ ∖ { 0 } ) ) |
13 |
12
|
adantl |
⊢ ( ( ⊤ ∧ 𝑛 ∈ ℕ ) → ( π / 𝑛 ) ∈ ( ℝ ∖ { 0 } ) ) |
14 |
|
eqidd |
⊢ ( ⊤ → ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) = ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) ) |
15 |
|
eqidd |
⊢ ( ⊤ → ( 𝑦 ∈ ( ℝ ∖ { 0 } ) ↦ ( ( sin ‘ 𝑦 ) / 𝑦 ) ) = ( 𝑦 ∈ ( ℝ ∖ { 0 } ) ↦ ( ( sin ‘ 𝑦 ) / 𝑦 ) ) ) |
16 |
|
fveq2 |
⊢ ( 𝑦 = ( π / 𝑛 ) → ( sin ‘ 𝑦 ) = ( sin ‘ ( π / 𝑛 ) ) ) |
17 |
|
id |
⊢ ( 𝑦 = ( π / 𝑛 ) → 𝑦 = ( π / 𝑛 ) ) |
18 |
16 17
|
oveq12d |
⊢ ( 𝑦 = ( π / 𝑛 ) → ( ( sin ‘ 𝑦 ) / 𝑦 ) = ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) |
19 |
13 14 15 18
|
fmptco |
⊢ ( ⊤ → ( ( 𝑦 ∈ ( ℝ ∖ { 0 } ) ↦ ( ( sin ‘ 𝑦 ) / 𝑦 ) ) ∘ ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) ) = ( 𝑛 ∈ ℕ ↦ ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) ) |
20 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) = ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) |
21 |
20 12
|
fmpti |
⊢ ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) : ℕ ⟶ ( ℝ ∖ { 0 } ) |
22 |
|
pire |
⊢ π ∈ ℝ |
23 |
22
|
recni |
⊢ π ∈ ℂ |
24 |
|
divcnv |
⊢ ( π ∈ ℂ → ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) ⇝ 0 ) |
25 |
23 24
|
mp1i |
⊢ ( ⊤ → ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) ⇝ 0 ) |
26 |
|
sinccvg |
⊢ ( ( ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) : ℕ ⟶ ( ℝ ∖ { 0 } ) ∧ ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) ⇝ 0 ) → ( ( 𝑦 ∈ ( ℝ ∖ { 0 } ) ↦ ( ( sin ‘ 𝑦 ) / 𝑦 ) ) ∘ ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) ) ⇝ 1 ) |
27 |
21 25 26
|
sylancr |
⊢ ( ⊤ → ( ( 𝑦 ∈ ( ℝ ∖ { 0 } ) ↦ ( ( sin ‘ 𝑦 ) / 𝑦 ) ) ∘ ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) ) ⇝ 1 ) |
28 |
19 27
|
eqbrtrrd |
⊢ ( ⊤ → ( 𝑛 ∈ ℕ ↦ ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) ⇝ 1 ) |
29 |
|
2re |
⊢ 2 ∈ ℝ |
30 |
29 22
|
remulcli |
⊢ ( 2 · π ) ∈ ℝ |
31 |
30 3
|
remulcli |
⊢ ( ( 2 · π ) · 𝑅 ) ∈ ℝ |
32 |
31
|
recni |
⊢ ( ( 2 · π ) · 𝑅 ) ∈ ℂ |
33 |
32
|
a1i |
⊢ ( ⊤ → ( ( 2 · π ) · 𝑅 ) ∈ ℂ ) |
34 |
|
nnex |
⊢ ℕ ∈ V |
35 |
34
|
mptex |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( 2 · 𝑛 ) · ( 𝑅 · ( sin ‘ ( 𝐴 / 2 ) ) ) ) ) ∈ V |
36 |
2 35
|
eqeltri |
⊢ 𝑃 ∈ V |
37 |
36
|
a1i |
⊢ ( ⊤ → 𝑃 ∈ V ) |
38 |
|
eqid |
⊢ ( 𝑦 ∈ ( ℝ ∖ { 0 } ) ↦ ( ( sin ‘ 𝑦 ) / 𝑦 ) ) = ( 𝑦 ∈ ( ℝ ∖ { 0 } ) ↦ ( ( sin ‘ 𝑦 ) / 𝑦 ) ) |
39 |
|
eldifi |
⊢ ( 𝑦 ∈ ( ℝ ∖ { 0 } ) → 𝑦 ∈ ℝ ) |
40 |
39
|
resincld |
⊢ ( 𝑦 ∈ ( ℝ ∖ { 0 } ) → ( sin ‘ 𝑦 ) ∈ ℝ ) |
41 |
|
eldifsni |
⊢ ( 𝑦 ∈ ( ℝ ∖ { 0 } ) → 𝑦 ≠ 0 ) |
42 |
40 39 41
|
redivcld |
⊢ ( 𝑦 ∈ ( ℝ ∖ { 0 } ) → ( ( sin ‘ 𝑦 ) / 𝑦 ) ∈ ℝ ) |
43 |
38 42
|
fmpti |
⊢ ( 𝑦 ∈ ( ℝ ∖ { 0 } ) ↦ ( ( sin ‘ 𝑦 ) / 𝑦 ) ) : ( ℝ ∖ { 0 } ) ⟶ ℝ |
44 |
|
fco |
⊢ ( ( ( 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 ) |
45 |
43 21 44
|
mp2an |
⊢ ( ( 𝑦 ∈ ( ℝ ∖ { 0 } ) ↦ ( ( sin ‘ 𝑦 ) / 𝑦 ) ) ∘ ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) ) : ℕ ⟶ ℝ |
