Step | Hyp | Ref
| Expression |
1 | | fourierdlem112.23 |
. . . . 5
⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) |
2 | | fveq2 6674 |
. . . . . . . 8
⊢ (𝑛 = 𝑗 → (𝐴‘𝑛) = (𝐴‘𝑗)) |
3 | | oveq1 7177 |
. . . . . . . . 9
⊢ (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋)) |
4 | 3 | fveq2d 6678 |
. . . . . . . 8
⊢ (𝑛 = 𝑗 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑗 · 𝑋))) |
5 | 2, 4 | oveq12d 7188 |
. . . . . . 7
⊢ (𝑛 = 𝑗 → ((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋)))) |
6 | | fveq2 6674 |
. . . . . . . 8
⊢ (𝑛 = 𝑗 → (𝐵‘𝑛) = (𝐵‘𝑗)) |
7 | 3 | fveq2d 6678 |
. . . . . . . 8
⊢ (𝑛 = 𝑗 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑗 · 𝑋))) |
8 | 6, 7 | oveq12d 7188 |
. . . . . . 7
⊢ (𝑛 = 𝑗 → ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋)))) |
9 | 5, 8 | oveq12d 7188 |
. . . . . 6
⊢ (𝑛 = 𝑗 → (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))) |
10 | 9 | cbvmptv 5133 |
. . . . 5
⊢ (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))) |
11 | 1, 10 | eqtri 2761 |
. . . 4
⊢ 𝑆 = (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))) |
12 | | seqeq3 13465 |
. . . 4
⊢ (𝑆 = (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))) → seq1( + , 𝑆) = seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))))) |
13 | 11, 12 | mp1i 13 |
. . 3
⊢ (𝜑 → seq1( + , 𝑆) = seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))))) |
14 | | nnuz 12363 |
. . . . 5
⊢ ℕ =
(ℤ≥‘1) |
15 | | 1zzd 12094 |
. . . . 5
⊢ (𝜑 → 1 ∈
ℤ) |
16 | | nfv 1921 |
. . . . . . 7
⊢
Ⅎ𝑛𝜑 |
17 | | nfcv 2899 |
. . . . . . . 8
⊢
Ⅎ𝑛ℕ |
18 | | nfcv 2899 |
. . . . . . . . 9
⊢
Ⅎ𝑛(-π(,)0) |
19 | | nfcv 2899 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(𝐹‘(𝑋 + 𝑠)) |
20 | | nfcv 2899 |
. . . . . . . . . 10
⊢
Ⅎ𝑛
· |
21 | | nfcv 2899 |
. . . . . . . . . 10
⊢
Ⅎ𝑛((𝐷‘𝑚)‘𝑠) |
22 | 19, 20, 21 | nfov 7200 |
. . . . . . . . 9
⊢
Ⅎ𝑛((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) |
23 | 18, 22 | nfitg 24527 |
. . . . . . . 8
⊢
Ⅎ𝑛∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠 |
24 | 17, 23 | nfmpt 5127 |
. . . . . . 7
⊢
Ⅎ𝑛(𝑚 ∈ ℕ ↦
∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠) |
25 | | nfcv 2899 |
. . . . . . . . 9
⊢
Ⅎ𝑛(0(,)π) |
26 | 25, 22 | nfitg 24527 |
. . . . . . . 8
⊢
Ⅎ𝑛∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠 |
27 | 17, 26 | nfmpt 5127 |
. . . . . . 7
⊢
Ⅎ𝑛(𝑚 ∈ ℕ ↦
∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠) |
28 | | fourierdlem112.z |
. . . . . . . 8
⊢ 𝑍 = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))) |
29 | | fourierdlem112.a |
. . . . . . . . . . . . 13
⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦
(∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) |
30 | | nfmpt1 5128 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛(𝑛 ∈ ℕ0 ↦
(∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) |
31 | 29, 30 | nfcxfr 2897 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑛𝐴 |
32 | | nfcv 2899 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑛0 |
33 | 31, 32 | nffv 6684 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛(𝐴‘0) |
34 | | nfcv 2899 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛
/ |
35 | | nfcv 2899 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛2 |
36 | 33, 34, 35 | nfov 7200 |
. . . . . . . . . 10
⊢
Ⅎ𝑛((𝐴‘0) / 2) |
37 | | nfcv 2899 |
. . . . . . . . . 10
⊢
Ⅎ𝑛
+ |
38 | | nfcv 2899 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛(1...𝑚) |
39 | 38 | nfsum1 15139 |
. . . . . . . . . 10
⊢
Ⅎ𝑛Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) |
40 | 36, 37, 39 | nfov 7200 |
. . . . . . . . 9
⊢
Ⅎ𝑛(((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) |
41 | 17, 40 | nfmpt 5127 |
. . . . . . . 8
⊢
Ⅎ𝑛(𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))) |
42 | 28, 41 | nfcxfr 2897 |
. . . . . . 7
⊢
Ⅎ𝑛𝑍 |
43 | | fourierdlem112.f |
. . . . . . . 8
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
44 | | fourierdlem112.25 |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ ℝ) |
45 | | eqid 2738 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ
↑m (0...𝑛))
∣ (((𝑝‘0) =
(-π + 𝑋) ∧ (𝑝‘𝑛) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑛)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑛) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
46 | | picn 25204 |
. . . . . . . . . . . . 13
⊢ π
∈ ℂ |
47 | 46 | 2timesi 11854 |
. . . . . . . . . . . 12
⊢ (2
· π) = (π + π) |
48 | | fourierdlem112.t |
. . . . . . . . . . . 12
⊢ 𝑇 = (2 ·
π) |
49 | 46, 46 | subnegi 11043 |
. . . . . . . . . . . 12
⊢ (π
− -π) = (π + π) |
50 | 47, 48, 49 | 3eqtr4i 2771 |
. . . . . . . . . . 11
⊢ 𝑇 = (π −
-π) |
51 | | fourierdlem112.p |
. . . . . . . . . . 11
⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
52 | | fourierdlem112.m |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℕ) |
53 | | fourierdlem112.q |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) |
54 | | pire 25203 |
. . . . . . . . . . . . . 14
⊢ π
∈ ℝ |
55 | 54 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → π ∈
ℝ) |
56 | 55 | renegcld 11145 |
. . . . . . . . . . . 12
⊢ (𝜑 → -π ∈
ℝ) |
57 | 56, 44 | readdcld 10748 |
. . . . . . . . . . 11
⊢ (𝜑 → (-π + 𝑋) ∈ ℝ) |
58 | 55, 44 | readdcld 10748 |
. . . . . . . . . . 11
⊢ (𝜑 → (π + 𝑋) ∈ ℝ) |
59 | | negpilt0 42356 |
. . . . . . . . . . . . . 14
⊢ -π
< 0 |
60 | | pipos 25205 |
. . . . . . . . . . . . . 14
⊢ 0 <
π |
61 | 54 | renegcli 11025 |
. . . . . . . . . . . . . . 15
⊢ -π
∈ ℝ |
62 | | 0re 10721 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
ℝ |
63 | 61, 62, 54 | lttri 10844 |
. . . . . . . . . . . . . 14
⊢ ((-π
< 0 ∧ 0 < π) → -π < π) |
64 | 59, 60, 63 | mp2an 692 |
. . . . . . . . . . . . 13
⊢ -π
< π |
65 | 64 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → -π <
π) |
66 | 56, 55, 44, 65 | ltadd1dd 11329 |
. . . . . . . . . . 11
⊢ (𝜑 → (-π + 𝑋) < (π + 𝑋)) |
67 | | oveq1 7177 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑥 → (𝑦 + (𝑘 · 𝑇)) = (𝑥 + (𝑘 · 𝑇))) |
68 | 67 | eleq1d 2817 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑥 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄)) |
69 | 68 | rexbidv 3207 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄)) |
70 | 69 | cbvrabv 3393 |
. . . . . . . . . . . 12
⊢ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑥 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} |
71 | 70 | uneq2i 4050 |
. . . . . . . . . . 11
⊢ ({(-π
+ 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑥 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) |
72 | | fourierdlem112.n |
. . . . . . . . . . 11
⊢ 𝑁 = ((♯‘({(-π +
𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1) |
73 | | fourierdlem112.v |
. . . . . . . . . . 11
⊢ 𝑉 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))) |
74 | 50, 51, 52, 53, 57, 58, 66, 45, 71, 72, 73 | fourierdlem54 43243 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑉 ∈ ((𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑛)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑛) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})‘𝑁)) ∧ 𝑉 Isom < , < ((0...𝑁), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})))) |
75 | 74 | simpld 498 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑉 ∈ ((𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑛)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑛) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})‘𝑁))) |
76 | 75 | simpld 498 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℕ) |
77 | 75 | simprd 499 |
. . . . . . . 8
⊢ (𝜑 → 𝑉 ∈ ((𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑛)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑛) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})‘𝑁)) |
78 | | fourierdlem112.xran |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ ran 𝑉) |
79 | 43 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℝ) |
80 | | fveq2 6674 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = 𝑗 → (𝑝‘𝑖) = (𝑝‘𝑗)) |
81 | | oveq1 7177 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1)) |
82 | 81 | fveq2d 6678 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = 𝑗 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑗 + 1))) |
83 | 80, 82 | breq12d 5043 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 = 𝑗 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))) |
84 | 83 | cbvralvw 3349 |
. . . . . . . . . . . . . 14
⊢
(∀𝑖 ∈
(0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑗 ∈ (0..^𝑛)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1))) |
85 | 84 | anbi2i 626 |
. . . . . . . . . . . . 13
⊢ ((((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))) |
86 | 85 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑝 ∈ (ℝ
↑m (0...𝑛))
→ ((((𝑝‘0) =
-π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1))))) |
87 | 86 | rabbiia 3373 |
. . . . . . . . . . 11
⊢ {𝑝 ∈ (ℝ
↑m (0...𝑛))
∣ (((𝑝‘0) =
-π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} = {𝑝 ∈ (ℝ ↑m
(0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))} |
88 | 87 | mpteq2i 5122 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ
↑m (0...𝑛))
∣ (((𝑝‘0) =
-π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))}) |
89 | 51, 88 | eqtri 2761 |
. . . . . . . . 9
⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))}) |
90 | 52 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ) |
91 | 53 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑄 ∈ (𝑃‘𝑀)) |
92 | | fourierdlem112.fper |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
93 | 92 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
94 | | eleq1w 2815 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀))) |
95 | 94 | anbi2d 632 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑗 ∈ (0..^𝑀)))) |
96 | | fveq2 6674 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 = 𝑗 → (𝑄‘𝑖) = (𝑄‘𝑗)) |
97 | 81 | fveq2d 6678 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1))) |
98 | 96, 97 | oveq12d 7188 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑗 → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) |
99 | 98 | reseq2d 5825 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1))))) |
100 | 98 | oveq1d 7185 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) = (((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ)) |
101 | 99, 100 | eleq12d 2827 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) ↔ (𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))) |
102 | 95, 101 | imbi12d 348 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) ↔ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ)))) |
103 | | fourierdlem112.fcn |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
104 | 102, 103 | chvarvv 2010 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ)) |
105 | 104 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ)) |
106 | 57 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (-π + 𝑋) ∈ ℝ) |
107 | 57 | rexrd 10769 |
. . . . . . . . . . 11
⊢ (𝜑 → (-π + 𝑋) ∈
ℝ*) |
108 | | pnfxr 10773 |
. . . . . . . . . . . 12
⊢ +∞
∈ ℝ* |
109 | 108 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → +∞ ∈
ℝ*) |
110 | 58 | ltpnfd 12599 |
. . . . . . . . . . 11
⊢ (𝜑 → (π + 𝑋) < +∞) |
111 | 107, 109,
58, 66, 110 | eliood 42576 |
. . . . . . . . . 10
⊢ (𝜑 → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞)) |
112 | 111 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞)) |
113 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^𝑁)) |
114 | 72 | oveq2i 7181 |
. . . . . . . . . . 11
⊢
(0..^𝑁) =
(0..^((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)) |
115 | 113, 114 | eleqtrdi 2843 |
. . . . . . . . . 10
