Proof of Theorem sqwvfourb
| Step | Hyp | Ref
| Expression |
| 1 | | pire 26500 |
. . . . . 6
⊢ π
∈ ℝ |
| 2 | 1 | renegcli 11570 |
. . . . 5
⊢ -π
∈ ℝ |
| 3 | 2 | a1i 11 |
. . . 4
⊢ (𝜑 → -π ∈
ℝ) |
| 4 | 1 | a1i 11 |
. . . 4
⊢ (𝜑 → π ∈
ℝ) |
| 5 | | 0re 11263 |
. . . . . 6
⊢ 0 ∈
ℝ |
| 6 | | negpilt0 45292 |
. . . . . . 7
⊢ -π
< 0 |
| 7 | 2, 5, 6 | ltleii 11384 |
. . . . . 6
⊢ -π
≤ 0 |
| 8 | | pipos 26502 |
. . . . . . 7
⊢ 0 <
π |
| 9 | 5, 1, 8 | ltleii 11384 |
. . . . . 6
⊢ 0 ≤
π |
| 10 | 2, 1 | elicc2i 13453 |
. . . . . 6
⊢ (0 ∈
(-π[,]π) ↔ (0 ∈ ℝ ∧ -π ≤ 0 ∧ 0 ≤
π)) |
| 11 | 5, 7, 9, 10 | mpbir3an 1342 |
. . . . 5
⊢ 0 ∈
(-π[,]π) |
| 12 | 11 | a1i 11 |
. . . 4
⊢ (𝜑 → 0 ∈
(-π[,]π)) |
| 13 | | elioore 13417 |
. . . . . . . 8
⊢ (𝑥 ∈ (-π(,)π) →
𝑥 ∈
ℝ) |
| 14 | 13 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)π)) → 𝑥 ∈
ℝ) |
| 15 | | 1re 11261 |
. . . . . . . 8
⊢ 1 ∈
ℝ |
| 16 | 15 | renegcli 11570 |
. . . . . . . 8
⊢ -1 ∈
ℝ |
| 17 | 15, 16 | ifcli 4573 |
. . . . . . 7
⊢ if((𝑥 mod 𝑇) < π, 1, -1) ∈
ℝ |
| 18 | | sqwvfourb.f |
. . . . . . . 8
⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if((𝑥 mod 𝑇) < π, 1, -1)) |
| 19 | 18 | fvmpt2 7027 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ ∧ if((𝑥 mod 𝑇) < π, 1, -1) ∈ ℝ) →
(𝐹‘𝑥) = if((𝑥 mod 𝑇) < π, 1, -1)) |
| 20 | 14, 17, 19 | sylancl 586 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)π)) → (𝐹‘𝑥) = if((𝑥 mod 𝑇) < π, 1, -1)) |
| 21 | 17 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)π)) → if((𝑥 mod 𝑇) < π, 1, -1) ∈
ℝ) |
| 22 | 21 | recnd 11289 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)π)) → if((𝑥 mod 𝑇) < π, 1, -1) ∈
ℂ) |
| 23 | 20, 22 | eqeltrd 2841 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)π)) → (𝐹‘𝑥) ∈ ℂ) |
| 24 | | sqwvfourb.n |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 25 | 24 | nncnd 12282 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 26 | 25 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)π)) → 𝑁 ∈
ℂ) |
| 27 | 14 | recnd 11289 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)π)) → 𝑥 ∈
ℂ) |
| 28 | 26, 27 | mulcld 11281 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)π)) → (𝑁 · 𝑥) ∈ ℂ) |
| 29 | 28 | sincld 16166 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)π)) →
(sin‘(𝑁 ·
𝑥)) ∈
ℂ) |
| 30 | 23, 29 | mulcld 11281 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)π)) → ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) ∈ ℂ) |
| 31 | | elioore 13417 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (-π(,)0) → 𝑥 ∈
ℝ) |
| 32 | 31, 17, 19 | sylancl 586 |
. . . . . . . . 9
⊢ (𝑥 ∈ (-π(,)0) →
(𝐹‘𝑥) = if((𝑥 mod 𝑇) < π, 1, -1)) |
| 33 | 1 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (-π(,)0) → π
∈ ℝ) |
| 34 | | sqwvfourb.t |
. . . . . . . . . . . . . 14
⊢ 𝑇 = (2 ·
π) |
| 35 | | 2rp 13039 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℝ+ |
| 36 | | pirp 26503 |
. . . . . . . . . . . . . . 15
⊢ π
∈ ℝ+ |
| 37 | | rpmulcl 13058 |
. . . . . . . . . . . . . . 15
⊢ ((2
∈ ℝ+ ∧ π ∈ ℝ+) → (2
· π) ∈ ℝ+) |
| 38 | 35, 36, 37 | mp2an 692 |
. . . . . . . . . . . . . 14
⊢ (2
· π) ∈ ℝ+ |
| 39 | 34, 38 | eqeltri 2837 |
. . . . . . . . . . . . 13
⊢ 𝑇 ∈
ℝ+ |
| 40 | 39 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (-π(,)0) → 𝑇 ∈
ℝ+) |
| 41 | 31, 40 | modcld 13915 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (-π(,)0) →
(𝑥 mod 𝑇) ∈ ℝ) |
| 42 | | picn 26501 |
. . . . . . . . . . . . . . . . . 18
⊢ π
∈ ℂ |
| 43 | 42 | 2timesi 12404 |
. . . . . . . . . . . . . . . . 17
⊢ (2
· π) = (π + π) |
| 44 | 34, 43 | eqtri 2765 |
. . . . . . . . . . . . . . . 16
⊢ 𝑇 = (π +
π) |
| 45 | 44 | oveq2i 7442 |
. . . . . . . . . . . . . . 15
⊢ (-π +
𝑇) = (-π + (π +
π)) |
| 46 | 2 | recni 11275 |
. . . . . . . . . . . . . . . 16
⊢ -π
∈ ℂ |
| 47 | 46, 42, 42 | addassi 11271 |
. . . . . . . . . . . . . . 15
⊢ ((-π +
π) + π) = (-π + (π + π)) |
| 48 | 42 | negidi 11578 |
. . . . . . . . . . . . . . . . . 18
⊢ (π +
-π) = 0 |
| 49 | 42, 46, 48 | addcomli 11453 |