46 |
19
|
mptru |
⊢ ( ( 𝑦 ∈ ( ℝ ∖ { 0 } ) ↦ ( ( sin ‘ 𝑦 ) / 𝑦 ) ) ∘ ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) ) = ( 𝑛 ∈ ℕ ↦ ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) |
47 |
46
|
feq1i |
⊢ ( ( ( 𝑦 ∈ ( ℝ ∖ { 0 } ) ↦ ( ( sin ‘ 𝑦 ) / 𝑦 ) ) ∘ ( 𝑛 ∈ ℕ ↦ ( π / 𝑛 ) ) ) : ℕ ⟶ ℝ ↔ ( 𝑛 ∈ ℕ ↦ ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) : ℕ ⟶ ℝ ) |
48 |
45 47
|
mpbi |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) : ℕ ⟶ ℝ |
49 |
48
|
ffvelrni |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) ‘ 𝑘 ) ∈ ℝ ) |
50 |
49
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) ‘ 𝑘 ) ∈ ℝ ) |
51 |
50
|
recnd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) ‘ 𝑘 ) ∈ ℂ ) |
52 |
29
|
recni |
⊢ 2 ∈ ℂ |
53 |
52
|
a1i |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → 2 ∈ ℂ ) |
54 |
23
|
a1i |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → π ∈ ℂ ) |
55 |
|
nncn |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℂ ) |
56 |
55
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℂ ) |
57 |
|
nnne0 |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ≠ 0 ) |
58 |
57
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → 𝑘 ≠ 0 ) |
59 |
53 54 56 58
|
divassd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 2 · π ) / 𝑘 ) = ( 2 · ( π / 𝑘 ) ) ) |
60 |
59
|
oveq1d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( ( 2 · π ) / 𝑘 ) / 2 ) = ( ( 2 · ( π / 𝑘 ) ) / 2 ) ) |
61 |
|
simpr |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℕ ) |
62 |
|
nndivre |
⊢ ( ( π ∈ ℝ ∧ 𝑘 ∈ ℕ ) → ( π / 𝑘 ) ∈ ℝ ) |
63 |
22 61 62
|
sylancr |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( π / 𝑘 ) ∈ ℝ ) |
64 |
63
|
recnd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( π / 𝑘 ) ∈ ℂ ) |
65 |
|
2ne0 |
⊢ 2 ≠ 0 |
66 |
65
|
a1i |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → 2 ≠ 0 ) |
67 |
64 53 66
|
divcan3d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 2 · ( π / 𝑘 ) ) / 2 ) = ( π / 𝑘 ) ) |
68 |
60 67
|
eqtrd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( ( 2 · π ) / 𝑘 ) / 2 ) = ( π / 𝑘 ) ) |
69 |
68
|
fveq2d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( sin ‘ ( ( ( 2 · π ) / 𝑘 ) / 2 ) ) = ( sin ‘ ( π / 𝑘 ) ) ) |
70 |
63
|
resincld |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( sin ‘ ( π / 𝑘 ) ) ∈ ℝ ) |
71 |
70
|
recnd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( sin ‘ ( π / 𝑘 ) ) ∈ ℂ ) |
72 |
|
nnrp |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℝ+ ) |
73 |
72
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℝ+ ) |
74 |
|
rpdivcl |
⊢ ( ( π ∈ ℝ+ ∧ 𝑘 ∈ ℝ+ ) → ( π / 𝑘 ) ∈ ℝ+ ) |
75 |
6 73 74
|
sylancr |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( π / 𝑘 ) ∈ ℝ+ ) |
76 |
75
|
rpne0d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( π / 𝑘 ) ≠ 0 ) |
77 |
71 64 76
|