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1))) |
116 | 115 | adantl 485 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1))) |
117 | 72 | oveq2i 7181 |
. . . . . . . . . . . 12
⊢
(0...𝑁) =
(0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)) |
118 | | isoeq4 7086 |
. . . . . . . . . . . 12
⊢
((0...𝑁) =
(0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)) → (𝑓 Isom < , < ((0...𝑁), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑓 Isom < , <
((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})))) |
119 | 117, 118 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (𝑓 Isom < , < ((0...𝑁), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑓 Isom < , <
((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))) |
120 | 119 | iotabii 6324 |
. . . . . . . . . 10
⊢
(℩𝑓𝑓 Isom < , < ((0...𝑁), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))) = (℩𝑓𝑓 Isom < , <
((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))) |
121 | 73, 120 | eqtri 2761 |
. . . . . . . . 9
⊢ 𝑉 = (℩𝑓𝑓 Isom < , <
((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))) |
122 | 79, 89, 50, 90, 91, 93, 105, 106, 112, 116, 121 | fourierdlem98 43287 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ)) |
123 | | fourierdlem112.fbd |
. . . . . . . . . 10
⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑤) |
124 | 123 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑤) |
125 | | nfra1 3131 |
. . . . . . . . . . 11
⊢
Ⅎ𝑡∀𝑡 ∈ ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑤 |
126 | | elioore 12851 |
. . . . . . . . . . . . 13
⊢ (𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) → 𝑡 ∈ ℝ) |
127 | | rspa 3119 |
. . . . . . . . . . . . 13
⊢
((∀𝑡 ∈
ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ 𝑡 ∈ ℝ) → (abs‘(𝐹‘𝑡)) ≤ 𝑤) |
128 | 126, 127 | sylan2 596 |
. . . . . . . . . . . 12
⊢
((∀𝑡 ∈
ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ 𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(𝐹‘𝑡)) ≤ 𝑤) |
129 | 128 | ex 416 |
. . . . . . . . . . 11
⊢
(∀𝑡 ∈
ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑤 → (𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘(𝐹‘𝑡)) ≤ 𝑤)) |
130 | 125, 129 | ralrimi 3128 |
. . . . . . . . . 10
⊢
(∀𝑡 ∈
ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑤 → ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(𝐹‘𝑡)) ≤ 𝑤) |
131 | 130 | reximi 3157 |
. . . . . . . . 9
⊢
(∃𝑤 ∈
ℝ ∀𝑡 ∈
ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑤 → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(𝐹‘𝑡)) ≤ 𝑤) |
132 | 124, 131 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(𝐹‘𝑡)) ≤ 𝑤) |
133 | | ssid 3899 |
. . . . . . . . . . . 12
⊢ ℝ
⊆ ℝ |
134 | | dvfre 24703 |
. . . . . . . . . . . 12
⊢ ((𝐹:ℝ⟶ℝ ∧
ℝ ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
135 | 43, 133, 134 | sylancl 589 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
136 | 135 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
137 | | eqid 2738 |
. . . . . . . . . . . . 13
⊢ (ℝ
D 𝐹) = (ℝ D 𝐹) |
138 | 54 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → π ∈
ℝ) |
139 | 61 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → -π ∈
ℝ) |
140 | 98 | reseq2d 5825 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑗 → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((ℝ D 𝐹) ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1))))) |
141 | 140, 100 | eleq12d 2827 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = 𝑗 → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) ↔ ((ℝ D 𝐹) ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))) |
142 | 95, 141 | imbi12d 348 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) ↔ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ)))) |
143 | | fourierdlem112.fdvcn |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
144 | 142, 143 | chvarvv 2010 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ)) |
145 | 144 | adantlr 715 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ)) |
146 | | fourierdlem112.x |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑋 ∈ ℝ) |
147 | 56, 146 | readdcld 10748 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (-π + 𝑋) ∈ ℝ) |
148 | 147 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (-π + 𝑋) ∈ ℝ) |
149 | 147 | rexrd 10769 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (-π + 𝑋) ∈
ℝ*) |
150 | 55, 146 | readdcld 10748 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (π + 𝑋) ∈ ℝ) |
151 | 56, 55, 146, 65 | ltadd1dd 11329 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (-π + 𝑋) < (π + 𝑋)) |
152 | 150 | ltpnfd 12599 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (π + 𝑋) < +∞) |
153 | 149, 109,
150, 151, 152 | eliood 42576 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞)) |
154 | 153 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞)) |
155 | | oveq1 7177 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = ℎ → (𝑘 · 𝑇) = (ℎ · 𝑇)) |
156 | 155 | oveq2d 7186 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = ℎ → (𝑦 + (𝑘 · 𝑇)) = (𝑦 + (ℎ · 𝑇))) |
157 | 156 | eleq1d 2817 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = ℎ → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄)) |
158 | 157 | cbvrexvw 3350 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∃𝑘 ∈
ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄) |
159 | 158 | rgenw 3065 |
. . . . . . . . . . . . . . . . . 18
⊢
∀𝑦 ∈
((-π + 𝑋)[,](π +
𝑋))(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄) |
160 | | rabbi 3286 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀𝑦 ∈
((-π + 𝑋)[,](π +
𝑋))(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄) ↔ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}) |
161 | 159, 160 | mpbi 233 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄} |
162 | 161 | uneq2i 4050 |
. . . . . . . . . . . . . . . 16
⊢ ({(-π
+ 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}) |
163 | | isoeq5 7087 |
. . . . . . . . . . . . . . . 16
⊢ (({(-π
+ 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}) → (𝑓 Isom < , <
((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑓 Isom < , <
((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄})))) |
164 | 162, 163 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 Isom < , <
((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑓 Isom < , <
((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}))) |
165 | 164 | iotabii 6324 |
. . . . . . . . . . . . . 14
⊢
(℩𝑓𝑓 Isom < , <
((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))) = (℩𝑓𝑓 Isom < , <
((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}))) |
166 | 121, 165 | eqtri 2761 |
. . . . . . . . . . . . 13
⊢ 𝑉 = (℩𝑓𝑓 Isom < , <
((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}))) |
167 | | eleq1w 2815 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = 𝑢 → (𝑣 ∈ dom (ℝ D 𝐹) ↔ 𝑢 ∈ dom (ℝ D 𝐹))) |
168 | | fveq2 6674 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = 𝑢 → ((ℝ D 𝐹)‘𝑣) = ((ℝ D 𝐹)‘𝑢)) |
169 | 167, 168 | ifbieq1d 4438 |
. . . . . . . . . . . . . 14
⊢ (𝑣 = 𝑢 → if(𝑣 ∈ dom (ℝ D 𝐹), ((ℝ D 𝐹)‘𝑣), 0) = if(𝑢 ∈ dom (ℝ D 𝐹), ((ℝ D 𝐹)‘𝑢), 0)) |
170 | 169 | cbvmptv 5133 |
. . . . . . . . . . . . 13
⊢ (𝑣 ∈ ℝ ↦ if(𝑣 ∈ dom (ℝ D 𝐹), ((ℝ D 𝐹)‘𝑣), 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ dom (ℝ D 𝐹), ((ℝ D 𝐹)‘𝑢), 0)) |
171 | 79, 137, 89, 138, 139, 50, 90, 91, 93, 145, 148, 154, 116, 166, 170 | fourierdlem97 43286 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ)) |
172 | | cncff 23645 |
. . . . . . . . . . . 12
⊢
(((ℝ D 𝐹)
↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ) → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ) |
173 | | fdm 6513 |
. . . . . . . . . . . 12
⊢
(((ℝ D 𝐹)
↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ → dom ((ℝ D
𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) |
174 | 171, 172,
173 | 3syl 18 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → dom ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) |
175 | | ssdmres 5848 |
. . . . . . . . . . 11
⊢ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) |
176 | 174, 175 | sylibr 237 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹)) |
177 | 136, 176 | fssresd 6545 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ) |
178 | | ax-resscn 10672 |
. . . . . . . . . . 11
⊢ ℝ
⊆ ℂ |
179 | 178 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ℝ ⊆
ℂ) |
180 | | cncffvrn 23650 |
. . . . . . . . . 10
⊢ ((ℝ
⊆ ℂ ∧ ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ)) → (((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ) ↔ ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)) |
181 | 179, 171,
180 | syl2anc 587 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ) ↔ ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)) |
182 | 177, 181 | mpbird 260 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ)) |
183 | | fourierdlem112.fdvbd |
. . . . . . . . . . 11
⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) |
184 | 183 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) |
185 | | nfv 1921 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑡(𝜑 ∧ 𝑖 ∈ (0..^𝑁)) |
186 | | nfra1 3131 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑡∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧 |
187 | 185, 186 | nfan 1906 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑡((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) |
188 | | fvres 6693 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) → (((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡) = ((ℝ D 𝐹)‘𝑡)) |
189 | 188 | adantl 485 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) → (((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡) = ((ℝ D 𝐹)‘𝑡)) |
190 | 189 | fveq2d 6678 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) = (abs‘((ℝ D 𝐹)‘𝑡))) |
191 | 190 | adantlr 715 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) = (abs‘((ℝ D 𝐹)‘𝑡))) |
192 | | simplr 769 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) → ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) |
193 | 176 | sselda 3877 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) → 𝑡 ∈ dom (ℝ D 𝐹)) |
194 | 193 | adantlr 715 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) → 𝑡 ∈ dom (ℝ D 𝐹)) |
195 | | rspa 3119 |
. . . . . . . . . . . . . . . 16
⊢
((∀𝑡 ∈
dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧 ∧ 𝑡 ∈ dom (ℝ D 𝐹)) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) |
196 | 192, 194,
195 | syl2anc 587 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) |
197 | 191, 196 | eqbrtrd 5052 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧) |
198 | 197 | ex 416 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) → (𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧)) |
199 | 187, 198 | ralrimi 3128 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) → ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧) |
200 | 199 | ex 416 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧 → ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧)) |
201 | 200 | reximdv 3183 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧)) |
202 | 184, 201 | mpd 15 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧) |
203 | | nfra1 3131 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑡∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 |
204 | 188 | eqcomd 2744 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) → ((ℝ D 𝐹)‘𝑡) = (((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) |
205 | 204 | fveq2d 6678 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘((ℝ D 𝐹)‘𝑡)) = (abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡))) |