. . . . . . . . . . . . . . . . 17
⊢ (-π +
π) = 0 |
| 50 | 49 | oveq1i 7441 |
. . . . . . . . . . . . . . . 16
⊢ ((-π +
π) + π) = (0 + π) |
| 51 | 42 | addlidi 11449 |
. . . . . . . . . . . . . . . 16
⊢ (0 +
π) = π |
| 52 | 50, 51 | eqtri 2765 |
. . . . . . . . . . . . . . 15
⊢ ((-π +
π) + π) = π |
| 53 | 45, 47, 52 | 3eqtr2ri 2772 |
. . . . . . . . . . . . . 14
⊢ π =
(-π + 𝑇) |
| 54 | 53 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (-π(,)0) → π
= (-π + 𝑇)) |
| 55 | 2 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (-π(,)0) → -π
∈ ℝ) |
| 56 | | 2re 12340 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ∈
ℝ |
| 57 | 56, 1 | remulcli 11277 |
. . . . . . . . . . . . . . . 16
⊢ (2
· π) ∈ ℝ |
| 58 | 34, 57 | eqeltri 2837 |
. . . . . . . . . . . . . . 15
⊢ 𝑇 ∈ ℝ |
| 59 | 58 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (-π(,)0) → 𝑇 ∈
ℝ) |
| 60 | 2 | rexri 11319 |
. . . . . . . . . . . . . . 15
⊢ -π
∈ ℝ* |
| 61 | | 0xr 11308 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
ℝ* |
| 62 | | ioogtlb 45508 |
. . . . . . . . . . . . . . 15
⊢ ((-π
∈ ℝ* ∧ 0 ∈ ℝ* ∧ 𝑥 ∈ (-π(,)0)) →
-π < 𝑥) |
| 63 | 60, 61, 62 | mp3an12 1453 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (-π(,)0) → -π
< 𝑥) |
| 64 | 55, 31, 59, 63 | ltadd1dd 11874 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (-π(,)0) →
(-π + 𝑇) < (𝑥 + 𝑇)) |
| 65 | 54, 64 | eqbrtrd 5165 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (-π(,)0) → π
< (𝑥 + 𝑇)) |
| 66 | 58 | recni 11275 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑇 ∈ ℂ |
| 67 | 66 | mullidi 11266 |
. . . . . . . . . . . . . . . 16
⊢ (1
· 𝑇) = 𝑇 |
| 68 | 67 | eqcomi 2746 |
. . . . . . . . . . . . . . 15
⊢ 𝑇 = (1 · 𝑇) |
| 69 | 68 | oveq2i 7442 |
. . . . . . . . . . . . . 14
⊢ (𝑥 + 𝑇) = (𝑥 + (1 · 𝑇)) |
| 70 | 69 | oveq1i 7441 |
. . . . . . . . . . . . 13
⊢ ((𝑥 + 𝑇) mod 𝑇) = ((𝑥 + (1 · 𝑇)) mod 𝑇) |
| 71 | 31, 59 | readdcld 11290 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (-π(,)0) →
(𝑥 + 𝑇) ∈ ℝ) |
| 72 | | 0red 11264 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (-π(,)0) → 0
∈ ℝ) |
| 73 | 8 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ (-π(,)0) → 0
< π) |
| 74 | 72, 33, 71, 73, 65 | lttrd 11422 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (-π(,)0) → 0
< (𝑥 + 𝑇)) |
| 75 | 72, 71, 74 | ltled 11409 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (-π(,)0) → 0
≤ (𝑥 + 𝑇)) |
| 76 | | iooltub 45523 |
. . . . . . . . . . . . . . . . 17
⊢ ((-π
∈ ℝ* ∧ 0 ∈ ℝ* ∧ 𝑥 ∈ (-π(,)0)) →
𝑥 < 0) |
| 77 | 60, 61, 76 | mp3an12 1453 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ (-π(,)0) → 𝑥 < 0) |
| 78 | 31, 72, 59, 77 | ltadd1dd 11874 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (-π(,)0) →
(𝑥 + 𝑇) < (0 + 𝑇)) |
| 79 | 59 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ (-π(,)0) → 𝑇 ∈
ℂ) |
| 80 | 79 | addlidd 11462 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (-π(,)0) → (0 +
𝑇) = 𝑇) |
| 81 | 78, 80 | breqtrd 5169 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (-π(,)0) →
(𝑥 + 𝑇) < 𝑇) |
| 82 | | modid 13936 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 + 𝑇) ∈ ℝ ∧ 𝑇 ∈ ℝ+) ∧ (0 ≤
(𝑥 + 𝑇) ∧ (𝑥 + 𝑇) < 𝑇)) → ((𝑥 + 𝑇) mod 𝑇) = (𝑥 + 𝑇)) |
| 83 | 71, 40, 75, 81, 82 | syl22anc 839 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (-π(,)0) →
((𝑥 + 𝑇) mod 𝑇) = (𝑥 + 𝑇)) |
| 84 | | 1zzd 12648 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (-π(,)0) → 1
∈ ℤ) |
| 85 | | modcyc 13946 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑇 ∈ ℝ+
∧ 1 ∈ ℤ) → ((𝑥 + (1 · 𝑇)) mod 𝑇) = (𝑥 mod 𝑇)) |
| 86 | 31, 40, 84, 85 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (-π(,)0) →
((𝑥 + (1 · 𝑇)) mod 𝑇) = (𝑥 mod 𝑇)) |
| 87 | 70, 83, 86 | 3eqtr3a 2801 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (-π(,)0) →
(𝑥 + 𝑇) = (𝑥 mod 𝑇)) |
| 88 | 65, 87 | breqtrd 5169 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (-π(,)0) → π
< (𝑥 mod 𝑇)) |
| 89 | 33, 41, 88 | ltnsymd 11410 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (-π(,)0) → ¬