divcan2d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( π / 𝑘 ) · ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) = ( sin ‘ ( π / 𝑘 ) ) ) |
78 |
69 77
|
eqtr4d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( sin ‘ ( ( ( 2 · π ) / 𝑘 ) / 2 ) ) = ( ( π / 𝑘 ) · ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) ) |
79 |
78
|
oveq2d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝑅 · ( sin ‘ ( ( ( 2 · π ) / 𝑘 ) / 2 ) ) ) = ( 𝑅 · ( ( π / 𝑘 ) · ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) ) ) |
80 |
3
|
recni |
⊢ 𝑅 ∈ ℂ |
81 |
80
|
a1i |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → 𝑅 ∈ ℂ ) |
82 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( π / 𝑛 ) = ( π / 𝑘 ) ) |
83 |
82
|
fveq2d |
⊢ ( 𝑛 = 𝑘 → ( sin ‘ ( π / 𝑛 ) ) = ( sin ‘ ( π / 𝑘 ) ) ) |
84 |
83 82
|
oveq12d |
⊢ ( 𝑛 = 𝑘 → ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) = ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) |
85 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) = ( 𝑛 ∈ ℕ ↦ ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) |
86 |
|
ovex |
⊢ ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ∈ V |
87 |
84 85 86
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) ‘ 𝑘 ) = ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) |
88 |
87
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) ‘ 𝑘 ) = ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) |
89 |
88 51
|
eqeltrrd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ∈ ℂ ) |
90 |
81 64 89
|
mulassd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 𝑅 · ( π / 𝑘 ) ) · ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) = ( 𝑅 · ( ( π / 𝑘 ) · ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) ) ) |
91 |
79 90
|
eqtr4d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝑅 · ( sin ‘ ( ( ( 2 · π ) / 𝑘 ) / 2 ) ) ) = ( ( 𝑅 · ( π / 𝑘 ) ) · ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) ) |
92 |
91
|
oveq2d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 2 · 𝑘 ) · ( 𝑅 · ( sin ‘ ( ( ( 2 · π ) / 𝑘 ) / 2 ) ) ) ) = ( ( 2 · 𝑘 ) · ( ( 𝑅 · ( π / 𝑘 ) ) · ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) ) ) |
93 |
|
mulcl |
⊢ ( ( 2 ∈ ℂ ∧ 𝑘 ∈ ℂ ) → ( 2 · 𝑘 ) ∈ ℂ ) |
94 |
52 56 93
|
sylancr |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 2 · 𝑘 ) ∈ ℂ ) |
95 |
|
mulcl |
⊢ ( ( 𝑅 ∈ ℂ ∧ ( π / 𝑘 ) ∈ ℂ ) → ( 𝑅 · ( π / 𝑘 ) ) ∈ ℂ ) |
96 |
80 64 95
|
sylancr |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝑅 · ( π / 𝑘 ) ) ∈ ℂ ) |
97 |
94 96 89
|
mulassd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( ( 2 · 𝑘 ) · ( 𝑅 · ( π / 𝑘 ) ) ) · ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) = ( ( 2 · 𝑘 ) · ( ( 𝑅 · ( π / 𝑘 ) ) · ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) ) ) |
98 |
53 56 81 64
|
mul4d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 2 · 𝑘 ) · ( 𝑅 · ( π / 𝑘 ) ) ) = ( ( 2 · 𝑅 ) · ( 𝑘 · ( π / 𝑘 ) ) ) ) |