206 | 205 | adantl 485 |
. . . . . . . . . . . . . 14
⊢
((∀𝑡 ∈
((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 ∧ 𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘((ℝ D 𝐹)‘𝑡)) = (abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡))) |
207 | | rspa 3119 |
. . . . . . . . . . . . . 14
⊢
((∀𝑡 ∈
((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 ∧ 𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧) |
208 | 206, 207 | eqbrtrd 5052 |
. . . . . . . . . . . . 13
⊢
((∀𝑡 ∈
((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 ∧ 𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) |
209 | 208 | ex 416 |
. . . . . . . . . . . 12
⊢
(∀𝑡 ∈
((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 → (𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)) |
210 | 203, 209 | ralrimi 3128 |
. . . . . . . . . . 11
⊢
(∀𝑡 ∈
((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 → ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) |
211 | 210 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 → ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)) |
212 | 211 | reximdv 3183 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)) |
213 | 202, 212 | mpd 15 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) |
214 | | nfv 1921 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ (0..^𝑀)) |
215 | | nfcsb1v 3814 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑖⦋𝑗 / 𝑖⦌𝐶 |
216 | 215 | nfel1 2915 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖⦋𝑗 / 𝑖⦌𝐶 ∈ ((𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) limℂ (𝑄‘𝑗)) |
217 | 214, 216 | nfim 1903 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ⦋𝑗 / 𝑖⦌𝐶 ∈ ((𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) limℂ (𝑄‘𝑗))) |
218 | | csbeq1a 3804 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → 𝐶 = ⦋𝑗 / 𝑖⦌𝐶) |
219 | 99, 96 | oveq12d 7188 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)) = ((𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) limℂ (𝑄‘𝑗))) |
220 | 218, 219 | eleq12d 2827 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑗 → (𝐶 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)) ↔ ⦋𝑗 / 𝑖⦌𝐶 ∈ ((𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) limℂ (𝑄‘𝑗)))) |
221 | 95, 220 | imbi12d 348 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) ↔ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ⦋𝑗 / 𝑖⦌𝐶 ∈ ((𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) limℂ (𝑄‘𝑗))))) |
222 | | fourierdlem112.c |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
223 | 217, 221,
222 | chvarfv 2242 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ⦋𝑗 / 𝑖⦌𝐶 ∈ ((𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) limℂ (𝑄‘𝑗))) |
224 | 223 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (0..^𝑀)) → ⦋𝑗 / 𝑖⦌𝐶 ∈ ((𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) limℂ (𝑄‘𝑗))) |
225 | 79, 89, 50, 90, 91, 93, 105, 224, 106, 112, 116, 121 | fourierdlem96 43285 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → if(((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘(𝑉‘𝑖))) = (𝑄‘((𝑦 ∈ ℝ ↦ sup({𝑓 ∈ (0..^𝑀) ∣ (𝑄‘𝑓) ≤ ((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉‘𝑖))), ((𝑗 ∈ (0..^𝑀) ↦ ⦋𝑗 / 𝑖⦌𝐶)‘((𝑦 ∈ ℝ ↦ sup({𝑓 ∈ (0..^𝑀) ∣ (𝑄‘𝑓) ≤ ((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉‘𝑖))), (𝐹‘((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘(𝑉‘𝑖))))) ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑉‘𝑖))) |
226 | | nfcsb1v 3814 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑖⦋𝑗 / 𝑖⦌𝑈 |
227 | 226 | nfel1 2915 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖⦋𝑗 / 𝑖⦌𝑈 ∈ ((𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) limℂ (𝑄‘(𝑗 + 1))) |
228 | 214, 227 | nfim 1903 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ⦋𝑗 / 𝑖⦌𝑈 ∈ ((𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) limℂ (𝑄‘(𝑗 + 1)))) |
229 | | csbeq1a 3804 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → 𝑈 = ⦋𝑗 / 𝑖⦌𝑈) |
230 | 99, 97 | oveq12d 7188 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) limℂ (𝑄‘(𝑗 + 1)))) |
231 | 229, 230 | eleq12d 2827 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑗 → (𝑈 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1))) ↔ ⦋𝑗 / 𝑖⦌𝑈 ∈ ((𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) limℂ (𝑄‘(𝑗 + 1))))) |
232 | 95, 231 | imbi12d 348 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) ↔ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ⦋𝑗 / 𝑖⦌𝑈 ∈ ((𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) limℂ (𝑄‘(𝑗 + 1)))))) |
233 | | fourierdlem112.u |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
234 | 228, 232,
233 | chvarfv 2242 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ⦋𝑗 / 𝑖⦌𝑈 ∈ ((𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) limℂ (𝑄‘(𝑗 + 1)))) |
235 | 234 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (0..^𝑀)) → ⦋𝑗 / 𝑖⦌𝑈 ∈ ((𝐹 ↾ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) limℂ (𝑄‘(𝑗 + 1)))) |
236 | 79, 89, 50, 90, 91, 93, 105, 235, 148, 154, 116, 121 | fourierdlem99 43288 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → if(((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘(𝑉‘(𝑖 + 1))) = (𝑄‘(((𝑦 ∈ ℝ ↦ sup({ℎ ∈ (0..^𝑀) ∣ (𝑄‘ℎ) ≤ ((𝑔 ∈ (-π(,]π) ↦ if(𝑔 = π, -π, 𝑔))‘((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉‘𝑖)) + 1)), ((𝑗 ∈ (0..^𝑀) ↦ ⦋𝑗 / 𝑖⦌𝑈)‘((𝑦 ∈ ℝ ↦ sup({ℎ ∈ (0..^𝑀) ∣ (𝑄‘ℎ) ≤ ((𝑔 ∈ (-π(,]π) ↦ if(𝑔 = π, -π, 𝑔))‘((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉‘𝑖))), (𝐹‘((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘(𝑉‘(𝑖 + 1))))) ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑉‘(𝑖 + 1)))) |
237 | | eqeq1 2742 |
. . . . . . . . . 10
⊢ (𝑔 = 𝑠 → (𝑔 = 0 ↔ 𝑠 = 0)) |
238 | | oveq2 7178 |
. . . . . . . . . . . . 13
⊢ (𝑔 = 𝑠 → (𝑋 + 𝑔) = (𝑋 + 𝑠)) |
239 | 238 | fveq2d 6678 |
. . . . . . . . . . . 12
⊢ (𝑔 = 𝑠 → (𝐹‘(𝑋 + 𝑔)) = (𝐹‘(𝑋 + 𝑠))) |
240 | | breq2 5034 |
. . . . . . . . . . . . 13
⊢ (𝑔 = 𝑠 → (0 < 𝑔 ↔ 0 < 𝑠)) |
241 | 240 | ifbid 4437 |
. . . . . . . . . . . 12
⊢ (𝑔 = 𝑠 → if(0 < 𝑔, 𝑅, 𝐿) = if(0 < 𝑠, 𝑅, 𝐿)) |
242 | 239, 241 | oveq12d 7188 |
. . . . . . . . . . 11
⊢ (𝑔 = 𝑠 → ((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) = ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑅, 𝐿))) |
243 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑔 = 𝑠 → 𝑔 = 𝑠) |
244 | 242, 243 | oveq12d 7188 |
. . . . . . . . . 10
⊢ (𝑔 = 𝑠 → (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑅, 𝐿)) / 𝑠)) |
245 | 237, 244 | ifbieq2d 4440 |
. . . . . . . . 9
⊢ (𝑔 = 𝑠 → if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑅, 𝐿)) / 𝑠))) |
246 | 245 | cbvmptv 5133 |
. . . . . . . 8
⊢ (𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔))) = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑅, 𝐿)) / 𝑠))) |
247 | | eqeq1 2742 |
. . . . . . . . . 10
⊢ (𝑜 = 𝑠 → (𝑜 = 0 ↔ 𝑠 = 0)) |
248 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑜 = 𝑠 → 𝑜 = 𝑠) |
249 | | oveq1 7177 |
. . . . . . . . . . . . 13
⊢ (𝑜 = 𝑠 → (𝑜 / 2) = (𝑠 / 2)) |
250 | 249 | fveq2d 6678 |
. . . . . . . . . . . 12
⊢ (𝑜 = 𝑠 → (sin‘(𝑜 / 2)) = (sin‘(𝑠 / 2))) |
251 | 250 | oveq2d 7186 |
. . . . . . . . . . 11
⊢ (𝑜 = 𝑠 → (2 · (sin‘(𝑜 / 2))) = (2 ·
(sin‘(𝑠 /
2)))) |
252 | 248, 251 | oveq12d 7188 |
. . . . . . . . . 10
⊢ (𝑜 = 𝑠 → (𝑜 / (2 · (sin‘(𝑜 / 2)))) = (𝑠 / (2 · (sin‘(𝑠 / 2))))) |
253 | 247, 252 | ifbieq2d 4440 |
. . . . . . . . 9
⊢ (𝑜 = 𝑠 → if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) |
254 | 253 | cbvmptv 5133 |
. . . . . . . 8
⊢ (𝑜 ∈ (-π[,]π) ↦
if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2)))))) = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) |
255 | | fveq2 6674 |
. . . . . . . . . 10
⊢ (𝑟 = 𝑠 → ((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) = ((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑠)) |
256 | | fveq2 6674 |
. . . . . . . . . 10
⊢ (𝑟 = 𝑠 → ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟) = ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑠)) |
257 | 255, 256 | oveq12d 7188 |
. . . . . . . . 9
⊢ (𝑟 = 𝑠 → (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)) = (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑠) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑠))) |
258 | 257 | cbvmptv 5133 |
. . . . . . . 8
⊢ (𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟))) = (𝑠 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑠) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑠))) |
259 | | oveq2 7178 |
. . . . . . . . . 10
⊢ (𝑑 = 𝑠 → ((𝑘 + (1 / 2)) · 𝑑) = ((𝑘 + (1 / 2)) · 𝑠)) |
260 | 259 | fveq2d 6678 |
. . . . . . . . 9
⊢ (𝑑 = 𝑠 → (sin‘((𝑘 + (1 / 2)) · 𝑑)) = (sin‘((𝑘 + (1 / 2)) · 𝑠))) |
261 | 260 | cbvmptv 5133 |
. . . . . . . 8
⊢ (𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑))) = (𝑠 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑠))) |
262 | | fveq2 6674 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑠 → ((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) = ((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠)) |
263 | | fveq2 6674 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑠 → ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧) = ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑠)) |
264 | 262, 263 | oveq12d 7188 |
. . . . . . . . 9
⊢ (𝑧 = 𝑠 → (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧)) = (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑠))) |
265 | 264 | cbvmptv 5133 |
. . . . . . . 8
⊢ (𝑧 ∈ (-π[,]π) ↦
(((𝑟 ∈ (-π[,]π)
↦ (((𝑔 ∈
(-π[,]π) ↦ if(𝑔
= 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧))) = (𝑠 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑠))) |
266 | | fveq2 6674 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑛 → (𝐷‘𝑚) = (𝐷‘𝑛)) |
267 | 266 | fveq1d 6676 |
. . . . . . . . . . . 12
⊢ (𝑚 = 𝑛 → ((𝐷‘𝑚)‘𝑠) = ((𝐷‘𝑛)‘𝑠)) |
268 | 267 | oveq2d 7186 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑛 → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) |
269 | 268 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝑚 = 𝑛 ∧ 𝑠 ∈ (-π(,)0)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) |
270 | 269 | itgeq2dv 24534 |
. . . . . . . . 9
⊢ (𝑚 = 𝑛 → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠 = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠) |
271 | 270 | cbvmptv 5133 |
. . . . . . . 8
⊢ (𝑚 ∈ ℕ ↦
∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠) = (𝑛 ∈ ℕ ↦
∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠) |
272 | | oveq1 7177 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑐 = 𝑘 → (𝑐 + (1 / 2)) = (𝑘 + (1 / 2))) |
273 | 272 | oveq1d 7185 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 = 𝑘 → ((𝑐 + (1 / 2)) · 𝑑) = ((𝑘 + (1 / 2)) · 𝑑)) |
274 | 273 | fveq2d 6678 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 = 𝑘 → (sin‘((𝑐 + (1 / 2)) · 𝑑)) = (sin‘((𝑘 + (1 / 2)) · 𝑑))) |
275 | 274 | mpteq2dv 5126 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐 = 𝑘 → (𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑐 + (1 / 2))
· 𝑑))) = (𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))) |
276 | 275 | fveq1d 6676 |
. . . . . . . . . . . . . . 15
⊢ (𝑐 = 𝑘 → ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑐 + (1 / 2))
· 𝑑)))‘𝑧) = ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧)) |
277 | 276 | oveq2d 7186 |
. . . . . . . . . . . . . 14
⊢ (𝑐 = 𝑘 → (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑐 + (1 / 2))
· 𝑑)))‘𝑧)) = (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧))) |
278 | 277 | mpteq2dv 5126 |
. . . . . . . . . . . . 13
⊢ (𝑐 = 𝑘 → (𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑐 + (1 / 2))
· 𝑑)))‘𝑧))) = (𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧)))) |
279 | 278 | fveq1d 6676 |
. . . . . . . . . . . 12
⊢ (𝑐 = 𝑘 → ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑐 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) = ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠)) |
280 | 279 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝑐 = 𝑘 ∧ 𝑠 ∈ (-π(,)0)) → ((𝑧 ∈ (-π[,]π) ↦
(((𝑟 ∈ (-π[,]π)
↦ (((𝑔 ∈
(-π[,]π) ↦ if(𝑔
= 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑐 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) = ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠)) |
281 | 280 | itgeq2dv 24534 |
. . . . . . . . . 10
⊢ (𝑐 = 𝑘 → ∫(-π(,)0)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑐 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) d𝑠 = ∫(-π(,)0)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) d𝑠) |
282 | 281 | oveq1d 7185 |
. . . . . . . . 9
⊢ (𝑐 = 𝑘 → (∫(-π(,)0)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑐 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π) = (∫(-π(,)0)((𝑧 ∈ (-π[,]π) ↦
(((𝑟 ∈ (-π[,]π)
↦ (((𝑔 ∈
(-π[,]π) ↦ if(𝑔
= 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π)) |
283 | 282 | cbvmptv 5133 |
. . . . . . . 8
⊢ (𝑐 ∈ ℕ ↦
(∫(-π(,)0)((𝑧 ∈
(-π[,]π) ↦ (((𝑟
∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑐 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π)) = (𝑘 ∈ ℕ ↦
(∫(-π(,)0)((𝑧 ∈
(-π[,]π) ↦ (((𝑟
∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π)) |
284 | | fourierdlem112.r |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) limℂ 𝑋)) |
285 | | fourierdlem112.l |
. . . . . . . 8
⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋)) |
286 | | fourierdlem112.e |
. . . . . . . 8
⊢ (𝜑 → 𝐸 ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) limℂ 𝑋)) |
287 | | fourierdlem112.i |
. . . . . . . 8
⊢ (𝜑 → 𝐼 ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) limℂ 𝑋)) |
288 | | fourierdlem112.d |
. . . . . . . . 9
⊢ 𝐷 = (𝑚 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0,
(((2 · 𝑚) + 1) / (2
· π)), ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) ·
(sin‘(𝑦 /
2))))))) |
289 | | oveq1 7177 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑠 → (𝑦 mod (2 · π)) = (𝑠 mod (2 · π))) |
290 | 289 | eqeq1d 2740 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑠 → ((𝑦 mod (2 · π)) = 0 ↔ (𝑠 mod (2 · π)) =
0)) |
291 | | oveq2 7178 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑠 → ((𝑚 + (1 / 2)) · 𝑦) = ((𝑚 + (1 / 2)) · 𝑠)) |
292 | 291 | fveq2d 6678 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑠 → (sin‘((𝑚 + (1 / 2)) · 𝑦)) = (sin‘((𝑚 + (1 / 2)) · 𝑠))) |
293 | | oveq1 7177 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑠 → (𝑦 / 2) = (𝑠 / 2)) |
294 | 293 | fveq2d 6678 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑠 → (sin‘(𝑦 / 2)) = (sin‘(𝑠 / 2))) |
295 | 294 | oveq2d 7186 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑠 → ((2 · π) ·
(sin‘(𝑦 / 2))) = ((2
· π) · (sin‘(𝑠 / 2)))) |
296 | 292, 295 | oveq12d 7188 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑠 → ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) ·
(sin‘(𝑦 / 2)))) =
((sin‘((𝑚 + (1 / 2))
· 𝑠)) / ((2 ·
π) · (sin‘(𝑠 / 2))))) |
297 | 290, 296 | ifbieq2d 4440 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑠 → if((𝑦 mod (2 · π)) = 0, (((2 ·
𝑚) + 1) / (2 ·
π)), ((sin‘((𝑚 +
(1 / 2)) · 𝑦)) / ((2
· π) · (sin‘(𝑦 / 2))))) = if((𝑠 mod (2 · π)) = 0, (((2 ·
𝑚) + 1) / (2 ·
π)), ((sin‘((𝑚 +
(1 / 2)) · 𝑠)) / ((2
· π) · (sin‘(𝑠 / 2)))))) |
298 | 297 | cbvmptv 5133 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℝ ↦
if((𝑦 mod (2 ·
π)) = 0, (((2 · 𝑚) + 1) / (2 · π)),
((sin‘((𝑚 + (1 / 2))
· 𝑦)) / ((2 ·
π) · (sin‘(𝑦 / 2)))))) = (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0,
(((2 · 𝑚) + 1) / (2
· π)), ((sin‘((𝑚 + (1 / 2)) · 𝑠)) / ((2 · π) ·
(sin‘(𝑠 /
2)))))) |
299 | | simpl 486 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 = 𝑘 ∧ 𝑠 ∈ ℝ) → 𝑚 = 𝑘) |
300 | 299 | oveq2d 7186 |
. . . . . . . . . . . . . . 15
⊢ ((𝑚 = 𝑘 ∧ 𝑠 ∈ ℝ) → (2 · 𝑚) = (2 · 𝑘)) |
301 | 300 | oveq1d 7185 |
. . . . . . . . . . . . . 14
⊢ ((𝑚 = 𝑘 ∧ 𝑠 ∈ ℝ) → ((2 · 𝑚) + 1) = ((2 · 𝑘) + 1)) |
302 | 301 | oveq1d 7185 |
. . . . . . . . . . . . 13
⊢ ((𝑚 = 𝑘 ∧ 𝑠 ∈ ℝ) → (((2 · 𝑚) + 1) / (2 · π)) =
(((2 · 𝑘) + 1) / (2
· π))) |
303 | 299 | oveq1d 7185 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 = 𝑘 ∧ 𝑠 ∈ ℝ) → (𝑚 + (1 / 2)) = (𝑘 + (1 / 2))) |
304 | 303 | oveq1d 7185 |
. . . . . . . . . . . . . . 15
⊢ ((𝑚 = 𝑘 ∧ 𝑠 ∈ ℝ) → ((𝑚 + (1 / 2)) · 𝑠) = ((𝑘 + (1 / 2)) · 𝑠)) |
305 | 304 | fveq2d 6678 |
. . . . . . . . . . . . . 14
⊢ ((𝑚 = 𝑘 ∧ 𝑠 ∈ ℝ) → (sin‘((𝑚 + (1 / 2)) · 𝑠)) = (sin‘((𝑘 + (1 / 2)) · 𝑠))) |
306 | 305 | oveq1d 7185 |
. . . . . . . . . . . . 13
⊢ ((𝑚 = 𝑘 ∧ 𝑠 ∈ ℝ) → ((sin‘((𝑚 + (1 / 2)) · 𝑠)) / ((2 · π) ·
(sin‘(𝑠 / 2)))) =
((sin‘((𝑘 + (1 / 2))
· 𝑠)) / ((2 ·
π) · (sin‘(𝑠 / 2))))) |
307 | 302, 306 | ifeq12d 4435 |
. . . . . . . . . . . 12
⊢ ((𝑚 = 𝑘 ∧ 𝑠 ∈ ℝ) → if((𝑠 mod (2 · π)) = 0,
(((2 · 𝑚) + 1) / (2
· π)), ((sin‘((𝑚 + (1 / 2)) · 𝑠)) / ((2 · π) ·
(sin‘(𝑠 / 2))))) =
if((𝑠 mod (2 ·
π)) = 0, (((2 · 𝑘) + 1) / (2 · π)),
((sin‘((𝑘 + (1 / 2))
· 𝑠)) / ((2 ·
π) · (sin‘(𝑠 / 2)))))) |
308 | 307 | mpteq2dva 5125 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑘 → (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0,
(((2 · 𝑚) + 1) / (2
· π)), ((sin‘((𝑚 + (1 / 2)) · 𝑠)) / ((2 · π) ·
(sin‘(𝑠 / 2)))))) =
(𝑠 ∈ ℝ ↦
if((𝑠 mod (2 ·
π)) = 0, (((2 · 𝑘) + 1) / (2 · π)),
((sin‘((𝑘 + (1 / 2))
· 𝑠)) / ((2 ·
π) · (sin‘(𝑠 / 2))))))) |
309 | 298, 308 | syl5eq 2785 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑘 → (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0,
(((2 · 𝑚) + 1) / (2
· π)), ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) ·
(sin‘(𝑦 / 2)))))) =
(𝑠 ∈ ℝ ↦
if((𝑠 mod (2 ·
π)) = 0, (((2 · 𝑘) + 1) / (2 · π)),
((sin‘((𝑘 + (1 / 2))
· 𝑠)) / ((2 ·
π) · (sin‘(𝑠 / 2))))))) |
310 | 309 | cbvmptv 5133 |
. . . . . . . . 9
⊢ (𝑚 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦
if((𝑦 mod (2 ·
π)) = 0, (((2 · 𝑚) + 1) / (2 · π)),
((sin‘((𝑚 + (1 / 2))
· 𝑦)) / ((2 ·
π) · (sin‘(𝑦 / 2))))))) = (𝑘 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0,
(((2 · 𝑘) + 1) / (2
· π)), ((sin‘((𝑘 + (1 / 2)) · 𝑠)) / ((2 · π) ·
(sin‘(𝑠 /
2))))))) |
311 | 288, 310 | eqtri 2761 |
. . . . . . . 8
⊢ 𝐷 = (𝑘 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0,
(((2 · 𝑘) + 1) / (2
· π)), ((sin‘((𝑘 + (1 / 2)) · 𝑠)) / ((2 · π) ·
(sin‘(𝑠 /
2))))))) |
312 | | eqid 2738 |
. . . . . . . 8
⊢ ((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟))) ↾ (-π[,]𝑙)) = ((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟))) ↾ (-π[,]𝑙)) |
313 | | eqid 2738 |
. . . . . . . 8
⊢ ({-π,
𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙))) = ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙))) |
314 | | eqid 2738 |
. . . . . . . 8
⊢
((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1) = ((♯‘({-π,
𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1) |
315 | | isoeq1 7083 |
. . . . . . . . 9
⊢ (𝑢 = 𝑤 → (𝑢 Isom < , <
((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) ↔ 𝑤 Isom < , <
((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))))) |
316 | 315 | cbviotavw 6305 |
. . . . . . . 8
⊢
(℩𝑢𝑢 Isom < , <
((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙))))) = (℩𝑤𝑤 Isom < , <
((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙))))) |
317 | | fveq2 6674 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑖 → (𝑉‘𝑗) = (𝑉‘𝑖)) |
318 | 317 | oveq1d 7185 |
. . . . . . . . 9
⊢ (𝑗 = 𝑖 → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘𝑖) − 𝑋)) |
319 | 318 | cbvmptv 5133 |
. . . . . . . 8
⊢ (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) = (𝑖 ∈ (0...𝑁) ↦ ((𝑉‘𝑖) − 𝑋)) |
320 | | eqid 2738 |
. . . . . . . 8
⊢
(℩𝑚
∈ (0..^𝑁)(((℩𝑢𝑢 Isom < , <
((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))))‘𝑏)(,)((℩𝑢𝑢 Isom < , <
((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))))‘(𝑏 + 1))) ⊆ (((𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋))‘𝑚)(,)((𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋))‘(𝑚 + 1)))) = (℩𝑚 ∈ (0..^𝑁)(((℩𝑢𝑢 Isom < , <
((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))))‘𝑏)(,)((℩𝑢𝑢 Isom < , <
((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (-π(,)𝑙)))))‘(𝑏 + 1))) ⊆ (((𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋))‘𝑚)(,)((𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋))‘(𝑚 + 1)))) |
321 | | fveq2 6674 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑠 → ((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) = ((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠)) |
322 | | oveq2 7178 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑠 → ((𝑏 + (1 / 2)) · 𝑎) = ((𝑏 + (1 / 2)) · 𝑠)) |
323 | 322 | fveq2d 6678 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑠 → (sin‘((𝑏 + (1 / 2)) · 𝑎)) = (sin‘((𝑏 + (1 / 2)) · 𝑠))) |
324 | 321, 323 | oveq12d 7188 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑠 → (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) = (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠)))) |
325 | 324 | cbvitgv 24529 |
. . . . . . . . . . . 12
⊢
∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎 = ∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠 |
326 | 325 | fveq2i 6677 |
. . . . . . . . . . 11
⊢
(abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) = (abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) |
327 | 326 | breq1i 5037 |
. . . . . . . . . 10
⊢
((abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑖 / 2) ↔ (abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑖 / 2)) |
328 | 327 | anbi2i 626 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑖 ∈ ℝ+)
∧ 𝑙 ∈ (-π(,)0))
∧ 𝑏 ∈ ℕ)
∧ (abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑖 / 2)) ↔ ((((𝜑 ∧ 𝑖 ∈ ℝ+) ∧ 𝑙 ∈ (-π(,)0)) ∧ 𝑏 ∈ ℕ) ∧
(abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑖 / 2))) |
329 | 324 | cbvitgv 24529 |
. . . . . . . . . . 11
⊢
∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎 = ∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠 |
330 | 329 | fveq2i 6677 |
. . . . . . . . . 10
⊢
(abs‘∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) = (abs‘∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) |
331 | 330 | breq1i 5037 |
. . . . . . . . 9
⊢
((abs‘∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑖 / 2) ↔ (abs‘∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑖 / 2)) |
332 | 328, 331 | anbi12i 630 |
. . . . . . . 8
⊢
((((((𝜑 ∧ 𝑖 ∈ ℝ+)
∧ 𝑙 ∈ (-π(,)0))
∧ 𝑏 ∈ ℕ)
∧ (abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑖 / 2)) ∧ (abs‘∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑖 / 2)) ↔ (((((𝜑 ∧ 𝑖 ∈ ℝ+) ∧ 𝑙 ∈ (-π(,)0)) ∧ 𝑏 ∈ ℕ) ∧
(abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑖 / 2)) ∧ (abs‘∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑖 / 2))) |
333 | 43, 44, 45, 76, 77, 78, 122, 132, 182, 213, 225, 236, 246, 254, 258, 261, 265, 271, 283, 284, 285, 286, 287, 311, 312, 313, 314, 316, 319, 320, 332 | fourierdlem103 43292 |
. . . . . . 7
⊢ (𝜑 → (𝑚 ∈ ℕ ↦
∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠) ⇝ (𝐿 / 2)) |
334 | | nnex 11722 |
. . . . . . . . . 10
⊢ ℕ
∈ V |
335 | 334 | mptex 6996 |
. . . . . . . . 9
⊢ (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))) ∈ V |
336 | 28, 335 | eqeltri 2829 |
. . . . . . . 8
⊢ 𝑍 ∈ V |
337 | 336 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 𝑍 ∈ V) |
338 | 268 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝑚 = 𝑛 ∧ 𝑠 ∈ (0(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) |