(𝑥 mod 𝑇) < π) |
| 90 | 89 | iffalsed 4536 |
. . . . . . . . 9
⊢ (𝑥 ∈ (-π(,)0) →
if((𝑥 mod 𝑇) < π, 1, -1) = -1) |
| 91 | 32, 90 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝑥 ∈ (-π(,)0) →
(𝐹‘𝑥) = -1) |
| 92 | 91 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)0)) → (𝐹‘𝑥) = -1) |
| 93 | 92 | oveq1d 7446 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)0)) → ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) = (-1 · (sin‘(𝑁 · 𝑥)))) |
| 94 | 93 | mpteq2dva 5242 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ (-π(,)0) ↦ ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥)))) = (𝑥 ∈ (-π(,)0) ↦ (-1 ·
(sin‘(𝑁 ·
𝑥))))) |
| 95 | | neg1cn 12380 |
. . . . . . 7
⊢ -1 ∈
ℂ |
| 96 | 95 | a1i 11 |
. . . . . 6
⊢ (𝜑 → -1 ∈
ℂ) |
| 97 | 24 | nnred 12281 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 98 | 97 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)0)) → 𝑁 ∈ ℝ) |
| 99 | 31 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)0)) → 𝑥 ∈
ℝ) |
| 100 | 98, 99 | remulcld 11291 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)0)) → (𝑁 · 𝑥) ∈ ℝ) |
| 101 | 100 | resincld 16179 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)0)) → (sin‘(𝑁 · 𝑥)) ∈ ℝ) |
| 102 | | ioossicc 13473 |
. . . . . . . 8
⊢
(-π(,)0) ⊆ (-π[,]0) |
| 103 | 102 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (-π(,)0) ⊆
(-π[,]0)) |
| 104 | | ioombl 25600 |
. . . . . . . 8
⊢
(-π(,)0) ∈ dom vol |
| 105 | 104 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (-π(,)0) ∈ dom
vol) |
| 106 | 97 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]0)) → 𝑁 ∈ ℝ) |
| 107 | | iccssre 13469 |
. . . . . . . . . . . 12
⊢ ((-π
∈ ℝ ∧ 0 ∈ ℝ) → (-π[,]0) ⊆
ℝ) |
| 108 | 2, 5, 107 | mp2an 692 |
. . . . . . . . . . 11
⊢
(-π[,]0) ⊆ ℝ |
| 109 | 108 | sseli 3979 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (-π[,]0) → 𝑥 ∈
ℝ) |
| 110 | 109 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]0)) → 𝑥 ∈
ℝ) |
| 111 | 106, 110 | remulcld 11291 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]0)) → (𝑁 · 𝑥) ∈ ℝ) |
| 112 | 111 | resincld 16179 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]0)) → (sin‘(𝑁 · 𝑥)) ∈ ℝ) |
| 113 | | 0red 11264 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℝ) |
| 114 | | sincn 26488 |
. . . . . . . . . 10
⊢ sin
∈ (ℂ–cn→ℂ) |
| 115 | 114 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → sin ∈
(ℂ–cn→ℂ)) |
| 116 | | ax-resscn 11212 |
. . . . . . . . . . . . 13
⊢ ℝ
⊆ ℂ |
| 117 | 108, 116 | sstri 3993 |
. . . . . . . . . . . 12
⊢
(-π[,]0) ⊆ ℂ |
| 118 | 117 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (-π[,]0) ⊆
ℂ) |
| 119 | | ssid 4006 |
. . . . . . . . . . . 12
⊢ ℂ
⊆ ℂ |
| 120 | 119 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → ℂ ⊆
ℂ) |
| 121 | 118, 25, 120 | constcncfg 45887 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ (-π[,]0) ↦ 𝑁) ∈ ((-π[,]0)–cn→ℂ)) |
| 122 | 118, 120 | idcncfg 45888 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ (-π[,]0) ↦ 𝑥) ∈
((-π[,]0)–cn→ℂ)) |
| 123 | 121, 122 | mulcncf 25480 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (-π[,]0) ↦ (𝑁 · 𝑥)) ∈ ((-π[,]0)–cn→ℂ)) |
| 124 | 115, 123 | cncfmpt1f 24940 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (-π[,]0) ↦ (sin‘(𝑁 · 𝑥))) ∈ ((-π[,]0)–cn→ℂ)) |
| 125 | | cniccibl 25876 |
. . . . . . . 8
⊢ ((-π
∈ ℝ ∧ 0 ∈ ℝ ∧ (𝑥 ∈ (-π[,]0) ↦ (sin‘(𝑁 · 𝑥))) ∈ ((-π[,]0)–cn→ℂ)) → (𝑥 ∈ (-π[,]0) ↦ (sin‘(𝑁 · 𝑥))) ∈
𝐿1) |
| 126 | 3, 113, 124, 125 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (-π[,]0) ↦ (sin‘(𝑁 · 𝑥))) ∈
𝐿1) |
| 127 | 103, 105,
112, 126 | iblss 25840 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (-π(,)0) ↦ (sin‘(𝑁 · 𝑥))) ∈
𝐿1) |
| 128 | 96, 101, 127 | iblmulc2 25866 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ (-π(,)0) ↦ (-1 ·
(sin‘(𝑁 ·
𝑥)))) ∈
𝐿1) |
| 129 | 94, 128 | eqeltrd 2841 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (-π(,)0) ↦ ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥)))) ∈
