Proof of Theorem fourierdlem66
| Step | Hyp | Ref
| Expression |
| 1 | | fourierdlem66.a |
. . . . . . . 8
⊢ 𝐴 = ((-π[,]π) ∖
{0}) |
| 2 | 1 | eqimssi 4044 |
. . . . . . 7
⊢ 𝐴 ⊆ ((-π[,]π) ∖
{0}) |
| 3 | | difss 4136 |
. . . . . . 7
⊢
((-π[,]π) ∖ {0}) ⊆ (-π[,]π) |
| 4 | 2, 3 | sstri 3993 |
. . . . . 6
⊢ 𝐴 ⊆
(-π[,]π) |
| 5 | 4 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝐴 ⊆ (-π[,]π)) |
| 6 | 5 | sselda 3983 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (-π[,]π)) |
| 7 | 6 | adantlr 715 |
. . 3
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (-π[,]π)) |
| 8 | | fourierdlem66.f |
. . . . . . . 8
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
| 9 | 8 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℝ⟶ℝ) |
| 10 | | fourierdlem66.x |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ ℝ) |
| 11 | 10 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ ℝ) |
| 12 | | fourierdlem66.y |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ∈ ℝ) |
| 13 | 12 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑌 ∈ ℝ) |
| 14 | | fourierdlem66.w |
. . . . . . . 8
⊢ (𝜑 → 𝑊 ∈ ℝ) |
| 15 | 14 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ) |
| 16 | | fourierdlem66.h |
. . . . . . 7
⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) |
| 17 | | fourierdlem66.k |
. . . . . . 7
⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) |
| 18 | | fourierdlem66.u |
. . . . . . 7
⊢ 𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) |
| 19 | 9, 11, 13, 15, 16, 17, 18 | fourierdlem55 46176 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑈:(-π[,]π)⟶ℝ) |
| 20 | 19 | adantr 480 |
. . . . 5
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑈:(-π[,]π)⟶ℝ) |
| 21 | 20, 7 | ffvelcdmd 7105 |
. . . 4
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) ∈ ℝ) |
| 22 | | nnre 12273 |
. . . . . . 7
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ) |
| 23 | | fourierdlem66.s |
. . . . . . . 8
⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦
(sin‘((𝑛 + (1 / 2))
· 𝑠))) |
| 24 | 23 | fourierdlem5 46127 |
. . . . . . 7
⊢ (𝑛 ∈ ℝ → 𝑆:(-π[,]π)⟶ℝ) |
| 25 | 22, 24 | syl 17 |
. . . . . 6
⊢ (𝑛 ∈ ℕ → 𝑆:(-π[,]π)⟶ℝ) |
| 26 | 25 | ad2antlr 727 |
. . . . 5
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑆:(-π[,]π)⟶ℝ) |
| 27 | 26, 7 | ffvelcdmd 7105 |
. . . 4
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑆‘𝑠) ∈ ℝ) |
| 28 | 21, 27 | remulcld 11291 |
. . 3
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ) |
| 29 | | fourierdlem66.g |
. . . 4
⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) |
| 30 | 29 | fvmpt2 7027 |
. . 3
⊢ ((𝑠 ∈ (-π[,]π) ∧
((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠))) |
| 31 | 7, 28, 30 | syl2anc 584 |
. 2
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠))) |
| 32 | 8, 10, 12, 14, 16 | fourierdlem9 46131 |
. . . . . . . . 9
⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ) |
| 33 | 32 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐻:(-π[,]π)⟶ℝ) |
| 34 | 33, 6 | ffvelcdmd 7105 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) ∈ ℝ) |
| 35 | 17 | fourierdlem43 46165 |
. . . . . . . . 9
⊢ 𝐾:(-π[,]π)⟶ℝ |
| 36 | 35 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐾:(-π[,]π)⟶ℝ) |
| 37 | 36, 6 | ffvelcdmd 7105 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) ∈ ℝ) |
| 38 | 34, 37 | remulcld 11291 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ) |
| 39 | 18 | fvmpt2 7027 |
. . . . . 6
⊢ ((𝑠 ∈ (-π[,]π) ∧
((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠))) |
| 40 | 6, 38, 39 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠))) |
