| Step | Hyp | Ref
| Expression |
| 1 | | nnnn0 12533 |
. . . . . 6
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℕ0) |
| 2 | | fourierdlem21.f |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
| 3 | 2 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐹:ℝ⟶ℝ) |
| 4 | | ioossre 13448 |
. . . . . . . . . . . 12
⊢
(-π(,)π) ⊆ ℝ |
| 5 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐶) |
| 6 | | fourierdlem21.c |
. . . . . . . . . . . . 13
⊢ 𝐶 =
(-π(,)π) |
| 7 | 5, 6 | eleqtrdi 2851 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ (-π(,)π)) |
| 8 | 4, 7 | sselid 3981 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ ℝ) |
| 9 | 8 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ℝ) |
| 10 | 3, 9 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐹‘𝑥) ∈ ℝ) |
| 11 | 10 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ 𝑥 ∈ 𝐶) → (𝐹‘𝑥) ∈ ℝ) |
| 12 | | nn0re 12535 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ0
→ 𝑛 ∈
ℝ) |
| 13 | 12 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ0
∧ 𝑥 ∈ 𝐶) → 𝑛 ∈ ℝ) |
| 14 | 8 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ0
∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ℝ) |
| 15 | 13, 14 | remulcld 11291 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ0
∧ 𝑥 ∈ 𝐶) → (𝑛 · 𝑥) ∈ ℝ) |
| 16 | 15 | resincld 16179 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ0
∧ 𝑥 ∈ 𝐶) → (sin‘(𝑛 · 𝑥)) ∈ ℝ) |
| 17 | 16 | adantll 714 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ 𝑥 ∈ 𝐶) → (sin‘(𝑛 · 𝑥)) ∈ ℝ) |
| 18 | 11, 17 | remulcld 11291 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ 𝑥 ∈ 𝐶) → ((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) ∈ ℝ) |
| 19 | | ioombl 25600 |
. . . . . . . . . . . 12
⊢
(-π(,)π) ∈ dom vol |
| 20 | 6, 19 | eqeltri 2837 |
. . . . . . . . . . 11
⊢ 𝐶 ∈ dom vol |
| 21 | 20 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐶 ∈ dom
vol) |
| 22 | | eqidd 2738 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) = (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))) |
| 23 | | eqidd 2738 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
| 24 | 21, 17, 11, 22, 23 | offval2 7717 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) ∘f · (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) = (𝑥 ∈ 𝐶 ↦ ((sin‘(𝑛 · 𝑥)) · (𝐹‘𝑥)))) |
| 25 | 17 | recnd 11289 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ 𝑥 ∈ 𝐶) → (sin‘(𝑛 · 𝑥)) ∈ ℂ) |
| 26 | 11 | recnd 11289 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ 𝑥 ∈ 𝐶) → (𝐹‘𝑥) ∈ ℂ) |
| 27 | 25, 26 | mulcomd 11282 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ 𝑥 ∈ 𝐶) → ((sin‘(𝑛 · 𝑥)) · (𝐹‘𝑥)) = ((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥)))) |
| 28 | 27 | mpteq2dva 5242 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑥 ∈ 𝐶 ↦ ((sin‘(𝑛 · 𝑥)) · (𝐹‘𝑥))) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))))) |
| 29 | 24, 28 | eqtr2d 2778 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥)))) = ((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) ∘f · (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))) |
| 30 | | sincn 26488 |
. . . . . . . . . . . 12
⊢ sin
∈ (ℂ–cn→ℂ) |
| 31 | 30 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → sin
∈ (ℂ–cn→ℂ)) |
| 32 | 6, 4 | eqsstri 4030 |
. . . . . . . . . . . . . . . 16
⊢ 𝐶 ⊆
ℝ |
| 33 | | ax-resscn 11212 |
. . . . . . . . . . . . . . . 16
⊢ ℝ
⊆ ℂ |
| 34 | 32, 33 | sstri 3993 |
. . . . . . . . . . . . . . 15
⊢ 𝐶 ⊆
ℂ |
| 35 | 34 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ0
→ 𝐶 ⊆
ℂ) |
| 36 | 12 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ0
→ 𝑛 ∈
ℂ) |
| 37 | | ssid 4006 |
. . . . . . . . . . . . . . 15
⊢ ℂ
⊆ ℂ |
| 38 | 37 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ0
→ ℂ ⊆ ℂ) |
