| Step | Hyp | Ref
| Expression |
| 1 | | fourierdlem62.k |
. . . 4
⊢ 𝐾 = (𝑦 ∈ (-π[,]π) ↦ if(𝑦 = 0, 1, (𝑦 / (2 · (sin‘(𝑦 / 2)))))) |
| 2 | | eqeq1 2741 |
. . . . . 6
⊢ (𝑦 = 𝑠 → (𝑦 = 0 ↔ 𝑠 = 0)) |
| 3 | | id 22 |
. . . . . . 7
⊢ (𝑦 = 𝑠 → 𝑦 = 𝑠) |
| 4 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑦 = 𝑠 → (𝑦 / 2) = (𝑠 / 2)) |
| 5 | 4 | fveq2d 6910 |
. . . . . . . 8
⊢ (𝑦 = 𝑠 → (sin‘(𝑦 / 2)) = (sin‘(𝑠 / 2))) |
| 6 | 5 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑦 = 𝑠 → (2 · (sin‘(𝑦 / 2))) = (2 ·
(sin‘(𝑠 /
2)))) |
| 7 | 3, 6 | oveq12d 7449 |
. . . . . 6
⊢ (𝑦 = 𝑠 → (𝑦 / (2 · (sin‘(𝑦 / 2)))) = (𝑠 / (2 · (sin‘(𝑠 / 2))))) |
| 8 | 2, 7 | ifbieq2d 4552 |
. . . . 5
⊢ (𝑦 = 𝑠 → if(𝑦 = 0, 1, (𝑦 / (2 · (sin‘(𝑦 / 2))))) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) |
| 9 | 8 | cbvmptv 5255 |
. . . 4
⊢ (𝑦 ∈ (-π[,]π) ↦
if(𝑦 = 0, 1, (𝑦 / (2 · (sin‘(𝑦 / 2)))))) = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) |
| 10 | 1, 9 | eqtri 2765 |
. . 3
⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) |
| 11 | 10 | fourierdlem43 46165 |
. 2
⊢ 𝐾:(-π[,]π)⟶ℝ |
| 12 | | ax-resscn 11212 |
. . 3
⊢ ℝ
⊆ ℂ |
| 13 | | fss 6752 |
. . . . . 6
⊢ ((𝐾:(-π[,]π)⟶ℝ
∧ ℝ ⊆ ℂ) → 𝐾:(-π[,]π)⟶ℂ) |
| 14 | 11, 12, 13 | mp2an 692 |
. . . . 5
⊢ 𝐾:(-π[,]π)⟶ℂ |
| 15 | 14 | a1i 11 |
. . . . . . . . 9
⊢ (𝑠 = 0 → 𝐾:(-π[,]π)⟶ℂ) |
| 16 | | difss 4136 |
. . . . . . . . . . . . . . . . 17
⊢
((-π(,)π) ∖ {0}) ⊆ (-π(,)π) |
| 17 | | elioore 13417 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ (-π(,)π) →
𝑠 ∈
ℝ) |
| 18 | 17 | ssriv 3987 |
. . . . . . . . . . . . . . . . 17
⊢
(-π(,)π) ⊆ ℝ |
| 19 | 16, 18 | sstri 3993 |
. . . . . . . . . . . . . . . 16
⊢
((-π(,)π) ∖ {0}) ⊆ ℝ |
| 20 | 19 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (⊤
→ ((-π(,)π) ∖ {0}) ⊆ ℝ) |
| 21 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ 𝑥) = (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ 𝑥) |
| 22 | 19 | sseli 3979 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) → 𝑥 ∈
ℝ) |
| 23 | 21, 22 | fmpti 7132 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ 𝑥):((-π(,)π) ∖
{0})⟶ℝ |
| 24 | 23 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (⊤
→ (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 𝑥):((-π(,)π) ∖
{0})⟶ℝ) |
| 25 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ (2 · (sin‘(𝑥 / 2)))) = (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2)))) |
| 26 | | 2re 12340 |
. . . . . . . . . . . . . . . . . . 19
⊢ 2 ∈
ℝ |
| 27 | 26 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) → 2 ∈ ℝ) |
| 28 | 22 | rehalfcld 12513 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) → (𝑥 / 2) ∈
ℝ) |
| 29 | 28 | resincld 16179 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) → (sin‘(𝑥 /
2)) ∈ ℝ) |
| 30 | 27, 29 | remulcld 11291 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) → (2 · (sin‘(𝑥 / 2))) ∈ ℝ) |
| 31 | 25, 30 | fmpti 7132 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ (2 · (sin‘(𝑥 / 2)))):((-π(,)π) ∖
{0})⟶ℝ |
| 32 | 31 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (⊤
→ (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2)))):((-π(,)π) ∖
{0})⟶ℝ) |
| 33 | | iooretop 24786 |
. . . . . . . . . . . . . . . 16
⊢
(-π(,)π) ∈ (topGen‘ran (,)) |
| 34 | 33 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (⊤
→ (-π(,)π) ∈ (topGen‘ran (,))) |
| 35 | | 0re 11263 |
. . . . . . . . . . . . . . . . 17
⊢ 0 ∈
ℝ |
| 36 | | negpilt0 45292 |
. . . . . . . . . . . . . . . . 17
⊢ -π
< 0 |
| 37 | | pipos 26502 |
. . . . . . . . . . . . . . . . 17
⊢ 0 <
π |
| 38 | | pire 26500 |
. . . . . . . . . . . . . . . . . . . 20
⊢ π
∈ ℝ |
| 39 | 38 | renegcli 11570 |
. . . . . . . . . . . . . . . . . . 19
⊢ -π
∈ ℝ |
| 40 | 39 | rexri 11319 |
. . . . . . . . . . . . . . . . . 18