| 41 | | 0red 11264 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 0 ∈ ℝ) |
| 42 | 8 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐹:ℝ⟶ℝ) |
| 43 | 10 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑋 ∈ ℝ) |
| 44 | | pire 26500 |
. . . . . . . . . . . . . . . . 17
⊢ π
∈ ℝ |
| 45 | 44 | renegcli 11570 |
. . . . . . . . . . . . . . . 16
⊢ -π
∈ ℝ |
| 46 | | iccssre 13469 |
. . . . . . . . . . . . . . . 16
⊢ ((-π
∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆
ℝ) |
| 47 | 45, 44, 46 | mp2an 692 |
. . . . . . . . . . . . . . 15
⊢
(-π[,]π) ⊆ ℝ |
| 48 | 4 | sseli 3979 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ (-π[,]π)) |
| 49 | 47, 48 | sselid 3981 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ ℝ) |
| 50 | 49 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ) |
| 51 | 43, 50 | readdcld 11290 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑋 + 𝑠) ∈ ℝ) |
| 52 | 42, 51 | ffvelcdmd 7105 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ) |
| 53 | 12, 14 | ifcld 4572 |
. . . . . . . . . . . 12
⊢ (𝜑 → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ) |
| 54 | 53 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ) |
| 55 | 52, 54 | resubcld 11691 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℝ) |
| 56 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴) |
| 57 | 2, 56 | sselid 3981 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ((-π[,]π) ∖
{0})) |
| 58 | 57 | eldifbd 3964 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ¬ 𝑠 ∈ {0}) |
| 59 | | velsn 4642 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ {0} ↔ 𝑠 = 0) |
| 60 | 58, 59 | sylnib 328 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ¬ 𝑠 = 0) |
| 61 | 60 | neqned 2947 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ≠ 0) |
| 62 | 55, 50, 61 | redivcld 12095 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) ∈ ℝ) |
| 63 | 41, 62 | ifcld 4572 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) ∈ ℝ) |
| 64 | 16 | fvmpt2 7027 |
. . . . . . . 8
⊢ ((𝑠 ∈ (-π[,]π) ∧
if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) ∈ ℝ) → (𝐻‘𝑠) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) |
| 65 | 6, 63, 64 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) |
| 66 | 60 | iffalsed 4536 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) |
| 67 | 65, 66 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) |
| 68 | | 1red 11262 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 1 ∈ ℝ) |
| 69 | | 2re 12340 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℝ |
| 70 | 69 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℝ) |
| 71 | 50 | rehalfcld 12513 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑠 / 2) ∈ ℝ) |
| 72 | 71 | resincld 16179 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℝ) |
| 73 | 70, 72 | remulcld 11291 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈
ℝ) |
| 74 | | 2cnd 12344 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℂ) |
| 75 | 72 | recnd 11289 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℂ) |
| 76 | | 2ne0 12370 |
. . . . . . . . . . . 12
⊢ 2 ≠
0 |
| 77 | 76 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ≠ 0) |
| 78 | | fourierdlem44 46166 |
. . . . . . . . . . . 12
⊢ ((𝑠 ∈ (-π[,]π) ∧
𝑠 ≠ 0) →
(sin‘(𝑠 / 2)) ≠
0) |
| 79 | 6, 61, 78 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ≠ 0) |
| 80 | 74, 75, 77, 79 | mulne0d 11915 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0) |
| 81 | 50, 73, 80 | redivcld 12095 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ ℝ) |
| 82 | 68, 81 | ifcld 4572 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ ℝ) |