| 39 | 35, 36, 38 | constcncfg 45887 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ0
→ (𝑥 ∈ 𝐶 ↦ 𝑛) ∈ (𝐶–cn→ℂ)) |
| 40 | 35, 38 | idcncfg 45888 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ0
→ (𝑥 ∈ 𝐶 ↦ 𝑥) ∈ (𝐶–cn→ℂ)) |
| 41 | 39, 40 | mulcncf 25480 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ0
→ (𝑥 ∈ 𝐶 ↦ (𝑛 · 𝑥)) ∈ (𝐶–cn→ℂ)) |
| 42 | 41 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑥 ∈ 𝐶 ↦ (𝑛 · 𝑥)) ∈ (𝐶–cn→ℂ)) |
| 43 | 31, 42 | cncfmpt1f 24940 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) ∈ (𝐶–cn→ℂ)) |
| 44 | | cnmbf 25694 |
. . . . . . . . . 10
⊢ ((𝐶 ∈ dom vol ∧ (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) ∈ (𝐶–cn→ℂ)) → (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) ∈ MblFn) |
| 45 | 20, 43, 44 | sylancr 587 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) ∈ MblFn) |
| 46 | 2 | feqmptd 6977 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥))) |
| 47 | 46 | reseq1d 5996 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ↾ 𝐶)) |
| 48 | | resmpt 6055 |
. . . . . . . . . . . . 13
⊢ (𝐶 ⊆ ℝ → ((𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
| 49 | 32, 48 | mp1i 13 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
| 50 | 47, 49 | eqtr2d 2778 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝐹 ↾ 𝐶)) |
| 51 | | fourierdlem21.fibl |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈
𝐿1) |
| 52 | 50, 51 | eqeltrd 2841 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈
𝐿1) |
| 53 | 52 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈
𝐿1) |
| 54 | | 1re 11261 |
. . . . . . . . . . 11
⊢ 1 ∈
ℝ |
| 55 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))) → 𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))) |
| 56 | | nfv 1914 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑥 𝑛 ∈
ℕ0 |
| 57 | | nfmpt1 5250 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑥(𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) |
| 58 | 57 | nfdm 5962 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑥dom
(𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) |
| 59 | 58 | nfcri 2897 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑥 𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) |
| 60 | 56, 59 | nfan 1899 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑥(𝑛 ∈ ℕ0
∧ 𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))) |
| 61 | 16 | ex 412 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ0
→ (𝑥 ∈ 𝐶 → (sin‘(𝑛 · 𝑥)) ∈ ℝ)) |
| 62 | 61 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))) → (𝑥 ∈ 𝐶 → (sin‘(𝑛 · 𝑥)) ∈ ℝ)) |
| 63 | 60, 62 | ralrimi 3257 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))) → ∀𝑥 ∈ 𝐶 (sin‘(𝑛 · 𝑥)) ∈ ℝ) |
| 64 | | dmmptg 6262 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑥 ∈
𝐶 (sin‘(𝑛 · 𝑥)) ∈ ℝ → dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) = 𝐶) |
| 65 | 63, 64 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))) → dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) = 𝐶) |
| 66 | 55, 65 | eleqtrd 2843 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))) → 𝑦 ∈ 𝐶) |
| 67 | | eqidd 2738 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) = (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))) |
| 68 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → (𝑛 · 𝑥) = (𝑛 · 𝑦)) |
| 69 | 68 | fveq2d 6910 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → (sin‘(𝑛 · 𝑥)) = (sin‘(𝑛 · 𝑦))) |
| 70 | 69 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑛 ∈ ℕ0
∧ 𝑦 ∈ 𝐶) ∧ 𝑥 = 𝑦) → (sin‘(𝑛 · 𝑥)) = (sin‘(𝑛 · 𝑦))) |
| 71 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ 𝐶) |
| 72 | 12 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ 𝐶) → 𝑛 ∈ ℝ) |