𝐿1) |
| 130 | 60 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (0(,)π) → -π
∈ ℝ*) |
| 131 | 1 | rexri 11319 |
. . . . . . . . . . . 12
⊢ π
∈ ℝ* |
| 132 | 131 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (0(,)π) → π
∈ ℝ*) |
| 133 | | elioore 13417 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈
ℝ) |
| 134 | 2 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (0(,)π) → -π
∈ ℝ) |
| 135 | | 0red 11264 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (0(,)π) → 0
∈ ℝ) |
| 136 | 6 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (0(,)π) → -π
< 0) |
| 137 | | ioogtlb 45508 |
. . . . . . . . . . . . 13
⊢ ((0
∈ ℝ* ∧ π ∈ ℝ* ∧ 𝑥 ∈ (0(,)π)) → 0
< 𝑥) |
| 138 | 61, 131, 137 | mp3an12 1453 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (0(,)π) → 0 <
𝑥) |
| 139 | 134, 135,
133, 136, 138 | lttrd 11422 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (0(,)π) → -π
< 𝑥) |
| 140 | | iooltub 45523 |
. . . . . . . . . . . 12
⊢ ((0
∈ ℝ* ∧ π ∈ ℝ* ∧ 𝑥 ∈ (0(,)π)) → 𝑥 < π) |
| 141 | 61, 131, 140 | mp3an12 1453 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (0(,)π) → 𝑥 < π) |
| 142 | 130, 132,
133, 139, 141 | eliood 45511 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈
(-π(,)π)) |
| 143 | 142, 20 | sylan2 593 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝐹‘𝑥) = if((𝑥 mod 𝑇) < π, 1, -1)) |
| 144 | 39 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (0(,)π) → 𝑇 ∈
ℝ+) |
| 145 | 135, 133,
138 | ltled 11409 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (0(,)π) → 0 ≤
𝑥) |
| 146 | 1 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (0(,)π) → π
∈ ℝ) |
| 147 | 58 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (0(,)π) → 𝑇 ∈
ℝ) |
| 148 | | 2timesgt 45300 |
. . . . . . . . . . . . . . . . 17
⊢ (π
∈ ℝ+ → π < (2 · π)) |
| 149 | 36, 148 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ π <
(2 · π) |
| 150 | 149, 34 | breqtrri 5170 |
. . . . . . . . . . . . . . 15
⊢ π <
𝑇 |
| 151 | 150 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (0(,)π) → π
< 𝑇) |
| 152 | 133, 146,
147, 141, 151 | lttrd 11422 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (0(,)π) → 𝑥 < 𝑇) |
| 153 | | modid 13936 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ ℝ ∧ 𝑇 ∈ ℝ+)
∧ (0 ≤ 𝑥 ∧ 𝑥 < 𝑇)) → (𝑥 mod 𝑇) = 𝑥) |
| 154 | 133, 144,
145, 152, 153 | syl22anc 839 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (0(,)π) → (𝑥 mod 𝑇) = 𝑥) |
| 155 | 154, 141 | eqbrtrd 5165 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (0(,)π) → (𝑥 mod 𝑇) < π) |
| 156 | 155 | iftrued 4533 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (0(,)π) →
if((𝑥 mod 𝑇) < π, 1, -1) = 1) |
| 157 | 156 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → if((𝑥 mod 𝑇) < π, 1, -1) = 1) |
| 158 | 143, 157 | eqtrd 2777 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝐹‘𝑥) = 1) |
| 159 | 158 | oveq1d 7446 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) = (1 · (sin‘(𝑁 · 𝑥)))) |
| 160 | 142, 29 | sylan2 593 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘(𝑁 · 𝑥)) ∈ ℂ) |
| 161 | 160 | mullidd 11279 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (1 ·
(sin‘(𝑁 ·
𝑥))) = (sin‘(𝑁 · 𝑥))) |
| 162 | 159, 161 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) = (sin‘(𝑁 · 𝑥))) |
| 163 | 162 | mpteq2dva 5242 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥)))) = (𝑥 ∈ (0(,)π) ↦ (sin‘(𝑁 · 𝑥)))) |
| 164 | | ioossicc 13473 |
. . . . . . 7
⊢
(0(,)π) ⊆ (0[,]π) |
| 165 | 164 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (0(,)π) ⊆
(0[,]π)) |
| 166 | | ioombl 25600 |
. . . . . . 7
⊢
(0(,)π) ∈ dom vol |
| 167 | 166 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (0(,)π) ∈ dom
vol) |
| 168 | 97 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑁 ∈ ℝ) |
| 169 | | iccssre 13469 |
. . . . . . . . . . 11
⊢ ((0
∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆
ℝ) |
| 170 | 5, 1, 169 | mp2an 692 |
. . . . . . . . . 10
⊢
(0[,]π) ⊆ ℝ |