⊢ -π
∈ ℝ* |
| 41 | 38 | rexri 11319 |
. . . . . . . . . . . . . . . . . 18
⊢ π
∈ ℝ* |
| 42 | | elioo2 13428 |
. . . . . . . . . . . . . . . . . 18
⊢ ((-π
∈ ℝ* ∧ π ∈ ℝ*) → (0
∈ (-π(,)π) ↔ (0 ∈ ℝ ∧ -π < 0 ∧ 0 <
π))) |
| 43 | 40, 41, 42 | mp2an 692 |
. . . . . . . . . . . . . . . . 17
⊢ (0 ∈
(-π(,)π) ↔ (0 ∈ ℝ ∧ -π < 0 ∧ 0 <
π)) |
| 44 | 35, 36, 37, 43 | mpbir3an 1342 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
(-π(,)π) |
| 45 | 44 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (⊤
→ 0 ∈ (-π(,)π)) |
| 46 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
((-π(,)π) ∖ {0}) = ((-π(,)π) ∖
{0}) |
| 47 | | 1ex 11257 |
. . . . . . . . . . . . . . . . . . 19
⊢ 1 ∈
V |
| 48 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ 1) = (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 1) |
| 49 | 47, 48 | dmmpti 6712 |
. . . . . . . . . . . . . . . . . 18
⊢ dom
(𝑥 ∈ ((-π(,)π)
∖ {0}) ↦ 1) = ((-π(,)π) ∖ {0}) |
| 50 | | reelprrecn 11247 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ℝ
∈ {ℝ, ℂ} |
| 51 | 50 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (⊤
→ ℝ ∈ {ℝ, ℂ}) |
| 52 | 12 | sseli 3979 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℂ) |
| 53 | 52 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((⊤ ∧ 𝑥
∈ ℝ) → 𝑥
∈ ℂ) |
| 54 | | 1red 11262 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((⊤ ∧ 𝑥
∈ ℝ) → 1 ∈ ℝ) |
| 55 | 51 | dvmptid 25995 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (⊤
→ (ℝ D (𝑥 ∈
ℝ ↦ 𝑥)) =
(𝑥 ∈ ℝ ↦
1)) |
| 56 | | tgioo4 24826 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(topGen‘ran (,)) = ((TopOpen‘ℂfld)
↾t ℝ) |
| 57 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
| 58 | | sncldre 45049 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (0 ∈
ℝ → {0} ∈ (Clsd‘(topGen‘ran (,)))) |
| 59 | 35, 58 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ {0}
∈ (Clsd‘(topGen‘ran (,))) |
| 60 | | retopon 24784 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(topGen‘ran (,)) ∈ (TopOn‘ℝ) |
| 61 | 60 | toponunii 22922 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ℝ =
∪ (topGen‘ran (,)) |
| 62 | 61 | difopn 23042 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((-π(,)π) ∈ (topGen‘ran (,)) ∧ {0} ∈
(Clsd‘(topGen‘ran (,)))) → ((-π(,)π) ∖ {0}) ∈
(topGen‘ran (,))) |
| 63 | 33, 59, 62 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((-π(,)π) ∖ {0}) ∈ (topGen‘ran
(,)) |
| 64 | 63 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (⊤
→ ((-π(,)π) ∖ {0}) ∈ (topGen‘ran
(,))) |
| 65 | 51, 53, 54, 55, 20, 56, 57, 64 | dvmptres 26001 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (⊤
→ (ℝ D (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 𝑥)) = (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
1)) |
| 66 | 65 | mptru 1547 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (ℝ
D (𝑥 ∈ ((-π(,)π)
∖ {0}) ↦ 𝑥)) =
(𝑥 ∈ ((-π(,)π)
∖ {0}) ↦ 1) |
| 67 | 66 | eqcomi 2746 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ 1) = (ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
𝑥)) |
| 68 | 67 | dmeqi 5915 |
. . . . . . . . . . . . . . . . . 18
⊢ dom
(𝑥 ∈ ((-π(,)π)
∖ {0}) ↦ 1) = dom (ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
𝑥)) |
| 69 | 49, 68 | eqtr3i 2767 |
. . . . . . . . . . . . . . . . 17
⊢
((-π(,)π) ∖ {0}) = dom (ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
𝑥)) |
| 70 | 69 | eqimssi 4044 |
. . . . . . . . . . . . . . . 16
⊢
((-π(,)π) ∖ {0}) ⊆ dom (ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
𝑥)) |
| 71 | 70 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (⊤
→ ((-π(,)π) ∖ {0}) ⊆ dom (ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
𝑥))) |
| 72 | | fvex 6919 |
. . . . . . . . . . . . . . . . . . 19
⊢
(cos‘(𝑥 / 2))