339 | 338 | itgeq2dv 24534 |
. . . . . . . . 9
⊢ (𝑚 = 𝑛 → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠 = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠) |
340 | 339 | cbvmptv 5133 |
. . . . . . . 8
⊢ (𝑚 ∈ ℕ ↦
∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠) = (𝑛 ∈ ℕ ↦
∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠) |
341 | 279 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝑐 = 𝑘 ∧ 𝑠 ∈ (0(,)π)) → ((𝑧 ∈ (-π[,]π) ↦
(((𝑟 ∈ (-π[,]π)
↦ (((𝑔 ∈
(-π[,]π) ↦ if(𝑔
= 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑐 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) = ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠)) |
342 | 341 | itgeq2dv 24534 |
. . . . . . . . . 10
⊢ (𝑐 = 𝑘 → ∫(0(,)π)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑐 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) d𝑠 = ∫(0(,)π)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) d𝑠) |
343 | 342 | oveq1d 7185 |
. . . . . . . . 9
⊢ (𝑐 = 𝑘 → (∫(0(,)π)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑐 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π) = (∫(0(,)π)((𝑧 ∈ (-π[,]π) ↦
(((𝑟 ∈ (-π[,]π)
↦ (((𝑔 ∈
(-π[,]π) ↦ if(𝑔
= 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π)) |
344 | 343 | cbvmptv 5133 |
. . . . . . . 8
⊢ (𝑐 ∈ ℕ ↦
(∫(0(,)π)((𝑧 ∈
(-π[,]π) ↦ (((𝑟
∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑐 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π)) = (𝑘 ∈ ℕ ↦
(∫(0(,)π)((𝑧 ∈
(-π[,]π) ↦ (((𝑟
∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦
(sin‘((𝑘 + (1 / 2))
· 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π)) |
345 | | eqid 2738 |
. . . . . . . 8
⊢ ((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟))) ↾ (𝑒[,]π)) = ((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟))) ↾ (𝑒[,]π)) |
346 | | eqid 2738 |
. . . . . . . 8
⊢ ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π))) = ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π))) |
347 | | eqid 2738 |
. . . . . . . 8
⊢
((♯‘({𝑒,
π} ∪ (ran (𝑗 ∈
(0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1) =
((♯‘({𝑒, π}
∪ (ran (𝑗 ∈
(0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1) |
348 | | isoeq1 7083 |
. . . . . . . . 9
⊢ (𝑢 = 𝑣 → (𝑢 Isom < , <
((0...((♯‘({𝑒,
π} ∪ (ran (𝑗 ∈
(0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))) ↔ 𝑣 Isom < , <
((0...((♯‘({𝑒,
π} ∪ (ran (𝑗 ∈
(0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))))) |
349 | 348 | cbviotavw 6305 |
. . . . . . . 8
⊢
(℩𝑢𝑢 Isom < , <
((0...((♯‘({𝑒,
π} ∪ (ran (𝑗 ∈
(0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π))))) = (℩𝑣𝑣 Isom < , <
((0...((♯‘({𝑒,
π} ∪ (ran (𝑗 ∈
(0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π))))) |
350 | | eqid 2738 |
. . . . . . . 8
⊢
(℩𝑎
∈ (0..^𝑁)(((℩𝑢𝑢 Isom < , <
((0...((♯‘({𝑒,
π} ∪ (ran (𝑗 ∈
(0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))))‘𝑏)(,)((℩𝑢𝑢 Isom < , <
((0...((♯‘({𝑒,
π} ∪ (ran (𝑗 ∈
(0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))))‘(𝑏 + 1))) ⊆ (((𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋))‘𝑎)(,)((𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋))‘(𝑎 + 1)))) = (℩𝑎 ∈ (0..^𝑁)(((℩𝑢𝑢 Isom < , <
((0...((♯‘({𝑒,
π} ∪ (ran (𝑗 ∈
(0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))))‘𝑏)(,)((℩𝑢𝑢 Isom < , <
((0...((♯‘({𝑒,
π} ∪ (ran (𝑗 ∈
(0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋)) ∩ (𝑒(,)π)))))‘(𝑏 + 1))) ⊆ (((𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋))‘𝑎)(,)((𝑗 ∈ (0...𝑁) ↦ ((𝑉‘𝑗) − 𝑋))‘(𝑎 + 1)))) |
351 | 324 | cbvitgv 24529 |
. . . . . . . . . . . 12
⊢
∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦
(((𝑔 ∈ (-π[,]π)
↦ if(𝑔 = 0, 0,
(((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎 = ∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠 |
352 | 351 | fveq2i 6677 |
. . . . . . . . . . 11
⊢
(abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) = (abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) |
353 | 352 | breq1i 5037 |
. . . . . . . . . 10
⊢
((abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑞 / 2) ↔ (abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑞 / 2)) |
354 | 353 | anbi2i 626 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑞 ∈ ℝ+)
∧ 𝑒 ∈ (0(,)π))
∧ 𝑏 ∈ ℕ)
∧ (abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑞 / 2)) ↔ ((((𝜑 ∧ 𝑞 ∈ ℝ+) ∧ 𝑒 ∈ (0(,)π)) ∧ 𝑏 ∈ ℕ) ∧
(abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑞 / 2))) |
355 | 324 | cbvitgv 24529 |
. . . . . . . . . . 11
⊢
∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎 = ∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠 |
356 | 355 | fveq2i 6677 |
. . . . . . . . . 10
⊢
(abs‘∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) = (abs‘∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) |
357 | 356 | breq1i 5037 |
. . . . . . . . 9
⊢
((abs‘∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑞 / 2) ↔ (abs‘∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑞 / 2)) |
358 | 354, 357 | anbi12i 630 |
. . . . . . . 8
⊢
((((((𝜑 ∧ 𝑞 ∈ ℝ+)
∧ 𝑒 ∈ (0(,)π))
∧ 𝑏 ∈ ℕ)
∧ (abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑞 / 2)) ∧ (abs‘∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑞 / 2)) ↔ (((((𝜑 ∧ 𝑞 ∈ ℝ+) ∧ 𝑒 ∈ (0(,)π)) ∧ 𝑏 ∈ ℕ) ∧
(abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑞 / 2)) ∧ (abs‘∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦
if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑞 / 2))) |
359 | 43, 44, 45, 76, 77, 78, 122, 132, 182, 213, 225, 236, 246, 254, 258, 261, 265, 340, 344, 284, 285, 286, 287, 311, 345, 346, 347, 349, 319, 350, 358 | fourierdlem104 43293 |
. . . . . . 7
⊢ (𝜑 → (𝑚 ∈ ℕ ↦
∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠) ⇝ (𝑅 / 2)) |
360 | | eqidd 2739 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑚 ∈ ℕ ↦
∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠) = (𝑚 ∈ ℕ ↦
∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠)) |
361 | 270 | adantl 485 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 = 𝑛) → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠 = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠) |
362 | | simpr 488 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) |
363 | | elioore 12851 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ (-π(,)0) → 𝑠 ∈
ℝ) |
364 | 43 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝐹:ℝ⟶ℝ) |
365 | 44 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑋 ∈ ℝ) |
366 | | simpr 488 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℝ) |
367 | 365, 366 | readdcld 10748 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝑋 + 𝑠) ∈ ℝ) |
368 | 364, 367 | ffvelrnd 6862 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ) |
369 | 368 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ) |
370 | 288 | dirkerre 43178 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ) |
371 | 370 | adantll 714 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ) |
372 | 369, 371 | remulcld 10749 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℝ) |
373 | 363, 372 | sylan2 596 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π(,)0)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℝ) |
374 | | ioossicc 12907 |
. . . . . . . . . . . . 13
⊢
(-π(,)0) ⊆ (-π[,]0) |
375 | 61 | leidi 11252 |
. . . . . . . . . . . . . 14
⊢ -π
≤ -π |
376 | 62, 54, 60 | ltleii 10841 |
. . . . . . . . . . . . . 14
⊢ 0 ≤
π |
377 | | iccss 12889 |
. . . . . . . . . . . . . 14
⊢ (((-π
∈ ℝ ∧ π ∈ ℝ) ∧ (-π ≤ -π ∧ 0 ≤
π)) → (-π[,]0) ⊆ (-π[,]π)) |
378 | 61, 54, 375, 376, 377 | mp4an 693 |
. . . . . . . . . . . . 13
⊢
(-π[,]0) ⊆ (-π[,]π) |
379 | 374, 378 | sstri 3886 |
. . . . . . . . . . . 12
⊢
(-π(,)0) ⊆ (-π[,]π) |
380 | 379 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (-π(,)0) ⊆
(-π[,]π)) |
381 | | ioombl 24317 |
. . . . . . . . . . . 12
⊢
(-π(,)0) ∈ dom vol |
382 | 381 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (-π(,)0) ∈
dom vol) |
383 | 43 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → 𝐹:ℝ⟶ℝ) |
384 | 44 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → 𝑋 ∈
ℝ) |
385 | 56, 55 | iccssred 12908 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (-π[,]π) ⊆
ℝ) |
386 | 385 | sselda 3877 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → 𝑠 ∈
ℝ) |
387 | 384, 386 | readdcld 10748 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑋 + 𝑠) ∈ ℝ) |
388 | 383, 387 | ffvelrnd 6862 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ) |
389 | 388 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ) |
390 | | iccssre 12903 |
. . . . . . . . . . . . . . . 16
⊢ ((-π
∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆
ℝ) |
391 | 61, 54, 390 | mp2an 692 |
. . . . . . . . . . . . . . 15
⊢
(-π[,]π) ⊆ ℝ |
392 | 391 | sseli 3873 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ (-π[,]π) →
𝑠 ∈
ℝ) |
393 | 392, 370 | sylan2 596 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ (-π[,]π)) →
((𝐷‘𝑛)‘𝑠) ∈ ℝ) |
394 | 393 | adantll 714 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ) |
395 | 389, 394 | remulcld 10749 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℝ) |
396 | 61 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -π ∈
ℝ) |
397 | 54 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ∈
ℝ) |
398 | 43 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℝ⟶ℝ) |
399 | 44 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ ℝ) |
400 | 76 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑁 ∈ ℕ) |
401 | 77 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑉 ∈ ((𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑛)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑛) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})‘𝑁)) |
402 | 122 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ)) |
403 | 225 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑁)) → if(((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘(𝑉‘𝑖))) = (𝑄‘((𝑦 ∈ ℝ ↦ sup({𝑓 ∈ (0..^𝑀) ∣ (𝑄‘𝑓) ≤ ((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉‘𝑖))), ((𝑗 ∈ (0..^𝑀) ↦ ⦋𝑗 / 𝑖⦌𝐶)‘((𝑦 ∈ ℝ ↦ sup({𝑓 ∈ (0..^𝑀) ∣ (𝑄‘𝑓) ≤ ((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉‘𝑖))), (𝐹‘((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘(𝑉‘𝑖))))) ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑉‘𝑖))) |
404 | 236 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑁)) → if(((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘(𝑉‘(𝑖 + 1))) = (𝑄‘(((𝑦 ∈ ℝ ↦ sup({ℎ ∈ (0..^𝑀) ∣ (𝑄‘ℎ) ≤ ((𝑔 ∈ (-π(,]π) ↦ if(𝑔 = π, -π, 𝑔))‘((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉‘𝑖)) + 1)), ((𝑗 ∈ (0..^𝑀) ↦ ⦋𝑗 / 𝑖⦌𝑈)‘((𝑦 ∈ ℝ ↦ sup({ℎ ∈ (0..^𝑀) ∣ (𝑄‘ℎ) ≤ ((𝑔 ∈ (-π(,]π) ↦ if(𝑔 = π, -π, 𝑔))‘((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉‘𝑖))), (𝐹‘((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘(𝑉‘(𝑖 + 1))))) ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑉‘(𝑖 + 1)))) |
405 | 288 | dirkercncf 43190 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛) ∈ (ℝ–cn→ℝ)) |
406 | 405 | adantl 485 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ (ℝ–cn→ℝ)) |
407 | | eqid 2738 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ (-π[,]π) ↦
((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) = (𝑠 ∈ (-π[,]π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) |
408 | 396, 397,
398, 399, 45, 400, 401, 402, 403, 404, 319, 51, 406, 407 | fourierdlem84 43273 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) ∈
𝐿1) |
409 | 380, 382,
395, 408 | iblss 24557 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (-π(,)0) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) ∈
𝐿1) |
410 | 373, 409 | itgcl 24536 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) →
∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 ∈ ℂ) |
411 | 360, 361,
362, 410 | fvmptd 6782 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦
∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠)‘𝑛) = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠) |
412 | 411, 410 | eqeltrd 2833 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦
∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠)‘𝑛) ∈ ℂ) |
413 | | eqidd 2739 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑚 ∈ ℕ ↦
∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠) = (𝑚 ∈ ℕ ↦
∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠)) |
414 | 339 | adantl 485 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 = 𝑛) → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠 = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠) |
415 | 43 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)π)) → 𝐹:ℝ⟶ℝ) |
416 | 44 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)π)) → 𝑋 ∈ ℝ) |
417 | | elioore 12851 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈ (0(,)π) → 𝑠 ∈
ℝ) |
418 | 417 | adantl 485 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)π)) → 𝑠 ∈ ℝ) |
419 | 416, 418 | readdcld 10748 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)π)) → (𝑋 + 𝑠) ∈ ℝ) |
420 | 415, 419 | ffvelrnd 6862 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ) |
421 | 420 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ) |
422 | 417, 370 | sylan2 596 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ (0(,)π)) →
((𝐷‘𝑛)‘𝑠) ∈ ℝ) |
423 | 422 | adantll 714 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ) |
424 | 421, 423 | remulcld 10749 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℝ) |
425 | | ioossicc 12907 |
. . . . . . . . . . . . 13
⊢
(0(,)π) ⊆ (0[,]π) |
426 | 61, 62, 59 | ltleii 10841 |
. . . . . . . . . . . . . 14
⊢ -π
≤ 0 |
427 | 54 | leidi 11252 |
. . . . . . . . . . . . . 14
⊢ π ≤
π |
428 | | iccss 12889 |
. . . . . . . . . . . . . 14
⊢ (((-π
∈ ℝ ∧ π ∈ ℝ) ∧ (-π ≤ 0 ∧ π ≤
π)) → (0[,]π) ⊆ (-π[,]π)) |
429 | 61, 54, 426, 427, 428 | mp4an 693 |
. . . . . . . . . . . . 13
⊢
(0[,]π) ⊆ (-π[,]π) |
430 | 425, 429 | sstri 3886 |
. . . . . . . . . . . 12
⊢
(0(,)π) ⊆ (-π[,]π) |
431 | 430 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0(,)π) ⊆
(-π[,]π)) |
432 | | ioombl 24317 |
. . . . . . . . . . . 12
⊢
(0(,)π) ∈ dom vol |
433 | 432 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0(,)π) ∈ dom
vol) |
434 | 431, 433,
395, 408 | iblss 24557 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (0(,)π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) ∈
𝐿1) |
435 | 424, 434 | itgcl 24536 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) →
∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 ∈ ℂ) |
436 | 413, 414,
362, 435 | fvmptd 6782 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦
∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠)‘𝑛) = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠) |
437 | 436, 435 | eqeltrd 2833 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦
∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠)‘𝑛) ∈ ℂ) |
438 | | eleq1w 2815 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑛 → (𝑚 ∈ ℕ ↔ 𝑛 ∈ ℕ)) |
439 | 438 | anbi2d 632 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑛 → ((𝜑 ∧ 𝑚 ∈ ℕ) ↔ (𝜑 ∧ 𝑛 ∈ ℕ))) |
440 | | fveq2 6674 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑛 → (𝑍‘𝑚) = (𝑍‘𝑛)) |
441 | 270, 339 | oveq12d 7188 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑛 → (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)) |
442 | 440, 441 | eqeq12d 2754 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑛 → ((𝑍‘𝑚) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠) ↔ (𝑍‘𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠))) |
443 | 439, 442 | imbi12d 348 |
. . . . . . . . 9
⊢ (𝑚 = 𝑛 → (((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑍‘𝑚) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠)) ↔ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑍‘𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)))) |
444 | | oveq1 7177 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑚 → (𝑛 · 𝑥) = (𝑚 · 𝑥)) |
445 | 444 | fveq2d 6678 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑚 → (cos‘(𝑛 · 𝑥)) = (cos‘(𝑚 · 𝑥))) |
446 | 445 | oveq2d 7186 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → ((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) = ((𝐹‘𝑥) · (cos‘(𝑚 · 𝑥)))) |
447 | 446 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 = 𝑚 ∧ 𝑥 ∈ (-π(,)π)) → ((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) = ((𝐹‘𝑥) · (cos‘(𝑚 · 𝑥)))) |
448 | 447 | itgeq2dv 24534 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → ∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 = ∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥) |
449 | 448 | oveq1d 7185 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑚 → (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π) = (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥 / π)) |
450 | 449 | cbvmptv 5133 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ0
↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) = (𝑚 ∈ ℕ0 ↦
(∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥 / π)) |
451 | 29, 450 | eqtri 2761 |
. . . . . . . . . 10
⊢ 𝐴 = (𝑚 ∈ ℕ0 ↦
(∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥 / π)) |
452 | | fourierdlem112.b |
. . . . . . . . . . 11
⊢ 𝐵 = (𝑛 ∈ ℕ ↦
(∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) |
453 | 444 | fveq2d 6678 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑚 → (sin‘(𝑛 · 𝑥)) = (sin‘(𝑚 · 𝑥))) |
454 | 453 | oveq2d 7186 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → ((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) = ((𝐹‘𝑥) · (sin‘(𝑚 · 𝑥)))) |
455 | 454 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 = 𝑚 ∧ 𝑥 ∈ (-π(,)π)) → ((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) = ((𝐹‘𝑥) · (sin‘(𝑚 · 𝑥)))) |
456 | 455 | itgeq2dv 24534 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → ∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 = ∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥) |
457 | 456 | oveq1d 7185 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑚 → (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π) = (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π)) |
458 | 457 | cbvmptv 5133 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ ↦
(∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) = (𝑚 ∈ ℕ ↦
(∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π)) |
459 | 452, 458 | eqtri 2761 |
. . . . . . . . . 10
⊢ 𝐵 = (𝑚 ∈ ℕ ↦
(∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π)) |
460 | | fveq2 6674 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑘 → (𝐴‘𝑛) = (𝐴‘𝑘)) |
461 | | oveq1 7177 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋)) |
462 | 461 | fveq2d 6678 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋))) |
463 | 460, 462 | oveq12d 7188 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → ((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋)))) |
464 | | fveq2 6674 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑘 → (𝐵‘𝑛) = (𝐵‘𝑘)) |
465 | 461 | fveq2d 6678 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋))) |
466 | 464, 465 | oveq12d 7188 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) |
467 | 463, 466 | oveq12d 7188 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑘 → (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
468 | 467 | cbvsumv 15146 |
. . . . . . . . . . . . 13
⊢
Σ𝑛 ∈
(1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) = Σ𝑘 ∈ (1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) |
469 | 468 | oveq2i 7181 |
. . . . . . . . . . . 12
⊢ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
470 | 469 | mpteq2i 5122 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))) = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))) |
471 | | oveq2 7178 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛)) |
472 | 471 | sumeq1d 15151 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑛 → Σ𝑘 ∈ (1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
473 | 472 | oveq2d 7186 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑛 → (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) = (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))) |
474 | 473 | cbvmptv 5133 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))) = (𝑛 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))) |
475 | | fveq2 6674 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑚 → (𝐴‘𝑘) = (𝐴‘𝑚)) |
476 | | oveq1 7177 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑚 → (𝑘 · 𝑋) = (𝑚 · 𝑋)) |
477 | 476 | fveq2d 6678 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑚 → (cos‘(𝑘 · 𝑋)) = (cos‘(𝑚 · 𝑋))) |
478 | 475, 477 | oveq12d 7188 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑚 → ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) = ((𝐴‘𝑚) · (cos‘(𝑚 · 𝑋)))) |
479 | | fveq2 6674 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑚 → (𝐵‘𝑘) = (𝐵‘𝑚)) |
480 | 476 | fveq2d 6678 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑚 → (sin‘(𝑘 · 𝑋)) = (sin‘(𝑚 · 𝑋))) |
481 | 479, 480 | oveq12d 7188 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑚 → ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))) = ((𝐵‘𝑚) · (sin‘(𝑚 · 𝑋)))) |
482 | 478, 481 | oveq12d 7188 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑚 → (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = (((𝐴‘𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵‘𝑚) · (sin‘(𝑚 · 𝑋))))) |
483 | 482 | cbvsumv 15146 |
. . . . . . . . . . . . . 14
⊢
Σ𝑘 ∈
(1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑚 ∈ (1...𝑛)(((𝐴‘𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵‘𝑚) · (sin‘(𝑚 · 𝑋)))) |
484 | 483 | oveq2i 7181 |
. . . . . . . . . . . . 13
⊢ (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) = (((𝐴‘0) / 2) + Σ𝑚 ∈ (1...𝑛)(((𝐴‘𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵‘𝑚) · (sin‘(𝑚 · 𝑋))))) |
485 | 484 | mpteq2i 5122 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))) = (𝑛 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑚 ∈ (1...𝑛)(((𝐴‘𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵‘𝑚) · (sin‘(𝑚 · 𝑋)))))) |
486 | 474, 485 | eqtri 2761 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))) = (𝑛 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑚 ∈ (1...𝑛)(((𝐴‘𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵‘𝑚) · (sin‘(𝑚 · 𝑋)))))) |
487 | 28, 470, 486 | 3eqtri 2765 |
. . . . . . . . . 10
⊢ 𝑍 = (𝑛 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑚 ∈ (1...𝑛)(((𝐴‘𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵‘𝑚) · (sin‘(𝑚 · 𝑋)))))) |
488 | | oveq2 7178 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑥 → (𝑋 + 𝑦) = (𝑋 + 𝑥)) |
489 | 488 | fveq2d 6678 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑥 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑥))) |
490 | | fveq2 6674 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑥 → ((𝐷‘𝑚)‘𝑦) = ((𝐷‘𝑚)‘𝑥)) |
491 | 489, 490 | oveq12d 7188 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑥 → ((𝐹‘(𝑋 + 𝑦)) · ((𝐷‘𝑚)‘𝑦)) = ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑚)‘𝑥))) |
492 | 491 | cbvmptv 5133 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑦)) · ((𝐷‘𝑚)‘𝑦))) = (𝑥 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑚)‘𝑥))) |
493 | | eqid 2738 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ
↑m (0...𝑛))
∣ (((𝑝‘0) =
(-π − 𝑋) ∧