| 171 | 170 | sseli 3979 |
. . . . . . . . 9
⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈
ℝ) |
| 172 | 171 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑥 ∈ ℝ) |
| 173 | 168, 172 | remulcld 11291 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝑁 · 𝑥) ∈ ℝ) |
| 174 | 173 | resincld 16179 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (sin‘(𝑁 · 𝑥)) ∈ ℝ) |
| 175 | 170, 116 | sstri 3993 |
. . . . . . . . . . 11
⊢
(0[,]π) ⊆ ℂ |
| 176 | 175 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → (0[,]π) ⊆
ℂ) |
| 177 | 176, 25, 120 | constcncfg 45887 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ 𝑁) ∈ ((0[,]π)–cn→ℂ)) |
| 178 | 176, 120 | idcncfg 45888 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ 𝑥) ∈ ((0[,]π)–cn→ℂ)) |
| 179 | 177, 178 | mulcncf 25480 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (𝑁 · 𝑥)) ∈ ((0[,]π)–cn→ℂ)) |
| 180 | 115, 179 | cncfmpt1f 24940 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (sin‘(𝑁 · 𝑥))) ∈ ((0[,]π)–cn→ℂ)) |
| 181 | | cniccibl 25876 |
. . . . . . 7
⊢ ((0
∈ ℝ ∧ π ∈ ℝ ∧ (𝑥 ∈ (0[,]π) ↦ (sin‘(𝑁 · 𝑥))) ∈ ((0[,]π)–cn→ℂ)) → (𝑥 ∈ (0[,]π) ↦ (sin‘(𝑁 · 𝑥))) ∈
𝐿1) |
| 182 | 113, 4, 180, 181 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (sin‘(𝑁 · 𝑥))) ∈
𝐿1) |
| 183 | 165, 167,
174, 182 | iblss 25840 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ (sin‘(𝑁 · 𝑥))) ∈
𝐿1) |
| 184 | 163, 183 | eqeltrd 2841 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥)))) ∈
𝐿1) |
| 185 | 3, 4, 12, 30, 129, 184 | itgsplitioo 25873 |
. . 3
⊢ (𝜑 → ∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 = (∫(-π(,)0)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 + ∫(0(,)π)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥)) |
| 186 | 185 | oveq1d 7446 |
. 2
⊢ (𝜑 →
(∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 / π) = ((∫(-π(,)0)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 + ∫(0(,)π)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥) / π)) |
| 187 | 91 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝑥 ∈ (-π(,)0) →
((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) = (-1 · (sin‘(𝑁 · 𝑥)))) |
| 188 | 187 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)0)) → ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) = (-1 · (sin‘(𝑁 · 𝑥)))) |
| 189 | 60 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (-π(,)0) → -π
∈ ℝ*) |
| 190 | 131 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (-π(,)0) → π
∈ ℝ*) |
| 191 | 31, 72, 33, 77, 73 | lttrd 11422 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (-π(,)0) → 𝑥 < π) |
| 192 | 189, 190,
31, 63, 191 | eliood 45511 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (-π(,)0) → 𝑥 ∈
(-π(,)π)) |
| 193 | 192, 29 | sylan2 593 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)0)) → (sin‘(𝑁 · 𝑥)) ∈ ℂ) |
| 194 | 193 | mulm1d 11715 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)0)) → (-1 ·
(sin‘(𝑁 ·
𝑥))) = -(sin‘(𝑁 · 𝑥))) |
| 195 | 188, 194 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (-π(,)0)) → ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) = -(sin‘(𝑁 · 𝑥))) |
| 196 | 195 | itgeq2dv 25817 |
. . . . . 6
⊢ (𝜑 → ∫(-π(,)0)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 = ∫(-π(,)0)-(sin‘(𝑁 · 𝑥)) d𝑥) |
| 197 | 101, 127 | itgneg 25839 |
. . . . . 6
⊢ (𝜑 →
-∫(-π(,)0)(sin‘(𝑁 · 𝑥)) d𝑥 = ∫(-π(,)0)-(sin‘(𝑁 · 𝑥)) d𝑥) |
| 198 | 24 | nnne0d 12316 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ≠ 0) |
| 199 | 7 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → -π ≤
0) |
| 200 | 25, 198, 3, 113, 199 | itgsincmulx 45989 |
. . . . . . . . 9
⊢ (𝜑 →
∫(-π(,)0)(sin‘(𝑁 · 𝑥)) d𝑥 = (((cos‘(𝑁 · -π)) − (cos‘(𝑁 · 0))) / 𝑁)) |
| 201 | 24 | nnzd 12640 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 202 | | cosknegpi 45884 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℤ →
(cos‘(𝑁 ·
-π)) = if(2 ∥ 𝑁,
1, -1)) |
| 203 | 201, 202 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (cos‘(𝑁 · -π)) = if(2 ∥