| 83 | 17 | fvmpt2 7027 |
. . . . . . . 8
⊢ ((𝑠 ∈ (-π[,]π) ∧
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ ℝ) →
(𝐾‘𝑠) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) |
| 84 | 6, 82, 83 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) |
| 85 | 60 | iffalsed 4536 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = (𝑠 / (2 · (sin‘(𝑠 / 2))))) |
| 86 | 84, 85 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) = (𝑠 / (2 · (sin‘(𝑠 / 2))))) |
| 87 | 67, 86 | oveq12d 7449 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐻‘𝑠) · (𝐾‘𝑠)) = ((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) · (𝑠 / (2 · (sin‘(𝑠 / 2)))))) |
| 88 | 55 | recnd 11289 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ) |
| 89 | 50 | recnd 11289 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℂ) |
| 90 | 74, 75 | mulcld 11281 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈
ℂ) |
| 91 | 88, 89, 90, 61, 80 | dmdcan2d 12073 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) · (𝑠 / (2 · (sin‘(𝑠 / 2))))) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2))))) |
| 92 | 40, 87, 91 | 3eqtrd 2781 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2))))) |
| 93 | 92 | adantlr 715 |
. . 3
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2))))) |
| 94 | 22 | ad2antlr 727 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑛 ∈ ℝ) |
| 95 | | 1red 11262 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 1 ∈ ℝ) |
| 96 | 95 | rehalfcld 12513 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (1 / 2) ∈
ℝ) |
| 97 | 94, 96 | readdcld 11290 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑛 + (1 / 2)) ∈ ℝ) |
| 98 | 49 | adantl 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ) |
| 99 | 97, 98 | remulcld 11291 |
. . . . 5
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝑛 + (1 / 2)) · 𝑠) ∈ ℝ) |
| 100 | 99 | resincld 16179 |
. . . 4
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℝ) |
| 101 | 23 | fvmpt2 7027 |
. . . 4
⊢ ((𝑠 ∈ (-π[,]π) ∧
(sin‘((𝑛 + (1 / 2))
· 𝑠)) ∈
ℝ) → (𝑆‘𝑠) = (sin‘((𝑛 + (1 / 2)) · 𝑠))) |
| 102 | 7, 100, 101 | syl2anc 584 |
. . 3
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑆‘𝑠) = (sin‘((𝑛 + (1 / 2)) · 𝑠))) |
| 103 | 93, 102 | oveq12d 7449 |
. 2
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝑈‘𝑠) · (𝑆‘𝑠)) = ((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))) ·
(sin‘((𝑛 + (1 / 2))
· 𝑠)))) |
| 104 | 88 | adantlr 715 |
. . . 4
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ) |
| 105 | 90 | adantlr 715 |
. . . 4
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈
ℂ) |
| 106 | 100 | recnd 11289 |
. . . 4
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℂ) |
| 107 | 80 | adantlr 715 |
. . . 4
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0) |
| 108 | 104, 105,
106, 107 | div32d 12066 |
. . 3
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))) ·
(sin‘((𝑛 + (1 / 2))
· 𝑠))) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))))) |
| 109 | 22 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 𝑛 ∈ ℝ) |
| 110 | | halfre 12480 |
. . . . . . . . . . . . . 14
⊢ (1 / 2)
∈ ℝ |
| 111 | 110 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (1 / 2) ∈
ℝ) |
| 112 | 109, 111 | readdcld 11290 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (𝑛 + (1 / 2)) ∈ ℝ) |
| 113 | 49 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ) |
| 114 | 112, 113 | remulcld 11291 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝑛 + (1 / 2)) · 𝑠) ∈ ℝ) |
| 115 | 114 | resincld 16179 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℝ) |