(𝑝‘𝑛) = (π − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑛)) ∣ (((𝑝‘0) = (-π − 𝑋) ∧ (𝑝‘𝑛) = (π − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
494 | | fveq2 6674 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑖 → (𝑄‘𝑗) = (𝑄‘𝑖)) |
495 | 494 | oveq1d 7185 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑖 → ((𝑄‘𝑗) − 𝑋) = ((𝑄‘𝑖) − 𝑋)) |
496 | 495 | cbvmptv 5133 |
. . . . . . . . . 10
⊢ (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) − 𝑋)) = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) − 𝑋)) |
497 | 451, 459,
487, 288, 51, 52, 53, 146, 43, 92, 492, 103, 222, 233, 48, 493, 496 | fourierdlem111 43300 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑍‘𝑚) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠)) |
498 | 443, 497 | chvarvv 2010 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑍‘𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)) |
499 | 411, 436 | oveq12d 7188 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑚 ∈ ℕ ↦
∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠)‘𝑛) + ((𝑚 ∈ ℕ ↦
∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠)‘𝑛)) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)) |
500 | 498, 499 | eqtr4d 2776 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑍‘𝑛) = (((𝑚 ∈ ℕ ↦
∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠)‘𝑛) + ((𝑚 ∈ ℕ ↦
∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠)‘𝑛))) |
501 | 16, 24, 27, 42, 14, 15, 333, 337, 359, 412, 437, 500 | climaddf 42698 |
. . . . . 6
⊢ (𝜑 → 𝑍 ⇝ ((𝐿 / 2) + (𝑅 / 2))) |
502 | | limccl 24627 |
. . . . . . . 8
⊢ ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) ⊆
ℂ |
503 | 502, 285 | sseldi 3875 |
. . . . . . 7
⊢ (𝜑 → 𝐿 ∈ ℂ) |
504 | | limccl 24627 |
. . . . . . . 8
⊢ ((𝐹 ↾ (𝑋(,)+∞)) limℂ 𝑋) ⊆
ℂ |
505 | 504, 284 | sseldi 3875 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ ℂ) |
506 | | 2cnd 11794 |
. . . . . . 7
⊢ (𝜑 → 2 ∈
ℂ) |
507 | | 2pos 11819 |
. . . . . . . . 9
⊢ 0 <
2 |
508 | 507 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 0 < 2) |
509 | 508 | gt0ne0d 11282 |
. . . . . . 7
⊢ (𝜑 → 2 ≠ 0) |
510 | 503, 505,
506, 509 | divdird 11532 |
. . . . . 6
⊢ (𝜑 → ((𝐿 + 𝑅) / 2) = ((𝐿 / 2) + (𝑅 / 2))) |
511 | 501, 510 | breqtrrd 5058 |
. . . . 5
⊢ (𝜑 → 𝑍 ⇝ ((𝐿 + 𝑅) / 2)) |
512 | | 0nn0 11991 |
. . . . . . . 8
⊢ 0 ∈
ℕ0 |
513 | 43 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 0 ∈
ℕ0) → 𝐹:ℝ⟶ℝ) |
514 | | eqid 2738 |
. . . . . . . . . 10
⊢
(-π(,)π) = (-π(,)π) |
515 | | ioossre 12882 |
. . . . . . . . . . . . . 14
⊢
(-π(,)π) ⊆ ℝ |
516 | 515 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (-π(,)π) ⊆
ℝ) |
517 | 43, 516 | feqresmpt 6738 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐹 ↾ (-π(,)π)) = (𝑥 ∈ (-π(,)π) ↦
(𝐹‘𝑥))) |
518 | | ioossicc 12907 |
. . . . . . . . . . . . . 14
⊢
(-π(,)π) ⊆ (-π[,]π) |
519 | 518 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (-π(,)π) ⊆
(-π[,]π)) |
520 | | ioombl 24317 |
. . . . . . . . . . . . . 14
⊢
(-π(,)π) ∈ dom vol |
521 | 520 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (-π(,)π) ∈ dom
vol) |
522 | 43 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π)) → 𝐹:ℝ⟶ℝ) |
523 | 385 | sselda 3877 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π)) → 𝑥 ∈
ℝ) |
524 | 522, 523 | ffvelrnd 6862 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π)) → (𝐹‘𝑥) ∈ ℝ) |
525 | 43, 385 | feqresmpt 6738 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐹 ↾ (-π[,]π)) = (𝑥 ∈ (-π[,]π) ↦
(𝐹‘𝑥))) |
526 | 178 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ℝ ⊆
ℂ) |
527 | 43, 526 | fssd 6522 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐹:ℝ⟶ℂ) |
528 | 527, 385 | fssresd 6545 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐹 ↾
(-π[,]π)):(-π[,]π)⟶ℂ) |
529 | | ioossicc 12907 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) |
530 | 61 | rexri 10777 |
. . . . . . . . . . . . . . . . . . . 20
⊢ -π
∈ ℝ* |
531 | 530 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈
ℝ*) |
532 | 54 | rexri 10777 |
. . . . . . . . . . . . . . . . . . . 20
⊢ π
∈ ℝ* |
533 | 532 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → π ∈
ℝ*) |
534 | 51, 52, 53 | fourierdlem15 43205 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π)) |
535 | 534 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π)) |
536 | | simpr 488 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀)) |
537 | 531, 533,
535, 536 | fourierdlem8 43198 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆
(-π[,]π)) |
538 | 529, 537 | sstrid 3888 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆
(-π[,]π)) |
539 | 538 | resabs1d 5856 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) |
540 | 539, 103 | eqeltrd 2833 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
541 | 539 | eqcomd 2744 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) |
542 | 541 | oveq1d 7185 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)) = (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
543 | 222, 542 | eleqtrd 2835 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
544 | 541 | oveq1d 7185 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1))) = (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
545 | 233, 544 | eleqtrd 2835 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
546 | 51, 52, 53, 528, 540, 543, 545 | fourierdlem69 43258 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐹 ↾ (-π[,]π)) ∈
𝐿1) |
547 | 525, 546 | eqeltrrd 2834 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑥 ∈ (-π[,]π) ↦ (𝐹‘𝑥)) ∈
𝐿1) |
548 | 519, 521,
524, 547 | iblss 24557 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑥 ∈ (-π(,)π) ↦ (𝐹‘𝑥)) ∈
𝐿1) |
549 | 517, 548 | eqeltrd 2833 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹 ↾ (-π(,)π)) ∈
𝐿1) |
550 | 549 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 0 ∈
ℕ0) → (𝐹 ↾ (-π(,)π)) ∈
𝐿1) |
551 | | simpr 488 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 0 ∈
ℕ0) → 0 ∈ ℕ0) |
552 | 513, 514,
550, 29, 551 | fourierdlem16 43206 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 ∈
ℕ0) → (((𝐴‘0) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦
(𝐹‘𝑥)) ∈ 𝐿1) ∧
∫(-π(,)π)((𝐹‘𝑥) · (cos‘(0 · 𝑥))) d𝑥 ∈ ℝ)) |
553 | 552 | simplld 768 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 ∈
ℕ0) → (𝐴‘0) ∈ ℝ) |
554 | 512, 553 | mpan2 691 |
. . . . . . 7
⊢ (𝜑 → (𝐴‘0) ∈ ℝ) |
555 | 554 | rehalfcld 11963 |
. . . . . 6
⊢ (𝜑 → ((𝐴‘0) / 2) ∈
ℝ) |
556 | 555 | recnd 10747 |
. . . . 5
⊢ (𝜑 → ((𝐴‘0) / 2) ∈
ℂ) |
557 | 334 | mptex 6996 |
. . . . . 6
⊢ (𝑛 ∈ ℕ ↦
Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) ∈ V |
558 | 557 | a1i 11 |
. . . . 5
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) ∈ V) |
559 | | simpr 488 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ) |
560 | 555 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐴‘0) / 2) ∈
ℝ) |
561 | | fzfid 13432 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1...𝑚) ∈ Fin) |
562 | | simpll 767 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → 𝜑) |
563 | | elfznn 13027 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ (1...𝑚) → 𝑛 ∈ ℕ) |
564 | 563 | adantl 485 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → 𝑛 ∈ ℕ) |
565 | | simpl 486 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝜑) |
566 | 362 | nnnn0d 12036 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0) |
567 | | eleq1w 2815 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → (𝑘 ∈ ℕ0 ↔ 𝑛 ∈
ℕ0)) |
568 | 567 | anbi2d 632 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → ((𝜑 ∧ 𝑘 ∈ ℕ0) ↔ (𝜑 ∧ 𝑛 ∈
ℕ0))) |
569 | | fveq2 6674 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → (𝐴‘𝑘) = (𝐴‘𝑛)) |
570 | 569 | eleq1d 2817 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → ((𝐴‘𝑘) ∈ ℝ ↔ (𝐴‘𝑛) ∈ ℝ)) |
571 | 568, 570 | imbi12d 348 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑛 → (((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℝ) ↔ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐴‘𝑛) ∈ ℝ))) |
572 | 43 | adantr 484 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐹:ℝ⟶ℝ) |
573 | 549 | adantr 484 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹 ↾ (-π(,)π)) ∈
𝐿1) |
574 | | simpr 488 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℕ0) |
575 | 572, 514,
573, 29, 574 | fourierdlem16 43206 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝐴‘𝑘) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ (𝐹‘𝑥)) ∈ 𝐿1) ∧
∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑘 · 𝑥))) d𝑥 ∈ ℝ)) |
576 | 575 | simplld 768 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℝ) |
577 | 571, 576 | chvarvv 2010 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐴‘𝑛) ∈ ℝ) |
578 | 565, 566,
577 | syl2anc 587 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ ℝ) |
579 | 362 | nnred 11731 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ) |
580 | 579, 399 | remulcld 10749 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 · 𝑋) ∈ ℝ) |
581 | 580 | recoscld 15589 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (cos‘(𝑛 · 𝑋)) ∈ ℝ) |
582 | 578, 581 | remulcld 10749 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) ∈ ℝ) |
583 | | eleq1w 2815 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → (𝑘 ∈ ℕ ↔ 𝑛 ∈ ℕ)) |
584 | 583 | anbi2d 632 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑛 → ((𝜑 ∧ 𝑘 ∈ ℕ) ↔ (𝜑 ∧ 𝑛 ∈ ℕ))) |
585 | | fveq2 6674 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → (𝐵‘𝑘) = (𝐵‘𝑛)) |
586 | 585 | eleq1d 2817 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑛 → ((𝐵‘𝑘) ∈ ℝ ↔ (𝐵‘𝑛) ∈ ℝ)) |
587 | 584, 586 | imbi12d 348 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑛 → (((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐵‘𝑘) ∈ ℝ) ↔ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵‘𝑛) ∈ ℝ))) |
588 | 43 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:ℝ⟶ℝ) |
589 | 549 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹 ↾ (-π(,)π)) ∈
𝐿1) |
590 | | simpr 488 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) |
591 | 588, 514,
589, 452, 590 | fourierdlem21 43211 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐵‘𝑘) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ ((𝐹‘𝑥) · (sin‘(𝑘 · 𝑥)))) ∈ 𝐿1) ∧
∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑘 · 𝑥))) d𝑥 ∈ ℝ)) |
592 | 591 | simplld 768 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐵‘𝑘) ∈ ℝ) |
593 | 587, 592 | chvarvv 2010 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵‘𝑛) ∈ ℝ) |
594 | 580 | resincld 15588 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (sin‘(𝑛 · 𝑋)) ∈ ℝ) |
595 | 593, 594 | remulcld 10749 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))) ∈ ℝ) |
596 | 582, 595 | readdcld 10748 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ) |
597 | 562, 564,
596 | syl2anc 587 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ) |
598 | 561, 597 | fsumrecl 15184 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ) |
599 | 560, 598 | readdcld 10748 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ ℝ) |
600 | 28 | fvmpt2 6786 |
. . . . . . . 8
⊢ ((𝑚 ∈ ℕ ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ ℝ) → (𝑍‘𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))) |
601 | 559, 599,
600 | syl2anc 587 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑍‘𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))) |
602 | 601, 599 | eqeltrd 2833 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑍‘𝑚) ∈ ℝ) |
603 | 602 | recnd 10747 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑍‘𝑚) ∈ ℂ) |
604 | | eqidd 2739 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))) |