𝑁, 1, -1)) |
| 204 | 25 | mul01d 11460 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑁 · 0) = 0) |
| 205 | 204 | fveq2d 6910 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (cos‘(𝑁 · 0)) =
(cos‘0)) |
| 206 | | cos0 16186 |
. . . . . . . . . . . . 13
⊢
(cos‘0) = 1 |
| 207 | 205, 206 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ (𝜑 → (cos‘(𝑁 · 0)) =
1) |
| 208 | 203, 207 | oveq12d 7449 |
. . . . . . . . . . 11
⊢ (𝜑 → ((cos‘(𝑁 · -π)) −
(cos‘(𝑁 · 0)))
= (if(2 ∥ 𝑁, 1, -1)
− 1)) |
| 209 | | 1m1e0 12338 |
. . . . . . . . . . . . 13
⊢ (1
− 1) = 0 |
| 210 | | iftrue 4531 |
. . . . . . . . . . . . . 14
⊢ (2
∥ 𝑁 → if(2
∥ 𝑁, 1, -1) =
1) |
| 211 | 210 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ (2
∥ 𝑁 → (if(2
∥ 𝑁, 1, -1) −
1) = (1 − 1)) |
| 212 | | iftrue 4531 |
. . . . . . . . . . . . 13
⊢ (2
∥ 𝑁 → if(2
∥ 𝑁, 0, -2) =
0) |
| 213 | 209, 211,
212 | 3eqtr4a 2803 |
. . . . . . . . . . . 12
⊢ (2
∥ 𝑁 → (if(2
∥ 𝑁, 1, -1) −
1) = if(2 ∥ 𝑁, 0,
-2)) |
| 214 | | iffalse 4534 |
. . . . . . . . . . . . . 14
⊢ (¬ 2
∥ 𝑁 → if(2
∥ 𝑁, 1, -1) =
-1) |
| 215 | 214 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ (¬ 2
∥ 𝑁 → (if(2
∥ 𝑁, 1, -1) −
1) = (-1 − 1)) |
| 216 | | ax-1cn 11213 |
. . . . . . . . . . . . . . . 16
⊢ 1 ∈
ℂ |
| 217 | | negdi2 11567 |
. . . . . . . . . . . . . . . 16
⊢ ((1
∈ ℂ ∧ 1 ∈ ℂ) → -(1 + 1) = (-1 −
1)) |
| 218 | 216, 216,
217 | mp2an 692 |
. . . . . . . . . . . . . . 15
⊢ -(1 + 1)
= (-1 − 1) |
| 219 | 218 | eqcomi 2746 |
. . . . . . . . . . . . . 14
⊢ (-1
− 1) = -(1 + 1) |
| 220 | 219 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (¬ 2
∥ 𝑁 → (-1
− 1) = -(1 + 1)) |
| 221 | | 1p1e2 12391 |
. . . . . . . . . . . . . . 15
⊢ (1 + 1) =
2 |
| 222 | 221 | negeqi 11501 |
. . . . . . . . . . . . . 14
⊢ -(1 + 1)
= -2 |
| 223 | | iffalse 4534 |
. . . . . . . . . . . . . 14
⊢ (¬ 2
∥ 𝑁 → if(2
∥ 𝑁, 0, -2) =
-2) |
| 224 | 222, 223 | eqtr4id 2796 |
. . . . . . . . . . . . 13
⊢ (¬ 2
∥ 𝑁 → -(1 + 1) =
if(2 ∥ 𝑁, 0,
-2)) |
| 225 | 215, 220,
224 | 3eqtrd 2781 |
. . . . . . . . . . . 12
⊢ (¬ 2
∥ 𝑁 → (if(2
∥ 𝑁, 1, -1) −
1) = if(2 ∥ 𝑁, 0,
-2)) |
| 226 | 213, 225 | pm2.61i 182 |
. . . . . . . . . . 11
⊢ (if(2
∥ 𝑁, 1, -1) −
1) = if(2 ∥ 𝑁, 0,
-2) |
| 227 | 208, 226 | eqtrdi 2793 |
. . . . . . . . . 10
⊢ (𝜑 → ((cos‘(𝑁 · -π)) −
(cos‘(𝑁 · 0)))
= if(2 ∥ 𝑁, 0,
-2)) |
| 228 | 227 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝜑 → (((cos‘(𝑁 · -π)) −
(cos‘(𝑁 · 0)))
/ 𝑁) = (if(2 ∥ 𝑁, 0, -2) / 𝑁)) |
| 229 | 200, 228 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 →
∫(-π(,)0)(sin‘(𝑁 · 𝑥)) d𝑥 = (if(2 ∥ 𝑁, 0, -2) / 𝑁)) |
| 230 | 229 | negeqd 11502 |
. . . . . . 7
⊢ (𝜑 →
-∫(-π(,)0)(sin‘(𝑁 · 𝑥)) d𝑥 = -(if(2 ∥ 𝑁, 0, -2) / 𝑁)) |
| 231 | | 0cn 11253 |
. . . . . . . . . 10
⊢ 0 ∈
ℂ |
| 232 | | 2cn 12341 |
. . . . . . . . . . 11
⊢ 2 ∈
ℂ |
| 233 | 232 | negcli 11577 |
. . . . . . . . . 10
⊢ -2 ∈
ℂ |
| 234 | 231, 233 | ifcli 4573 |
. . . . . . . . 9
⊢ if(2
∥ 𝑁, 0, -2) ∈
ℂ |
| 235 | 234 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → if(2 ∥ 𝑁, 0, -2) ∈
ℂ) |
| 236 | 235, 25, 198 | divnegd 12056 |
. . . . . . 7
⊢ (𝜑 → -(if(2 ∥ 𝑁, 0, -2) / 𝑁) = (-if(2 ∥ 𝑁, 0, -2) / 𝑁)) |
| 237 | | neg0 11555 |
. . . . . . . . . . 11
⊢ -0 =
0 |
| 238 | 212 | negeqd 11502 |
. . . . . . . . . . 11
⊢ (2
∥ 𝑁 → -if(2
∥ 𝑁, 0, -2) =
-0) |
| 239 | | iftrue 4531 |
. . . . . . . . . . 11
⊢ (2
∥ 𝑁 → if(2
∥ 𝑁, 0, 2) =
0) |
| 240 | 237, 238,
239 | 3eqtr4a 2803 |
. . . . . . . . . 10
⊢ (2
∥ 𝑁 → -if(2
∥ 𝑁, 0, -2) = if(2
∥ 𝑁, 0,
2)) |
| 241 | 232 | negnegi 11579 |
. . . . . . . . . . 11
⊢ --2 =
2 |
| 242 | 223 | negeqd 11502 |
. . . . . . . . . . 11
⊢ (¬ 2
∥ 𝑁 → -if(2
∥ 𝑁, 0, -2) =
--2) |
| 243 | | iffalse 4534 |
. . . . . . . . . . 11
⊢ (¬ 2
∥ 𝑁 → if(2
∥ 𝑁, 0, 2) =
2) |
| 244 | 241, 242,
243 | 3eqtr4a 2803 |
. . . . . . . . . 10
⊢ (¬ 2
∥ 𝑁 → -if(2
∥ 𝑁, 0, -2) = if(2
∥ 𝑁, 0,
2)) |
| 245 | 240, 244 | pm2.61i 182 |
. . . . . . . . 9
⊢ -if(2
∥ 𝑁, 0, -2) = if(2
∥ 𝑁, 0,
2) |
| 246 | 245 | oveq1i 7441 |
. . . . . . . 8