| 116 | 115 | recnd 11289 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℂ) |
| 117 | 69 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℝ) |
| 118 | 113 | rehalfcld 12513 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (𝑠 / 2) ∈ ℝ) |
| 119 | 118 | resincld 16179 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℝ) |
| 120 | 117, 119 | remulcld 11291 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈
ℝ) |
| 121 | 120 | recnd 11289 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈
ℂ) |
| 122 | | picn 26501 |
. . . . . . . . . 10
⊢ π
∈ ℂ |
| 123 | 122 | a1i 11 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → π ∈
ℂ) |
| 124 | | 2cnd 12344 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ 𝐴 → 2 ∈ ℂ) |
| 125 | | rehalfcl 12492 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ ℝ → (𝑠 / 2) ∈
ℝ) |
| 126 | | resincl 16176 |
. . . . . . . . . . . . 13
⊢ ((𝑠 / 2) ∈ ℝ →
(sin‘(𝑠 / 2)) ∈
ℝ) |
| 127 | 49, 125, 126 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ∈ ℝ) |
| 128 | 127 | recnd 11289 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ∈ ℂ) |
| 129 | 76 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ 𝐴 → 2 ≠ 0) |
| 130 | | eldifsni 4790 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → 𝑠 ≠
0) |
| 131 | 130, 1 | eleq2s 2859 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ 𝐴 → 𝑠 ≠ 0) |
| 132 | 48, 131, 78 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ≠ 0) |
| 133 | 124, 128,
129, 132 | mulne0d 11915 |
. . . . . . . . . 10
⊢ (𝑠 ∈ 𝐴 → (2 · (sin‘(𝑠 / 2))) ≠ 0) |
| 134 | 133 | adantl 481 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0) |
| 135 | | 0re 11263 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
| 136 | | pipos 26502 |
. . . . . . . . . . 11
⊢ 0 <
π |
| 137 | 135, 136 | gtneii 11373 |
. . . . . . . . . 10
⊢ π ≠
0 |
| 138 | 137 | a1i 11 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → π ≠ 0) |
| 139 | 116, 121,
123, 134, 138 | divdiv1d 12074 |
. . . . . . . 8
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) / π) =
((sin‘((𝑛 + (1 / 2))
· 𝑠)) / ((2 ·
(sin‘(𝑠 / 2)))
· π))) |
| 140 | | 2cnd 12344 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℂ) |
| 141 | 128 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℂ) |
| 142 | 140, 141,
123 | mulassd 11284 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((2 · (sin‘(𝑠 / 2))) · π) = (2
· ((sin‘(𝑠 /
2)) · π))) |
| 143 | 142 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · (sin‘(𝑠 / 2))) · π)) =
((sin‘((𝑛 + (1 / 2))
· 𝑠)) / (2 ·
((sin‘(𝑠 / 2))
· π)))) |
| 144 | 141, 123 | mulcomd 11282 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘(𝑠 / 2)) · π) = (π ·
(sin‘(𝑠 /
2)))) |
| 145 | 144 | oveq2d 7447 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · ((sin‘(𝑠 / 2)) · π)) = (2
· (π · (sin‘(𝑠 / 2))))) |
| 146 | 140, 123,
141 | mulassd 11284 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((2 · π) ·
(sin‘(𝑠 / 2))) = (2
· (π · (sin‘(𝑠 / 2))))) |
| 147 | 145, 146 | eqtr4d 2780 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · ((sin‘(𝑠 / 2)) · π)) = ((2
· π) · (sin‘(𝑠 / 2)))) |
| 148 | 147 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · ((sin‘(𝑠 / 2)) · π))) =
((sin‘((𝑛 + (1 / 2))
· 𝑠)) / ((2 ·
π) · (sin‘(𝑠 / 2))))) |
| 149 | 139, 143,
148 | 3eqtrd 2781 |
. . . . . . 7
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) / π) =
((sin‘((𝑛 + (1 / 2))
· 𝑠)) / ((2 ·
π) · (sin‘(𝑠 / 2))))) |
| 150 | 149 | oveq2d 7447 |
. . . . . 6
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (π · (((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 ·
(sin‘(𝑠 / 2)))) /
π)) = (π · ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) ·