605 | | oveq2 7178 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚)) |
606 | 605 | sumeq1d 15151 |
. . . . . . . 8
⊢ (𝑛 = 𝑚 → Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
607 | 606 | adantl 485 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 = 𝑚) → Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
608 | | sumex 15137 |
. . . . . . . 8
⊢
Σ𝑘 ∈
(1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ V |
609 | 608 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ V) |
610 | 604, 607,
559, 609 | fvmptd 6782 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))‘𝑚) = Σ𝑘 ∈ (1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
611 | 560 | recnd 10747 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐴‘0) / 2) ∈
ℂ) |
612 | 598 | recnd 10747 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℂ) |
613 | 611, 612 | pncan2d 11077 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) − ((𝐴‘0) / 2)) = Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) |
614 | 613, 468 | eqtr2di 2790 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = ((((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) − ((𝐴‘0) / 2))) |
615 | | ovex 7203 |
. . . . . . . . 9
⊢ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ V |
616 | 28 | fvmpt2 6786 |
. . . . . . . . 9
⊢ ((𝑚 ∈ ℕ ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ V) → (𝑍‘𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))) |
617 | 559, 615,
616 | sylancl 589 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑍‘𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))) |
618 | 617 | eqcomd 2744 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑍‘𝑚)) |
619 | 618 | oveq1d 7185 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) − ((𝐴‘0) / 2)) = ((𝑍‘𝑚) − ((𝐴‘0) / 2))) |
620 | 610, 614,
619 | 3eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))‘𝑚) = ((𝑍‘𝑚) − ((𝐴‘0) / 2))) |
621 | 14, 15, 511, 556, 558, 603, 620 | climsubc1 15085 |
. . . 4
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))) |
622 | | seqex 13462 |
. . . . . 6
⊢ seq1( + ,
(𝑗 ∈ ℕ ↦
(((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋)))))) ∈ V |
623 | 622 | a1i 11 |
. . . . 5
⊢ (𝜑 → seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋)))))) ∈ V) |
624 | | eqidd 2739 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))) |
625 | | oveq2 7178 |
. . . . . . . . 9
⊢ (𝑛 = 𝑙 → (1...𝑛) = (1...𝑙)) |
626 | 625 | sumeq1d 15151 |
. . . . . . . 8
⊢ (𝑛 = 𝑙 → Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑙)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
627 | 626 | adantl 485 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ 𝑛 = 𝑙) → Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑙)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
628 | | simpr 488 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → 𝑙 ∈ ℕ) |
629 | | fzfid 13432 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → (1...𝑙) ∈ Fin) |
630 | | elfznn 13027 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ) |
631 | 630 | nnnn0d 12036 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ0) |
632 | 631, 576 | sylan2 596 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑙)) → (𝐴‘𝑘) ∈ ℝ) |
633 | 630 | nnred 11731 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℝ) |
634 | 633 | adantl 485 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑙)) → 𝑘 ∈ ℝ) |
635 | 146 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑙)) → 𝑋 ∈ ℝ) |
636 | 634, 635 | remulcld 10749 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑙)) → (𝑘 · 𝑋) ∈ ℝ) |
637 | 636 | recoscld 15589 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑙)) → (cos‘(𝑘 · 𝑋)) ∈ ℝ) |
638 | 632, 637 | remulcld 10749 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑙)) → ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ) |
639 | 630, 592 | sylan2 596 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑙)) → (𝐵‘𝑘) ∈ ℝ) |
640 | 636 | resincld 15588 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑙)) → (sin‘(𝑘 · 𝑋)) ∈ ℝ) |
641 | 639, 640 | remulcld 10749 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑙)) → ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))) ∈ ℝ) |
642 | 638, 641 | readdcld 10748 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑙)) → (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ) |
643 | 642 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ) |
644 | 629, 643 | fsumrecl 15184 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → Σ𝑘 ∈ (1...𝑙)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ) |
645 | 624, 627,
628, 644 | fvmptd 6782 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))‘𝑙) = Σ𝑘 ∈ (1...𝑙)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
646 | | eleq1w 2815 |
. . . . . . . . 9
⊢ (𝑛 = 𝑙 → (𝑛 ∈ ℕ ↔ 𝑙 ∈ ℕ)) |
647 | 646 | anbi2d 632 |
. . . . . . . 8
⊢ (𝑛 = 𝑙 → ((𝜑 ∧ 𝑛 ∈ ℕ) ↔ (𝜑 ∧ 𝑙 ∈ ℕ))) |
648 | | fveq2 6674 |
. . . . . . . . 9
⊢ (𝑛 = 𝑙 → (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑛) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑙)) |
649 | 626, 648 | eqeq12d 2754 |
. . . . . . . 8
⊢ (𝑛 = 𝑙 → (Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑛) ↔ Σ𝑘 ∈ (1...𝑙)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑙))) |
650 | 647, 649 | imbi12d 348 |
. . . . . . 7
⊢ (𝑛 = 𝑙 → (((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑛)) ↔ ((𝜑 ∧ 𝑙 ∈ ℕ) → Σ𝑘 ∈ (1...𝑙)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑙)))) |
651 | | eqidd 2739 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))) = (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋)))))) |
652 | | fveq2 6674 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → (𝐴‘𝑗) = (𝐴‘𝑘)) |
653 | | oveq1 7177 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑘 → (𝑗 · 𝑋) = (𝑘 · 𝑋)) |
654 | 653 | fveq2d 6678 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → (cos‘(𝑗 · 𝑋)) = (cos‘(𝑘 · 𝑋))) |
655 | 652, 654 | oveq12d 7188 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → ((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) = ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋)))) |
656 | | fveq2 6674 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → (𝐵‘𝑗) = (𝐵‘𝑘)) |
657 | 653 | fveq2d 6678 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → (sin‘(𝑗 · 𝑋)) = (sin‘(𝑘 · 𝑋))) |
658 | 656, 657 | oveq12d 7188 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))) = ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) |
659 | 655, 658 | oveq12d 7188 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑘 → (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
660 | 659 | adantl 485 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑗 = 𝑘) → (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
661 | | elfznn 13027 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ) |
662 | 661 | adantl 485 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ) |
663 | | simpll 767 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑) |
664 | | nnnn0 11983 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℕ0) |
665 | | nn0re 11985 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℝ) |
666 | 665 | adantl 485 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℝ) |
667 | 146 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑋 ∈
ℝ) |
668 | 666, 667 | remulcld 10749 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 · 𝑋) ∈ ℝ) |
669 | 668 | recoscld 15589 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) →
(cos‘(𝑘 ·
𝑋)) ∈
ℝ) |
670 | 576, 669 | remulcld 10749 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ) |
671 | 664, 670 | sylan2 596 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ) |
672 | 664, 668 | sylan2 596 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 · 𝑋) ∈ ℝ) |
673 | 672 | resincld 15588 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (sin‘(𝑘 · 𝑋)) ∈ ℝ) |
674 | 592, 673 | remulcld 10749 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))) ∈ ℝ) |
675 | 671, 674 | readdcld 10748 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ) |
676 | 663, 662,
675 | syl2anc 587 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ) |
677 | 651, 660,
662, 676 | fvmptd 6782 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋)))))‘𝑘) = (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
678 | 362, 14 | eleqtrdi 2843 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
(ℤ≥‘1)) |
679 | 676 | recnd 10747 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℂ) |
680 | 677, 678,
679 | fsumser 15180 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑛)) |
681 | 650, 680 | chvarvv 2010 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → Σ𝑘 ∈ (1...𝑙)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑙)) |
682 | 645, 681 | eqtrd 2773 |
. . . . 5
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))‘𝑙) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑙)) |
683 | 14, 558, 623, 15, 682 | climeq 15014 |
. . . 4
⊢ (𝜑 → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ↔ seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋)))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))) |
684 | 621, 683 | mpbid 235 |
. . 3
⊢ (𝜑 → seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋)))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))) |
685 | 13, 684 | eqbrtrd 5052 |
. 2
⊢ (𝜑 → seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))) |
686 | | eqidd 2739 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))))) = (𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋)))))) |
687 | | fveq2 6674 |
. . . . . . . . 9
⊢ (𝑗 = 𝑛 → (𝐴‘𝑗) = (𝐴‘𝑛)) |
688 | | oveq1 7177 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑛 → (𝑗 · 𝑋) = (𝑛 · 𝑋)) |
689 | 688 | fveq2d 6678 |
. . . . . . . . 9
⊢ (𝑗 = 𝑛 → (cos‘(𝑗 · 𝑋)) = (cos‘(𝑛 · 𝑋))) |
690 | 687, 689 | oveq12d 7188 |
. . . . . . . 8
⊢ (𝑗 = 𝑛 → ((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) = ((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋)))) |
691 | | fveq2 6674 |
. . . . . . . . 9
⊢ (𝑗 = 𝑛 → (𝐵‘𝑗) = (𝐵‘𝑛)) |
692 | 688 | fveq2d 6678 |
. . . . . . . . 9
⊢ (𝑗 = 𝑛 → (sin‘(𝑗 · 𝑋)) = (sin‘(𝑛 · 𝑋))) |
693 | 691, 692 | oveq12d 7188 |
. . . . . . . 8
⊢ (𝑗 = 𝑛 → ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋))) = ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) |
694 | 690, 693 | oveq12d 7188 |
. . . . . . 7
⊢ (𝑗 = 𝑛 → (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) |
695 | 694 | adantl 485 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 = 𝑛) → (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) |
696 | 686, 695,
362, 596 | fvmptd 6782 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (((𝐴‘𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵‘𝑗) · (sin‘(𝑗 · 𝑋)))))‘𝑛) = (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) |
697 | 596 | recnd 10747 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℂ) |
698 | 14, 15, 696, 697, 684 | isumclim 15205 |
. . . 4
⊢ (𝜑 → Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))) |
699 | 698 | oveq2d 7186 |
. . 3
⊢ (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = (((𝐴‘0) / 2) + (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))) |
700 | 503, 505 | addcld 10738 |
. . . . 5
⊢ (𝜑 → (𝐿 + 𝑅) ∈ ℂ) |
701 | 700 | halfcld 11961 |
. . . 4
⊢ (𝜑 → ((𝐿 + 𝑅) / 2) ∈ ℂ) |
702 | 556, 701 | pncan3d 11078 |
. . 3
⊢ (𝜑 → (((𝐴‘0) / 2) + (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))) = ((𝐿 + 𝑅) / 2)) |
703 | 699, 702 | eqtrd 2773 |
. 2
⊢ (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)) |
704 | 685, 703 | jca 515 |
1
⊢ (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2))) |