∈ V |
| 73 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ (cos‘(𝑥
/ 2))) = (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (cos‘(𝑥 / 2))) |
| 74 | 72, 73 | dmmpti 6712 |
. . . . . . . . . . . . . . . . . 18
⊢ dom
(𝑥 ∈ ((-π(,)π)
∖ {0}) ↦ (cos‘(𝑥 / 2))) = ((-π(,)π) ∖
{0}) |
| 75 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((⊤ ∧ 𝑥
∈ ℝ) → 2 ∈ ℂ) |
| 76 | 53 | halfcld 12511 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((⊤ ∧ 𝑥
∈ ℝ) → (𝑥 /
2) ∈ ℂ) |
| 77 | 76 | sincld 16166 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((⊤ ∧ 𝑥
∈ ℝ) → (sin‘(𝑥 / 2)) ∈ ℂ) |
| 78 | 75, 77 | mulcld 11281 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((⊤ ∧ 𝑥
∈ ℝ) → (2 · (sin‘(𝑥 / 2))) ∈ ℂ) |
| 79 | 76 | coscld 16167 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((⊤ ∧ 𝑥
∈ ℝ) → (cos‘(𝑥 / 2)) ∈ ℂ) |
| 80 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑥 ∈ ℝ → 2 ∈
ℂ) |
| 81 | | 2ne0 12370 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ 2 ≠
0 |
| 82 | 81 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑥 ∈ ℝ → 2 ≠
0) |
| 83 | 52, 80, 82 | divrec2d 12047 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑥 ∈ ℝ → (𝑥 / 2) = ((1 / 2) · 𝑥)) |
| 84 | 83 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 ∈ ℝ →
(sin‘(𝑥 / 2)) =
(sin‘((1 / 2) · 𝑥))) |
| 85 | 84 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 ∈ ℝ → (2
· (sin‘(𝑥 /
2))) = (2 · (sin‘((1 / 2) · 𝑥)))) |
| 86 | 85 | mpteq2ia 5245 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 ∈ ℝ ↦ (2
· (sin‘(𝑥 /
2)))) = (𝑥 ∈ ℝ
↦ (2 · (sin‘((1 / 2) · 𝑥)))) |
| 87 | 86 | oveq2i 7442 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (ℝ
D (𝑥 ∈ ℝ ↦
(2 · (sin‘(𝑥 /
2))))) = (ℝ D (𝑥
∈ ℝ ↦ (2 · (sin‘((1 / 2) · 𝑥))))) |
| 88 | | resmpt 6055 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (ℝ
⊆ ℂ → ((𝑥
∈ ℂ ↦ (2 · (sin‘((1 / 2) · 𝑥)))) ↾ ℝ) = (𝑥 ∈ ℝ ↦ (2
· (sin‘((1 / 2) · 𝑥))))) |
| 89 | 12, 88 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑥 ∈ ℂ ↦ (2
· (sin‘((1 / 2) · 𝑥)))) ↾ ℝ) = (𝑥 ∈ ℝ ↦ (2 ·
(sin‘((1 / 2) · 𝑥)))) |
| 90 | 89 | eqcomi 2746 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 ∈ ℝ ↦ (2
· (sin‘((1 / 2) · 𝑥)))) = ((𝑥 ∈ ℂ ↦ (2 ·
(sin‘((1 / 2) · 𝑥)))) ↾ ℝ) |
| 91 | 90 | oveq2i 7442 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (ℝ
D (𝑥 ∈ ℝ ↦
(2 · (sin‘((1 / 2) · 𝑥))))) = (ℝ D ((𝑥 ∈ ℂ ↦ (2 ·
(sin‘((1 / 2) · 𝑥)))) ↾ ℝ)) |
| 92 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 ∈ ℂ ↦ (2
· (sin‘((1 / 2) · 𝑥)))) = (𝑥 ∈ ℂ ↦ (2 ·
(sin‘((1 / 2) · 𝑥)))) |
| 93 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑥 ∈ ℂ → 2 ∈
ℂ) |
| 94 | | halfcn 12481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (1 / 2)
∈ ℂ |
| 95 | 94 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑥 ∈ ℂ → (1 / 2)
∈ ℂ) |
| 96 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑥 ∈ ℂ → 𝑥 ∈
ℂ) |
| 97 | 95, 96 | mulcld 11281 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑥 ∈ ℂ → ((1 / 2)
· 𝑥) ∈
ℂ) |
| 98 | 97 | sincld 16166 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑥 ∈ ℂ →
(sin‘((1 / 2) · 𝑥)) ∈ ℂ) |
| 99 | 93, 98 | mulcld 11281 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 ∈ ℂ → (2
· (sin‘((1 / 2) · 𝑥))) ∈ ℂ) |
| 100 | 92, 99 | fmpti 7132 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 ∈ ℂ ↦ (2
· (sin‘((1 / 2) · 𝑥)))):ℂ⟶ℂ |
| 101 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑥 ∈ ℂ ↦ ((2
· (1 / 2)) · (cos‘((1 / 2) · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((2 · (1 / 2))
· (cos‘((1 / 2) · 𝑥)))) |
| 102 | | 2cn 12341 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ 2 ∈
ℂ |
| 103 | 102, 94 | mulcli 11268 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (2
· (1 / 2)) ∈ ℂ |