⊢ (-if(2
∥ 𝑁, 0, -2) / 𝑁) = (if(2 ∥ 𝑁, 0, 2) / 𝑁) |
| 247 | 246 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (-if(2 ∥ 𝑁, 0, -2) / 𝑁) = (if(2 ∥ 𝑁, 0, 2) / 𝑁)) |
| 248 | 230, 236,
247 | 3eqtrd 2781 |
. . . . . 6
⊢ (𝜑 →
-∫(-π(,)0)(sin‘(𝑁 · 𝑥)) d𝑥 = (if(2 ∥ 𝑁, 0, 2) / 𝑁)) |
| 249 | 196, 197,
248 | 3eqtr2d 2783 |
. . . . 5
⊢ (𝜑 → ∫(-π(,)0)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 = (if(2 ∥ 𝑁, 0, 2) / 𝑁)) |
| 250 | 133, 17, 19 | sylancl 586 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (0(,)π) → (𝐹‘𝑥) = if((𝑥 mod 𝑇) < π, 1, -1)) |
| 251 | 250, 156 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (0(,)π) → (𝐹‘𝑥) = 1) |
| 252 | 251 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝑥 ∈ (0(,)π) →
((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) = (1 · (sin‘(𝑁 · 𝑥)))) |
| 253 | 252 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) = (1 · (sin‘(𝑁 · 𝑥)))) |
| 254 | 253, 161 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) = (sin‘(𝑁 · 𝑥))) |
| 255 | 254 | itgeq2dv 25817 |
. . . . . 6
⊢ (𝜑 → ∫(0(,)π)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 = ∫(0(,)π)(sin‘(𝑁 · 𝑥)) d𝑥) |
| 256 | 9 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 0 ≤
π) |
| 257 | 25, 198, 113, 4, 256 | itgsincmulx 45989 |
. . . . . 6
⊢ (𝜑 →
∫(0(,)π)(sin‘(𝑁 · 𝑥)) d𝑥 = (((cos‘(𝑁 · 0)) − (cos‘(𝑁 · π))) / 𝑁)) |
| 258 | | coskpi2 45881 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℤ →
(cos‘(𝑁 ·
π)) = if(2 ∥ 𝑁, 1,
-1)) |
| 259 | 201, 258 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (cos‘(𝑁 · π)) = if(2 ∥
𝑁, 1, -1)) |
| 260 | 207, 259 | oveq12d 7449 |
. . . . . . . 8
⊢ (𝜑 → ((cos‘(𝑁 · 0)) −
(cos‘(𝑁 ·
π))) = (1 − if(2 ∥ 𝑁, 1, -1))) |
| 261 | 210 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (2
∥ 𝑁 → (1 −
if(2 ∥ 𝑁, 1, -1)) =
(1 − 1)) |
| 262 | 209, 261,
239 | 3eqtr4a 2803 |
. . . . . . . . 9
⊢ (2
∥ 𝑁 → (1 −
if(2 ∥ 𝑁, 1, -1)) =
if(2 ∥ 𝑁, 0,
2)) |
| 263 | 214 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (¬ 2
∥ 𝑁 → (1 −
if(2 ∥ 𝑁, 1, -1)) =
(1 − -1)) |
| 264 | 216, 216 | subnegi 11588 |
. . . . . . . . . . 11
⊢ (1
− -1) = (1 + 1) |
| 265 | 264 | a1i 11 |
. . . . . . . . . 10
⊢ (¬ 2
∥ 𝑁 → (1 −
-1) = (1 + 1)) |
| 266 | 221, 243 | eqtr4id 2796 |
. . . . . . . . . 10
⊢ (¬ 2
∥ 𝑁 → (1 + 1) =
if(2 ∥ 𝑁, 0,
2)) |
| 267 | 263, 265,
266 | 3eqtrd 2781 |
. . . . . . . . 9
⊢ (¬ 2
∥ 𝑁 → (1 −
if(2 ∥ 𝑁, 1, -1)) =
if(2 ∥ 𝑁, 0,
2)) |
| 268 | 262, 267 | pm2.61i 182 |
. . . . . . . 8
⊢ (1
− if(2 ∥ 𝑁, 1,
-1)) = if(2 ∥ 𝑁, 0,
2) |
| 269 | 260, 268 | eqtrdi 2793 |
. . . . . . 7
⊢ (𝜑 → ((cos‘(𝑁 · 0)) −
(cos‘(𝑁 ·
π))) = if(2 ∥ 𝑁,
0, 2)) |
| 270 | 269 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → (((cos‘(𝑁 · 0)) −
(cos‘(𝑁 ·
π))) / 𝑁) = (if(2
∥ 𝑁, 0, 2) / 𝑁)) |
| 271 | 255, 257,
270 | 3eqtrd 2781 |
. . . . 5
⊢ (𝜑 → ∫(0(,)π)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 = (if(2 ∥ 𝑁, 0, 2) / 𝑁)) |
| 272 | 249, 271 | oveq12d 7449 |
. . . 4
⊢ (𝜑 → (∫(-π(,)0)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 + ∫(0(,)π)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥) = ((if(2 ∥ 𝑁, 0, 2) / 𝑁) + (if(2 ∥ 𝑁, 0, 2) / 𝑁))) |
| 273 | 231, 232 | ifcli 4573 |
. . . . . 6
⊢ if(2
∥ 𝑁, 0, 2) ∈
ℂ |
| 274 | 273 | a1i 11 |
. . . . 5
⊢ (𝜑 → if(2 ∥ 𝑁, 0, 2) ∈
ℂ) |
| 275 | 274, 274,
25, 198 | divdird 12081 |
. . . 4
⊢ (𝜑 → ((if(2 ∥ 𝑁, 0, 2) + if(2 ∥ 𝑁, 0, 2)) / 𝑁) = ((if(2 ∥ 𝑁, 0, 2) / 𝑁) + (if(2 ∥ 𝑁, 0, 2) / 𝑁))) |
| 276 | 239, 239 | oveq12d 7449 |
. . . . . . . . 9
⊢ (2
∥ 𝑁 → (if(2
∥ 𝑁, 0, 2) + if(2
∥ 𝑁, 0, 2)) = (0 +
0)) |
| 277 | | 00id 11436 |
. . . . . . . . 9
⊢ (0 + 0) =
0 |
| 278 | 276, 277 | eqtrdi 2793 |
. . . . . . . 8
⊢ (2
∥ 𝑁 → (if(2
∥ 𝑁, 0, 2) + if(2
∥ 𝑁, 0, 2)) =
0) |
| 279 | 278 | oveq1d 7446 |
. . . . . . 7
⊢ (2
∥ 𝑁 → ((if(2
∥ 𝑁, 0, 2) + if(2
∥ 𝑁, 0, 2)) / 𝑁) = (0 / 𝑁)) |
| 280 | 279 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → ((if(2 ∥ 𝑁, 0, 2) + if(2 ∥ 𝑁, 0, 2)) / 𝑁) = (0 / 𝑁)) |
| 281 | 25, 198 | div0d 12042 |
. . . . . . 7
⊢ (𝜑 → (0 / 𝑁) = 0) |