| 73 | 32, 71 | sselid 3981 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ ℝ) |
| 74 | 72, 73 | remulcld 11291 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ 𝐶) → (𝑛 · 𝑦) ∈ ℝ) |
| 75 | 74 | resincld 16179 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ 𝐶) → (sin‘(𝑛 · 𝑦)) ∈ ℝ) |
| 76 | 67, 70, 71, 75 | fvmptd 7023 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ 𝐶) → ((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))‘𝑦) = (sin‘(𝑛 · 𝑦))) |
| 77 | 76 | fveq2d 6910 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ 𝐶) → (abs‘((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))‘𝑦)) = (abs‘(sin‘(𝑛 · 𝑦)))) |
| 78 | | abssinbd 45307 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 · 𝑦) ∈ ℝ →
(abs‘(sin‘(𝑛
· 𝑦))) ≤
1) |
| 79 | 74, 78 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ 𝐶) →
(abs‘(sin‘(𝑛
· 𝑦))) ≤
1) |
| 80 | 77, 79 | eqbrtrd 5165 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ 𝐶) → (abs‘((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))‘𝑦)) ≤ 1) |
| 81 | 66, 80 | syldan 591 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ ℕ0
∧ 𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))) → (abs‘((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))‘𝑦)) ≤ 1) |
| 82 | 81 | ralrimiva 3146 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ0
→ ∀𝑦 ∈ dom
(𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))(abs‘((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))‘𝑦)) ≤ 1) |
| 83 | | breq2 5147 |
. . . . . . . . . . . . 13
⊢ (𝑏 = 1 → ((abs‘((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))‘𝑦)) ≤ 𝑏 ↔ (abs‘((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))‘𝑦)) ≤ 1)) |
| 84 | 83 | ralbidv 3178 |
. . . . . . . . . . . 12
⊢ (𝑏 = 1 → (∀𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))(abs‘((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))‘𝑦)) ≤ 𝑏 ↔ ∀𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))(abs‘((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))‘𝑦)) ≤ 1)) |
| 85 | 84 | rspcev 3622 |
. . . . . . . . . . 11
⊢ ((1
∈ ℝ ∧ ∀𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))(abs‘((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))‘𝑦)) ≤ 1) → ∃𝑏 ∈ ℝ ∀𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))(abs‘((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))‘𝑦)) ≤ 𝑏) |
| 86 | 54, 82, 85 | sylancr 587 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ0
→ ∃𝑏 ∈
ℝ ∀𝑦 ∈
dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))(abs‘((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))‘𝑦)) ≤ 𝑏) |
| 87 | 86 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) →
∃𝑏 ∈ ℝ
∀𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))(abs‘((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))‘𝑦)) ≤ 𝑏) |
| 88 | | bddmulibl 25874 |
. . . . . . . . 9
⊢ (((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) ∈ MblFn ∧ (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ 𝐿1 ∧
∃𝑏 ∈ ℝ
∀𝑦 ∈ dom (𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))(abs‘((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥)))‘𝑦)) ≤ 𝑏) → ((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) ∘f · (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) ∈
𝐿1) |
| 89 | 45, 53, 87, 88 | syl3anc 1373 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑥 ∈ 𝐶 ↦ (sin‘(𝑛 · 𝑥))) ∘f · (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) ∈
𝐿1) |
| 90 | 29, 89 | eqeltrd 2841 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥)))) ∈
𝐿1) |
| 91 | 18, 90 | itgrecl 25833 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) →
∫𝐶((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 ∈ ℝ) |
| 92 | 1, 91 | sylan2 593 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐶((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 ∈ ℝ) |