| 104 | 103 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑥 ∈ ℂ → (2
· (1 / 2)) ∈ ℂ) |
| 105 | 97 | coscld 16167 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑥 ∈ ℂ →
(cos‘((1 / 2) · 𝑥)) ∈ ℂ) |
| 106 | 104, 105 | mulcld 11281 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑥 ∈ ℂ → ((2
· (1 / 2)) · (cos‘((1 / 2) · 𝑥))) ∈ ℂ) |
| 107 | 106 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
((⊤ ∧ 𝑥
∈ ℂ) → ((2 · (1 / 2)) · (cos‘((1 / 2)
· 𝑥))) ∈
ℂ) |
| 108 | 101, 107 | dmmptd 6713 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (⊤
→ dom (𝑥 ∈
ℂ ↦ ((2 · (1 / 2)) · (cos‘((1 / 2) ·
𝑥)))) =
ℂ) |
| 109 | 108 | mptru 1547 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ dom
(𝑥 ∈ ℂ ↦
((2 · (1 / 2)) · (cos‘((1 / 2) · 𝑥)))) = ℂ |
| 110 | 12, 109 | sseqtrri 4033 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ℝ
⊆ dom (𝑥 ∈
ℂ ↦ ((2 · (1 / 2)) · (cos‘((1 / 2) ·
𝑥)))) |
| 111 | | dvasinbx 45935 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((2
∈ ℂ ∧ (1 / 2) ∈ ℂ) → (ℂ D (𝑥 ∈ ℂ ↦ (2
· (sin‘((1 / 2) · 𝑥))))) = (𝑥 ∈ ℂ ↦ ((2 · (1 / 2))
· (cos‘((1 / 2) · 𝑥))))) |
| 112 | 102, 94, 111 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (ℂ
D (𝑥 ∈ ℂ ↦
(2 · (sin‘((1 / 2) · 𝑥))))) = (𝑥 ∈ ℂ ↦ ((2 · (1 / 2))
· (cos‘((1 / 2) · 𝑥)))) |
| 113 | 112 | dmeqi 5915 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ dom
(ℂ D (𝑥 ∈
ℂ ↦ (2 · (sin‘((1 / 2) · 𝑥))))) = dom (𝑥 ∈ ℂ ↦ ((2 · (1 / 2))
· (cos‘((1 / 2) · 𝑥)))) |
| 114 | 110, 113 | sseqtrri 4033 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ℝ
⊆ dom (ℂ D (𝑥
∈ ℂ ↦ (2 · (sin‘((1 / 2) · 𝑥))))) |
| 115 | | dvcnre 45931 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑥 ∈ ℂ ↦ (2
· (sin‘((1 / 2) · 𝑥)))):ℂ⟶ℂ ∧ ℝ
⊆ dom (ℂ D (𝑥
∈ ℂ ↦ (2 · (sin‘((1 / 2) · 𝑥)))))) → (ℝ D ((𝑥 ∈ ℂ ↦ (2
· (sin‘((1 / 2) · 𝑥)))) ↾ ℝ)) = ((ℂ D (𝑥 ∈ ℂ ↦ (2
· (sin‘((1 / 2) · 𝑥))))) ↾ ℝ)) |
| 116 | 100, 114,
115 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (ℝ
D ((𝑥 ∈ ℂ
↦ (2 · (sin‘((1 / 2) · 𝑥)))) ↾ ℝ)) = ((ℂ D (𝑥 ∈ ℂ ↦ (2
· (sin‘((1 / 2) · 𝑥))))) ↾ ℝ) |
| 117 | 112 | reseq1i 5993 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((ℂ
D (𝑥 ∈ ℂ ↦
(2 · (sin‘((1 / 2) · 𝑥))))) ↾ ℝ) = ((𝑥 ∈ ℂ ↦ ((2 · (1 / 2))
· (cos‘((1 / 2) · 𝑥)))) ↾ ℝ) |
| 118 | | resmpt 6055 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (ℝ
⊆ ℂ → ((𝑥
∈ ℂ ↦ ((2 · (1 / 2)) · (cos‘((1 / 2)
· 𝑥)))) ↾
ℝ) = (𝑥 ∈
ℝ ↦ ((2 · (1 / 2)) · (cos‘((1 / 2) ·
𝑥))))) |
| 119 | 12, 118 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑥 ∈ ℂ ↦ ((2
· (1 / 2)) · (cos‘((1 / 2) · 𝑥)))) ↾ ℝ) = (𝑥 ∈ ℝ ↦ ((2 · (1 / 2))
· (cos‘((1 / 2) · 𝑥)))) |
| 120 | 102, 81 | recidi 11998 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (2
· (1 / 2)) = 1 |
| 121 | 120 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑥 ∈ ℝ → (2
· (1 / 2)) = 1) |
| 122 | 83 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑥 ∈ ℝ → ((1 / 2)
· 𝑥) = (𝑥 / 2)) |
| 123 | 122 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑥 ∈ ℝ →
(cos‘((1 / 2) · 𝑥)) = (cos‘(𝑥 / 2))) |
| 124 | 121, 123 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑥 ∈ ℝ → ((2
· (1 / 2)) · (cos‘((1 / 2) · 𝑥))) = (1 · (cos‘(𝑥 / 2)))) |
| 125 | 52 | halfcld 12511 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑥 ∈ ℝ → (𝑥 / 2) ∈
ℂ) |
| 126 | 125 | coscld 16167 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑥 ∈ ℝ →
(cos‘(𝑥 / 2)) ∈
ℂ) |
| 127 | 126 | mullidd 11279 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑥 ∈ ℝ → (1
· (cos‘(𝑥 /
2))) = (cos‘(𝑥 /
2))) |
| 128 | 124, 127 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 ∈ ℝ → ((2
· (1 / 2)) · (cos‘((1 / 2) · 𝑥))) = (cos‘(𝑥 / 2))) |
| 129 | 128 | mpteq2ia 5245 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 ∈ ℝ ↦ ((2
· (1 / 2)) · (cos‘((1 / 2) · 𝑥)))) = (𝑥 ∈ ℝ ↦ (cos‘(𝑥 / 2))) |