| 282 | 281 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (0 / 𝑁) = 0) |
| 283 | | iftrue 4531 |
. . . . . . . 8
⊢ (2
∥ 𝑁 → if(2
∥ 𝑁, 0, (4 / 𝑁)) = 0) |
| 284 | 283 | eqcomd 2743 |
. . . . . . 7
⊢ (2
∥ 𝑁 → 0 = if(2
∥ 𝑁, 0, (4 / 𝑁))) |
| 285 | 284 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → 0 = if(2 ∥ 𝑁, 0, (4 / 𝑁))) |
| 286 | 280, 282,
285 | 3eqtrd 2781 |
. . . . 5
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → ((if(2 ∥ 𝑁, 0, 2) + if(2 ∥ 𝑁, 0, 2)) / 𝑁) = if(2 ∥ 𝑁, 0, (4 / 𝑁))) |
| 287 | 243, 243 | oveq12d 7449 |
. . . . . . . . 9
⊢ (¬ 2
∥ 𝑁 → (if(2
∥ 𝑁, 0, 2) + if(2
∥ 𝑁, 0, 2)) = (2 +
2)) |
| 288 | | 2p2e4 12401 |
. . . . . . . . 9
⊢ (2 + 2) =
4 |
| 289 | 287, 288 | eqtrdi 2793 |
. . . . . . . 8
⊢ (¬ 2
∥ 𝑁 → (if(2
∥ 𝑁, 0, 2) + if(2
∥ 𝑁, 0, 2)) =
4) |
| 290 | 289 | oveq1d 7446 |
. . . . . . 7
⊢ (¬ 2
∥ 𝑁 → ((if(2
∥ 𝑁, 0, 2) + if(2
∥ 𝑁, 0, 2)) / 𝑁) = (4 / 𝑁)) |
| 291 | | iffalse 4534 |
. . . . . . 7
⊢ (¬ 2
∥ 𝑁 → if(2
∥ 𝑁, 0, (4 / 𝑁)) = (4 / 𝑁)) |
| 292 | 290, 291 | eqtr4d 2780 |
. . . . . 6
⊢ (¬ 2
∥ 𝑁 → ((if(2
∥ 𝑁, 0, 2) + if(2
∥ 𝑁, 0, 2)) / 𝑁) = if(2 ∥ 𝑁, 0, (4 / 𝑁))) |
| 293 | 292 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → ((if(2 ∥ 𝑁, 0, 2) + if(2 ∥ 𝑁, 0, 2)) / 𝑁) = if(2 ∥ 𝑁, 0, (4 / 𝑁))) |
| 294 | 286, 293 | pm2.61dan 813 |
. . . 4
⊢ (𝜑 → ((if(2 ∥ 𝑁, 0, 2) + if(2 ∥ 𝑁, 0, 2)) / 𝑁) = if(2 ∥ 𝑁, 0, (4 / 𝑁))) |
| 295 | 272, 275,
294 | 3eqtr2d 2783 |
. . 3
⊢ (𝜑 → (∫(-π(,)0)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 + ∫(0(,)π)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥) = if(2 ∥ 𝑁, 0, (4 / 𝑁))) |
| 296 | 295 | oveq1d 7446 |
. 2
⊢ (𝜑 → ((∫(-π(,)0)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 + ∫(0(,)π)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥) / π) = (if(2 ∥ 𝑁, 0, (4 / 𝑁)) / π)) |
| 297 | 283 | oveq1d 7446 |
. . . . 5
⊢ (2
∥ 𝑁 → (if(2
∥ 𝑁, 0, (4 / 𝑁)) / π) = (0 /
π)) |
| 298 | 297 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (if(2 ∥ 𝑁, 0, (4 / 𝑁)) / π) = (0 / π)) |
| 299 | 5, 8 | gtneii 11373 |
. . . . . 6
⊢ π ≠
0 |
| 300 | 42, 299 | div0i 12001 |
. . . . 5
⊢ (0 /
π) = 0 |
| 301 | 300 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (0 / π) = 0) |
| 302 | | iftrue 4531 |
. . . . . 6
⊢ (2
∥ 𝑁 → if(2
∥ 𝑁, 0, (4 / (𝑁 · π))) =
0) |
| 303 | 302 | eqcomd 2743 |
. . . . 5
⊢ (2
∥ 𝑁 → 0 = if(2
∥ 𝑁, 0, (4 / (𝑁 ·
π)))) |
| 304 | 303 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → 0 = if(2 ∥ 𝑁, 0, (4 / (𝑁 · π)))) |
| 305 | 298, 301,
304 | 3eqtrd 2781 |
. . 3
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (if(2 ∥ 𝑁, 0, (4 / 𝑁)) / π) = if(2 ∥ 𝑁, 0, (4 / (𝑁 · π)))) |
| 306 | 291 | oveq1d 7446 |
. . . . 5
⊢ (¬ 2
∥ 𝑁 → (if(2
∥ 𝑁, 0, (4 / 𝑁)) / π) = ((4 / 𝑁) / π)) |
| 307 | 306 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → (if(2 ∥ 𝑁, 0, (4 / 𝑁)) / π) = ((4 / 𝑁) / π)) |
| 308 | | 4cn 12351 |
. . . . . . 7
⊢ 4 ∈
ℂ |
| 309 | 308 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 4 ∈
ℂ) |
| 310 | 42 | a1i 11 |
. . . . . 6
⊢ (𝜑 → π ∈
ℂ) |
| 311 | 299 | a1i 11 |
. . . . . 6
⊢ (𝜑 → π ≠
0) |
| 312 | 309, 25, 310, 198, 311 | divdiv1d 12074 |
. . . . 5
⊢ (𝜑 → ((4 / 𝑁) / π) = (4 / (𝑁 · π))) |
| 313 | 312 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → ((4 / 𝑁) / π) = (4 / (𝑁 · π))) |
| 314 | | iffalse 4534 |
. . . . . 6
⊢ (¬ 2
∥ 𝑁 → if(2
∥ 𝑁, 0, (4 / (𝑁 · π))) = (4 / (𝑁 ·
π))) |
| 315 | 314 | eqcomd 2743 |
. . . . 5
⊢ (¬ 2
∥ 𝑁 → (4 /
(𝑁 · π)) = if(2
∥ 𝑁, 0, (4 / (𝑁 ·
π)))) |
| 316 | 315 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → (4 / (𝑁 · π)) = if(2 ∥ 𝑁, 0, (4 / (𝑁 · π)))) |
| 317 | 307, 313,
316 | 3eqtrd 2781 |
. . 3
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → (if(2 ∥ 𝑁, 0, (4 / 𝑁)) / π) = if(2 ∥ 𝑁, 0, (4 / (𝑁 · π)))) |
| 318 | 305, 317 | pm2.61dan 813 |
. 2
⊢ (𝜑 → (if(2 ∥ 𝑁, 0, (4 / 𝑁)) / π) = if(2 ∥ 𝑁, 0, (4 / (𝑁 · π)))) |
| 319 | 186, 296,
318 | 3eqtrd 2781 |
1
⊢ (𝜑 →
(∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 / π) = if(2 ∥ 𝑁, 0, (4 / (𝑁 · π)))) |