(sin‘(𝑠 /
2)))))) |
| 151 | 115, 120,
134 | redivcld 12095 |
. . . . . . . 8
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) ∈
ℝ) |
| 152 | 151 | recnd 11289 |
. . . . . . 7
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) ∈
ℂ) |
| 153 | 152, 123,
138 | divcan2d 12045 |
. . . . . 6
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (π · (((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 ·
(sin‘(𝑠 / 2)))) /
π)) = ((sin‘((𝑛 +
(1 / 2)) · 𝑠)) / (2
· (sin‘(𝑠 /
2))))) |
| 154 | | fourierdlem66.d |
. . . . . . . . . 10
⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0,
(((2 · 𝑛) + 1) / (2
· π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) ·
(sin‘(𝑠 /
2))))))) |
| 155 | 154 | dirkerval2 46109 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘𝑠) = if((𝑠 mod (2 · π)) = 0, (((2 ·
𝑛) + 1) / (2 ·
π)), ((sin‘((𝑛 +
(1 / 2)) · 𝑠)) / ((2
· π) · (sin‘(𝑠 / 2)))))) |
| 156 | 49, 155 | sylan2 593 |
. . . . . . . 8
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) = if((𝑠 mod (2 · π)) = 0, (((2 ·
𝑛) + 1) / (2 ·
π)), ((sin‘((𝑛 +
(1 / 2)) · 𝑠)) / ((2
· π) · (sin‘(𝑠 / 2)))))) |
| 157 | | fourierdlem24 46146 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → (𝑠 mod (2
· π)) ≠ 0) |
| 158 | 157, 1 | eleq2s 2859 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ 𝐴 → (𝑠 mod (2 · π)) ≠
0) |
| 159 | 158 | neneqd 2945 |
. . . . . . . . . 10
⊢ (𝑠 ∈ 𝐴 → ¬ (𝑠 mod (2 · π)) = 0) |
| 160 | 159 | adantl 481 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ¬ (𝑠 mod (2 · π)) = 0) |
| 161 | 160 | iffalsed 4536 |
. . . . . . . 8
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → if((𝑠 mod (2 · π)) = 0, (((2 ·
𝑛) + 1) / (2 ·
π)), ((sin‘((𝑛 +
(1 / 2)) · 𝑠)) / ((2
· π) · (sin‘(𝑠 / 2))))) = ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) ·
(sin‘(𝑠 /
2))))) |
| 162 | 156, 161 | eqtr2d 2778 |
. . . . . . 7
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) ·
(sin‘(𝑠 / 2)))) =
((𝐷‘𝑛)‘𝑠)) |
| 163 | 162 | oveq2d 7447 |
. . . . . 6
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (π · ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) ·
(sin‘(𝑠 / 2))))) =
(π · ((𝐷‘𝑛)‘𝑠))) |
| 164 | 150, 153,
163 | 3eqtr3d 2785 |
. . . . 5
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) = (π ·
((𝐷‘𝑛)‘𝑠))) |
| 165 | 164 | oveq2d 7447 |
. . . 4
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2))))) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · (π · ((𝐷‘𝑛)‘𝑠)))) |
| 166 | 165 | adantll 714 |
. . 3
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2))))) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · (π · ((𝐷‘𝑛)‘𝑠)))) |
| 167 | 122 | a1i 11 |
. . . 4
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → π ∈
ℂ) |
| 168 | 154 | dirkerre 46110 |
. . . . . . 7
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ) |
| 169 | 49, 168 | sylan2 593 |
. . . . . 6
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ) |
| 170 | 169 | recnd 11289 |
. . . . 5
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ) |
| 171 | 170 | adantll 714 |
. . . 4
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ) |
| 172 | 104, 167,
171 | mul12d 11470 |
. . 3
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · (π · ((𝐷‘𝑛)‘𝑠))) = (π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)))) |
| 173 | 108, 166,
172 | 3eqtrd 2781 |
. 2
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))) ·
(sin‘((𝑛 + (1 / 2))
· 𝑠))) = (π
· (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)))) |
| 174 | 31, 103, 173 | 3eqtrd 2781 |
1
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) = (π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)))) |