| 130 | 117, 119,
129 | 3eqtri 2769 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((ℂ
D (𝑥 ∈ ℂ ↦
(2 · (sin‘((1 / 2) · 𝑥))))) ↾ ℝ) = (𝑥 ∈ ℝ ↦ (cos‘(𝑥 / 2))) |
| 131 | 91, 116, 130 | 3eqtri 2769 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (ℝ
D (𝑥 ∈ ℝ ↦
(2 · (sin‘((1 / 2) · 𝑥))))) = (𝑥 ∈ ℝ ↦ (cos‘(𝑥 / 2))) |
| 132 | 87, 131 | eqtri 2765 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (ℝ
D (𝑥 ∈ ℝ ↦
(2 · (sin‘(𝑥 /
2))))) = (𝑥 ∈ ℝ
↦ (cos‘(𝑥 /
2))) |
| 133 | 132 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (⊤
→ (ℝ D (𝑥 ∈
ℝ ↦ (2 · (sin‘(𝑥 / 2))))) = (𝑥 ∈ ℝ ↦ (cos‘(𝑥 / 2)))) |
| 134 | 51, 78, 79, 133, 20, 56, 57, 64 | dvmptres 26001 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (⊤
→ (ℝ D (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2))))) = (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(cos‘(𝑥 /
2)))) |
| 135 | 134 | mptru 1547 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (ℝ
D (𝑥 ∈ ((-π(,)π)
∖ {0}) ↦ (2 · (sin‘(𝑥 / 2))))) = (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(cos‘(𝑥 /
2))) |
| 136 | 135 | eqcomi 2746 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ (cos‘(𝑥
/ 2))) = (ℝ D (𝑥
∈ ((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2))))) |
| 137 | 136 | dmeqi 5915 |
. . . . . . . . . . . . . . . . . 18
⊢ dom
(𝑥 ∈ ((-π(,)π)
∖ {0}) ↦ (cos‘(𝑥 / 2))) = dom (ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))) |
| 138 | 74, 137 | eqtr3i 2767 |
. . . . . . . . . . . . . . . . 17
⊢
((-π(,)π) ∖ {0}) = dom (ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))) |
| 139 | 138 | eqimssi 4044 |
. . . . . . . . . . . . . . . 16
⊢
((-π(,)π) ∖ {0}) ⊆ dom (ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))) |
| 140 | 139 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (⊤
→ ((-π(,)π) ∖ {0}) ⊆ dom (ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2)))))) |
| 141 | 17 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈ (-π(,)π) →
𝑠 ∈
ℂ) |
| 142 | 141 | ssriv 3987 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(-π(,)π) ⊆ ℂ |
| 143 | 142 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (⊤
→ (-π(,)π) ⊆ ℂ) |
| 144 | | ssid 4006 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ℂ
⊆ ℂ |
| 145 | 144 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (⊤
→ ℂ ⊆ ℂ) |
| 146 | 143, 145 | idcncfg 45888 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (⊤
→ (𝑥 ∈
(-π(,)π) ↦ 𝑥)
∈ ((-π(,)π)–cn→ℂ)) |
| 147 | 146 | mptru 1547 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (-π(,)π) ↦
𝑥) ∈
((-π(,)π)–cn→ℂ) |
| 148 | | cnlimc 25923 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((-π(,)π) ⊆ ℂ → ((𝑥 ∈ (-π(,)π) ↦ 𝑥) ∈
((-π(,)π)–cn→ℂ)
↔ ((𝑥 ∈
(-π(,)π) ↦ 𝑥):(-π(,)π)⟶ℂ ∧
∀𝑦 ∈
(-π(,)π)((𝑥 ∈
(-π(,)π) ↦ 𝑥)‘𝑦) ∈ ((𝑥 ∈ (-π(,)π) ↦ 𝑥) limℂ 𝑦)))) |
| 149 | 142, 148 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ (-π(,)π) ↦
𝑥) ∈
((-π(,)π)–cn→ℂ)
↔ ((𝑥 ∈
(-π(,)π) ↦ 𝑥):(-π(,)π)⟶ℂ ∧
∀𝑦 ∈
(-π(,)π)((𝑥 ∈
(-π(,)π) ↦ 𝑥)‘𝑦) ∈ ((𝑥 ∈ (-π(,)π) ↦ 𝑥) limℂ 𝑦))) |
| 150 | 147, 149 | mpbi 230 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ (-π(,)π) ↦
𝑥):(-π(,)π)⟶ℂ ∧
∀𝑦 ∈
(-π(,)π)((𝑥 ∈
(-π(,)π) ↦ 𝑥)‘𝑦) ∈ ((𝑥 ∈ (-π(,)π) ↦ 𝑥) limℂ 𝑦)) |
| 151 | 150 | simpri 485 |
. . . . . . . . . . . . . . . . . 18
⊢
∀𝑦 ∈
(-π(,)π)((𝑥 ∈
(-π(,)π) ↦ 𝑥)‘𝑦) ∈ ((𝑥 ∈ (-π(,)π) ↦ 𝑥) limℂ 𝑦) |
| 152 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 0 → ((𝑥 ∈ (-π(,)π) ↦ 𝑥)‘𝑦) = ((𝑥 ∈ (-π(,)π) ↦ 𝑥)‘0)) |
| 153 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 0 → ((𝑥 ∈ (-π(,)π) ↦ 𝑥) limℂ 𝑦) = ((𝑥 ∈ (-π(,)π) ↦ 𝑥) limℂ
0)) |
| 154 | 152, 153 | eleq12d 2835 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 0 → (((𝑥 ∈ (-π(,)π) ↦
𝑥)‘𝑦) ∈ ((𝑥 ∈ (-π(,)π) ↦ 𝑥) limℂ 𝑦) ↔ ((𝑥 ∈ (-π(,)π) ↦ 𝑥)‘0) ∈ ((𝑥 ∈ (-π(,)π) ↦