| 93 | | pire 26500 |
. . . . . 6
⊢ π
∈ ℝ |
| 94 | 93 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ∈
ℝ) |
| 95 | | 0re 11263 |
. . . . . . 7
⊢ 0 ∈
ℝ |
| 96 | | pipos 26502 |
. . . . . . 7
⊢ 0 <
π |
| 97 | 95, 96 | gtneii 11373 |
. . . . . 6
⊢ π ≠
0 |
| 98 | 97 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ≠
0) |
| 99 | 92, 94, 98 | redivcld 12095 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫𝐶((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π) ∈ ℝ) |
| 100 | | fourierdlem21.b |
. . . 4
⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫𝐶((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) |
| 101 | 99, 100 | fmptd 7134 |
. . 3
⊢ (𝜑 → 𝐵:ℕ⟶ℝ) |
| 102 | | fourierdlem21.n |
. . 3
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 103 | 101, 102 | ffvelcdmd 7105 |
. 2
⊢ (𝜑 → (𝐵‘𝑁) ∈ ℝ) |
| 104 | 102 | nnnn0d 12587 |
. . 3
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 105 | | eleq1 2829 |
. . . . . . 7
⊢ (𝑛 = 𝑁 → (𝑛 ∈ ℕ0 ↔ 𝑁 ∈
ℕ0)) |
| 106 | 105 | anbi2d 630 |
. . . . . 6
⊢ (𝑛 = 𝑁 → ((𝜑 ∧ 𝑛 ∈ ℕ0) ↔ (𝜑 ∧ 𝑁 ∈
ℕ0))) |
| 107 | | simpl 482 |
. . . . . . . . . . 11
⊢ ((𝑛 = 𝑁 ∧ 𝑥 ∈ 𝐶) → 𝑛 = 𝑁) |
| 108 | 107 | oveq1d 7446 |
. . . . . . . . . 10
⊢ ((𝑛 = 𝑁 ∧ 𝑥 ∈ 𝐶) → (𝑛 · 𝑥) = (𝑁 · 𝑥)) |
| 109 | 108 | fveq2d 6910 |
. . . . . . . . 9
⊢ ((𝑛 = 𝑁 ∧ 𝑥 ∈ 𝐶) → (sin‘(𝑛 · 𝑥)) = (sin‘(𝑁 · 𝑥))) |
| 110 | 109 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝑛 = 𝑁 ∧ 𝑥 ∈ 𝐶) → ((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) = ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥)))) |
| 111 | 110 | mpteq2dva 5242 |
. . . . . . 7
⊢ (𝑛 = 𝑁 → (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥)))) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))))) |
| 112 | 111 | eleq1d 2826 |
. . . . . 6
⊢ (𝑛 = 𝑁 → ((𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥)))) ∈ 𝐿1 ↔
(𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥)))) ∈
𝐿1)) |
| 113 | 106, 112 | imbi12d 344 |
. . . . 5
⊢ (𝑛 = 𝑁 → (((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥)))) ∈ 𝐿1) ↔
((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥)))) ∈
𝐿1))) |
| 114 | 113, 90 | vtoclg 3554 |
. . . 4
⊢ (𝑁 ∈ ℕ0
→ ((𝜑 ∧ 𝑁 ∈ ℕ0)
→ (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥)))) ∈
𝐿1)) |
| 115 | 114 | anabsi7 671 |
. . 3
⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥)))) ∈
𝐿1) |
| 116 | 104, 115 | mpdan 687 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥)))) ∈
𝐿1) |
| 117 | 102 | ancli 548 |
. . 3
⊢ (𝜑 → (𝜑 ∧ 𝑁 ∈ ℕ)) |
| 118 | | eleq1 2829 |
. . . . . 6
⊢ (𝑛 = 𝑁 → (𝑛 ∈ ℕ ↔ 𝑁 ∈ ℕ)) |
| 119 | 118 | anbi2d 630 |
. . . . 5
⊢ (𝑛 = 𝑁 → ((𝜑 ∧ 𝑛 ∈ ℕ) ↔ (𝜑 ∧ 𝑁 ∈ ℕ))) |
| 120 | 110 | itgeq2dv 25817 |
. . . . . 6
⊢ (𝑛 = 𝑁 → ∫𝐶((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 = ∫𝐶((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥) |
| 121 | 120 | eleq1d 2826 |
. . . . 5
⊢ (𝑛 = 𝑁 → (∫𝐶((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 ∈ ℝ ↔ ∫𝐶((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 ∈ ℝ)) |
| 122 | 119, 121 | imbi12d 344 |
. . . 4
⊢ (𝑛 = 𝑁 → (((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐶((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 ∈ ℝ) ↔ ((𝜑 ∧ 𝑁 ∈ ℕ) → ∫𝐶((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 ∈ ℝ))) |
| 123 | 122, 92 | vtoclg 3554 |
. . 3
⊢ (𝑁 ∈ ℕ → ((𝜑 ∧ 𝑁 ∈ ℕ) → ∫𝐶((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 ∈ ℝ)) |
| 124 | 102, 117,
123 | sylc 65 |
. 2
⊢ (𝜑 → ∫𝐶((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 ∈ ℝ) |
| 125 | 103, 116,
124 | jca31 514 |
1
⊢ (𝜑 → (((𝐵‘𝑁) ∈ ℝ ∧ (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥)))) ∈ 𝐿1) ∧
∫𝐶((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 ∈ ℝ)) |