99 |
54 56 58
|
divcan2d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝑘 · ( π / 𝑘 ) ) = π ) |
100 |
99
|
oveq2d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 2 · 𝑅 ) · ( 𝑘 · ( π / 𝑘 ) ) ) = ( ( 2 · 𝑅 ) · π ) ) |
101 |
53 81 54
|
mul32d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 2 · 𝑅 ) · π ) = ( ( 2 · π ) · 𝑅 ) ) |
102 |
100 101
|
eqtrd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 2 · 𝑅 ) · ( 𝑘 · ( π / 𝑘 ) ) ) = ( ( 2 · π ) · 𝑅 ) ) |
103 |
98 102
|
eqtrd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 2 · 𝑘 ) · ( 𝑅 · ( π / 𝑘 ) ) ) = ( ( 2 · π ) · 𝑅 ) ) |
104 |
103
|
oveq1d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( ( 2 · 𝑘 ) · ( 𝑅 · ( π / 𝑘 ) ) ) · ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) = ( ( ( 2 · π ) · 𝑅 ) · ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) ) |
105 |
92 97 104
|
3eqtr2d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 2 · 𝑘 ) · ( 𝑅 · ( sin ‘ ( ( ( 2 · π ) / 𝑘 ) / 2 ) ) ) ) = ( ( ( 2 · π ) · 𝑅 ) · ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) ) |
106 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( 2 · 𝑛 ) = ( 2 · 𝑘 ) ) |
107 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( ( 2 · π ) / 𝑛 ) = ( ( 2 · π ) / 𝑘 ) ) |
108 |
1 107
|
syl5eq |
⊢ ( 𝑛 = 𝑘 → 𝐴 = ( ( 2 · π ) / 𝑘 ) ) |
109 |
108
|
oveq1d |
⊢ ( 𝑛 = 𝑘 → ( 𝐴 / 2 ) = ( ( ( 2 · π ) / 𝑘 ) / 2 ) ) |
110 |
109
|
fveq2d |
⊢ ( 𝑛 = 𝑘 → ( sin ‘ ( 𝐴 / 2 ) ) = ( sin ‘ ( ( ( 2 · π ) / 𝑘 ) / 2 ) ) ) |
111 |
110
|
oveq2d |
⊢ ( 𝑛 = 𝑘 → ( 𝑅 · ( sin ‘ ( 𝐴 / 2 ) ) ) = ( 𝑅 · ( sin ‘ ( ( ( 2 · π ) / 𝑘 ) / 2 ) ) ) ) |
112 |
106 111
|
oveq12d |
⊢ ( 𝑛 = 𝑘 → ( ( 2 · 𝑛 ) · ( 𝑅 · ( sin ‘ ( 𝐴 / 2 ) ) ) ) = ( ( 2 · 𝑘 ) · ( 𝑅 · ( sin ‘ ( ( ( 2 · π ) / 𝑘 ) / 2 ) ) ) ) ) |
113 |
|
ovex |
⊢ ( ( 2 · 𝑘 ) · ( 𝑅 · ( sin ‘ ( ( ( 2 · π ) / 𝑘 ) / 2 ) ) ) ) ∈ V |
114 |
112 2 113
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( 𝑃 ‘ 𝑘 ) = ( ( 2 · 𝑘 ) · ( 𝑅 · ( sin ‘ ( ( ( 2 · π ) / 𝑘 ) / 2 ) ) ) ) ) |
115 |
114
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝑃 ‘ 𝑘 ) = ( ( 2 · 𝑘 ) · ( 𝑅 · ( sin ‘ ( ( ( 2 · π ) / 𝑘 ) / 2 ) ) ) ) ) |
116 |
88
|
oveq2d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( ( 2 · π ) · 𝑅 ) · ( ( 𝑛 ∈ ℕ ↦ ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) ‘ 𝑘 ) ) = ( ( ( 2 · π ) · 𝑅 ) · ( ( sin ‘ ( π / 𝑘 ) ) / ( π / 𝑘 ) ) ) ) |
117 |
105 115 116
|
3eqtr4d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝑃 ‘ 𝑘 ) = ( ( ( 2 · π ) · 𝑅 ) · ( ( 𝑛 ∈ ℕ ↦ ( ( sin ‘ ( π / 𝑛 ) ) / ( π / 𝑛 ) ) ) ‘ 𝑘 ) ) ) |
118 |
4 5 28 33 37 51 117
|
climmulc2 |
⊢ ( ⊤ → 𝑃 ⇝ ( ( ( 2 · π ) · 𝑅 ) · 1 ) ) |
119 |
118
|
mptru |
⊢ 𝑃 ⇝ ( ( ( 2 · π ) · 𝑅 ) · 1 ) |
120 |
32
|
mulid1i |
⊢ ( ( ( 2 · π ) · 𝑅 ) · 1 ) = ( ( 2 · π ) · 𝑅 ) |
121 |
119 120
|
breqtri |
⊢ 𝑃 ⇝ ( ( 2 · π ) · 𝑅 ) |