𝑥) limℂ
0))) |
| 155 | 154 | rspccva 3621 |
. . . . . . . . . . . . . . . . . 18
⊢
((∀𝑦 ∈
(-π(,)π)((𝑥 ∈
(-π(,)π) ↦ 𝑥)‘𝑦) ∈ ((𝑥 ∈ (-π(,)π) ↦ 𝑥) limℂ 𝑦) ∧ 0 ∈ (-π(,)π))
→ ((𝑥 ∈
(-π(,)π) ↦ 𝑥)‘0) ∈ ((𝑥 ∈ (-π(,)π) ↦ 𝑥) limℂ
0)) |
| 156 | 151, 44, 155 | mp2an 692 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ (-π(,)π) ↦
𝑥)‘0) ∈ ((𝑥 ∈ (-π(,)π) ↦
𝑥) limℂ
0) |
| 157 | | id 22 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 0 → 𝑥 = 0) |
| 158 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ (-π(,)π) ↦
𝑥) = (𝑥 ∈ (-π(,)π) ↦ 𝑥) |
| 159 | | c0ex 11255 |
. . . . . . . . . . . . . . . . . . 19
⊢ 0 ∈
V |
| 160 | 157, 158,
159 | fvmpt 7016 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 ∈
(-π(,)π) → ((𝑥
∈ (-π(,)π) ↦ 𝑥)‘0) = 0) |
| 161 | 44, 160 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ (-π(,)π) ↦
𝑥)‘0) =
0 |
| 162 | | elioore 13417 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 ∈ (-π(,)π) →
𝑥 ∈
ℝ) |
| 163 | 162 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ (-π(,)π) →
𝑥 ∈
ℂ) |
| 164 | 158, 163 | fmpti 7132 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (-π(,)π) ↦
𝑥):(-π(,)π)⟶ℂ |
| 165 | 164 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (⊤
→ (𝑥 ∈
(-π(,)π) ↦ 𝑥):(-π(,)π)⟶ℂ) |
| 166 | 165 | limcdif 25911 |
. . . . . . . . . . . . . . . . . . 19
⊢ (⊤
→ ((𝑥 ∈
(-π(,)π) ↦ 𝑥)
limℂ 0) = (((𝑥 ∈ (-π(,)π) ↦ 𝑥) ↾ ((-π(,)π)
∖ {0})) limℂ 0)) |
| 167 | 166 | mptru 1547 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ (-π(,)π) ↦
𝑥) limℂ 0)
= (((𝑥 ∈
(-π(,)π) ↦ 𝑥)
↾ ((-π(,)π) ∖ {0})) limℂ 0) |
| 168 | | resmpt 6055 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((-π(,)π) ∖ {0}) ⊆ (-π(,)π) → ((𝑥 ∈ (-π(,)π) ↦
𝑥) ↾ ((-π(,)π)
∖ {0})) = (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 𝑥)) |
| 169 | 16, 168 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ (-π(,)π) ↦
𝑥) ↾ ((-π(,)π)
∖ {0})) = (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 𝑥) |
| 170 | 169 | oveq1i 7441 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ (-π(,)π) ↦
𝑥) ↾ ((-π(,)π)
∖ {0})) limℂ 0) = ((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
𝑥) limℂ
0) |
| 171 | 167, 170 | eqtri 2765 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ (-π(,)π) ↦
𝑥) limℂ 0)
= ((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 𝑥) limℂ 0) |
| 172 | 156, 161,
171 | 3eltr3i 2853 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
((𝑥 ∈ ((-π(,)π)
∖ {0}) ↦ 𝑥)
limℂ 0) |
| 173 | 172 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (⊤
→ 0 ∈ ((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 𝑥) limℂ 0)) |
| 174 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 ∈ ℂ ↦ 2) =
(𝑥 ∈ ℂ ↦
2) |
| 175 | 144 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (2 ∈
ℂ → ℂ ⊆ ℂ) |
| 176 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (2 ∈
ℂ → 2 ∈ ℂ) |
| 177 | 175, 176,
175 | constcncfg 45887 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (2 ∈
ℂ → (𝑥 ∈
ℂ ↦ 2) ∈ (ℂ–cn→ℂ)) |
| 178 | 102, 177 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (⊤
→ (𝑥 ∈ ℂ
↦ 2) ∈ (ℂ–cn→ℂ)) |
| 179 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((⊤ ∧ 𝑥
∈ (-π(,)π)) → 2 ∈ ℂ) |
| 180 | 174, 178,
143, 145, 179 | cncfmptssg 45886 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (⊤
→ (𝑥 ∈
(-π(,)π) ↦ 2) ∈ ((-π(,)π)–cn→ℂ)) |
| 181 | | sincn 26488 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ sin
∈ (ℂ–cn→ℂ) |
| 182 | 181 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (⊤
→ sin ∈ (ℂ–cn→ℂ)) |
| 183 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 ∈ ℂ ↦ (𝑥 / 2)) = (𝑥 ∈ ℂ ↦ (𝑥 / 2)) |
| 184 | 183 | divccncf 24932 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((2
∈ ℂ ∧ 2 ≠ 0) → (𝑥 ∈ ℂ ↦ (𝑥 / 2)) ∈ (ℂ–cn→ℂ)) |
| 185 | 102, 81, 184 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 ∈ ℂ ↦ (𝑥 / 2)) ∈
(ℂ–cn→ℂ) |
| 186 | 185 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (⊤
→ (𝑥 ∈ ℂ
↦ (𝑥 / 2)) ∈
(ℂ–cn→ℂ)) |
| 187 | 163 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((⊤ ∧ 𝑥
∈ (-π(,)π)) → 𝑥 ∈ ℂ) |
| 188 | 187 | halfcld 12511 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((⊤ ∧ 𝑥
∈ (-π(,)π)) → (𝑥 / 2) ∈ ℂ) |
| 189 | 183, 186,
143, 145, 188 | cncfmptssg 45886 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (⊤
→ (𝑥 ∈
(-π(,)π) ↦ (𝑥 /
2)) ∈ ((-π(,)π)–cn→ℂ)) |
| 190 | 182, 189 | cncfmpt1f 24940 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (⊤
→ (𝑥 ∈
(-π(,)π) ↦ (sin‘(𝑥 / 2))) ∈ ((-π(,)π)–cn→ℂ)) |
| 191 | 180, 190 | mulcncf 25480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (⊤
→ (𝑥 ∈
(-π(,)π) ↦ (2 · (sin‘(𝑥 / 2)))) ∈ ((-π(,)π)–cn→ℂ)) |
| 192 | 191 | mptru 1547 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2)))) ∈ ((-π(,)π)–cn→ℂ) |
| 193 | | cnlimc 25923 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((-π(,)π) ⊆ ℂ → ((𝑥 ∈ (-π(,)π) ↦ (2 ·
(sin‘(𝑥 / 2))))
∈ ((-π(,)π)–cn→ℂ) ↔ ((𝑥 ∈ (-π(,)π) ↦ (2 ·
(sin‘(𝑥 /
2)))):(-π(,)π)⟶ℂ ∧ ∀𝑦 ∈ (-π(,)π)((𝑥 ∈ (-π(,)π) ↦ (2 ·
(sin‘(𝑥 /
2))))‘𝑦) ∈
((𝑥 ∈ (-π(,)π)
↦ (2 · (sin‘(𝑥 / 2)))) limℂ 𝑦)))) |
| 194 | 142, 193 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2)))) ∈ ((-π(,)π)–cn→ℂ) ↔ ((𝑥 ∈ (-π(,)π) ↦ (2 ·
(sin‘(𝑥 /
2)))):(-π(,)π)⟶ℂ ∧ ∀𝑦 ∈ (-π(,)π)((𝑥 ∈ (-π(,)π) ↦ (2 ·
(sin‘(𝑥 /
2))))‘𝑦) ∈
((𝑥 ∈ (-π(,)π)
↦ (2 · (sin‘(𝑥 / 2)))) limℂ 𝑦))) |
| 195 | 192, 194 | mpbi 230 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2)))):(-π(,)π)⟶ℂ ∧ ∀𝑦 ∈ (-π(,)π)((𝑥 ∈ (-π(,)π) ↦ (2 ·
(sin‘(𝑥 /
2))))‘𝑦) ∈
((𝑥 ∈ (-π(,)π)
↦ (2 · (sin‘(𝑥 / 2)))) limℂ 𝑦)) |
| 196 | 195 | simpri 485 |
. . . . . . . . . . . . . . . . . 18
⊢
∀𝑦 ∈
(-π(,)π)((𝑥 ∈
(-π(,)π) ↦ (2 · (sin‘(𝑥 / 2))))‘𝑦) ∈ ((𝑥 ∈ (-π(,)π) ↦ (2 ·
(sin‘(𝑥 / 2))))
limℂ 𝑦) |
| 197 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 0 → ((𝑥 ∈ (-π(,)π) ↦ (2 ·
(sin‘(𝑥 /
2))))‘𝑦) = ((𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2))))‘0)) |
| 198 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 0 → ((𝑥 ∈ (-π(,)π) ↦ (2 ·
(sin‘(𝑥 / 2))))
limℂ 𝑦) =
((𝑥 ∈ (-π(,)π)
↦ (2 · (sin‘(𝑥 / 2)))) limℂ
0)) |
| 199 | 197, 198 | eleq12d 2835 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 0 → (((𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2))))‘𝑦) ∈
((𝑥 ∈ (-π(,)π)
↦ (2 · (sin‘(𝑥 / 2)))) limℂ 𝑦) ↔ ((𝑥 ∈ (-π(,)π) ↦ (2 ·
(sin‘(𝑥 /
2))))‘0) ∈ ((𝑥
∈ (-π(,)π) ↦ (2 · (sin‘(𝑥 / 2)))) limℂ
0))) |
| 200 | 199 | rspccva 3621 |
. . . . . . . . . . . . . . . . . 18
⊢
((∀𝑦 ∈
(-π(,)π)((𝑥 ∈
(-π(,)π) ↦ (2 · (sin‘(𝑥 / 2))))‘𝑦) ∈ ((𝑥 ∈ (-π(,)π) ↦ (2 ·
(sin‘(𝑥 / 2))))
limℂ 𝑦)
∧ 0 ∈ (-π(,)π)) → ((𝑥 ∈ (-π(,)π) ↦ (2 ·
(sin‘(𝑥 /
2))))‘0) ∈ ((𝑥
∈ (-π(,)π) ↦ (2 · (sin‘(𝑥 / 2)))) limℂ
0)) |
| 201 | 196, 44, 200 | mp2an 692 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2))))‘0) ∈ ((𝑥
∈ (-π(,)π) ↦ (2 · (sin‘(𝑥 / 2)))) limℂ
0) |
| 202 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 0 → (𝑥 / 2) = (0 / 2)) |
| 203 | 102, 81 | div0i 12001 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (0 / 2) =
0 |
| 204 | 202, 203 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 0 → (𝑥 / 2) = 0) |
| 205 | 204 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 0 → (sin‘(𝑥 / 2)) =
(sin‘0)) |
| 206 | | sin0 16185 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(sin‘0) = 0 |
| 207 | 205, 206 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 0 → (sin‘(𝑥 / 2)) = 0) |
| 208 | 207 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 0 → (2 ·
(sin‘(𝑥 / 2))) = (2
· 0)) |
| 209 | | 2t0e0 12435 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (2
· 0) = 0 |
| 210 | 208, 209 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 0 → (2 ·
(sin‘(𝑥 / 2))) =
0) |
| 211 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2)))) = (𝑥 ∈
(-π(,)π) ↦ (2 · (sin‘(𝑥 / 2)))) |
| 212 | 210, 211,
159 | fvmpt 7016 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 ∈
(-π(,)π) → ((𝑥
∈ (-π(,)π) ↦ (2 · (sin‘(𝑥 / 2))))‘0) = 0) |
| 213 | 44, 212 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2))))‘0) = 0 |
| 214 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 ∈ (-π(,)π) → 2
∈ ℂ) |
| 215 | 163 | halfcld 12511 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 ∈ (-π(,)π) →
(𝑥 / 2) ∈
ℂ) |
| 216 | 215 | sincld 16166 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 ∈ (-π(,)π) →
(sin‘(𝑥 / 2)) ∈
ℂ) |
| 217 | 214, 216 | mulcld 11281 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ (-π(,)π) → (2
· (sin‘(𝑥 /
2))) ∈ ℂ) |
| 218 | 211, 217 | fmpti 7132 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2)))):(-π(,)π)⟶ℂ |
| 219 | 218 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (⊤
→ (𝑥 ∈
(-π(,)π) ↦ (2 · (sin‘(𝑥 /
2)))):(-π(,)π)⟶ℂ) |
| 220 | 219 | limcdif 25911 |
. . . . . . . . . . . . . . . . . . 19
⊢ (⊤
→ ((𝑥 ∈
(-π(,)π) ↦ (2 · (sin‘(𝑥 / 2)))) limℂ 0) = (((𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2)))) ↾ ((-π(,)π) ∖ {0})) limℂ
0)) |
| 221 | 220 | mptru 1547 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2)))) limℂ 0) = (((𝑥 ∈ (-π(,)π) ↦ (2 ·
(sin‘(𝑥 / 2))))
↾ ((-π(,)π) ∖ {0})) limℂ 0) |
| 222 | | resmpt 6055 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((-π(,)π) ∖ {0}) ⊆ (-π(,)π) → ((𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2)))) ↾ ((-π(,)π) ∖ {0})) = (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))) |
| 223 | 16, 222 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2)))) ↾ ((-π(,)π) ∖ {0})) = (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2)))) |
| 224 | 223 | oveq1i 7441 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2)))) ↾ ((-π(,)π) ∖ {0})) limℂ 0) = ((𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ (2 · (sin‘(𝑥 / 2)))) limℂ
0) |
| 225 | 221, 224 | eqtri 2765 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ (-π(,)π) ↦
(2 · (sin‘(𝑥 /
2)))) limℂ 0) = ((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2)))) limℂ 0) |
| 226 | 201, 213,
225 | 3eltr3i 2853 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
((𝑥 ∈ ((-π(,)π)
∖ {0}) ↦ (2 · (sin‘(𝑥 / 2)))) limℂ
0) |
| 227 | 226 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (⊤
→ 0 ∈ ((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2)))) limℂ
0)) |
| 228 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2)))) = (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))) |
| 229 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 𝑦 → (𝑥 / 2) = (𝑦 / 2)) |
| 230 | 229 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 𝑦 → (sin‘(𝑥 / 2)) = (sin‘(𝑦 / 2))) |
| 231 | 230 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑦 → (2 · (sin‘(𝑥 / 2))) = (2 ·
(sin‘(𝑦 /
2)))) |
| 232 | 231 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑦 ∈ ((-π(,)π) ∖
{0}) ∧ 𝑥 = 𝑦) → (2 ·
(sin‘(𝑥 / 2))) = (2
· (sin‘(𝑦 /
2)))) |
| 233 | | id 22 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → 𝑦 ∈
((-π(,)π) ∖ {0})) |
| 234 | 26 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → 2 ∈ ℝ) |
| 235 | 19 | sseli 3979 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → 𝑦 ∈
ℝ) |
| 236 | 235 | rehalfcld 12513 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (𝑦 / 2) ∈
ℝ) |
| 237 | 236 | resincld 16179 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (sin‘(𝑦 /
2)) ∈ ℝ) |
| 238 | 234, 237 | remulcld 11291 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (2 · (sin‘(𝑦 / 2))) ∈ ℝ) |
| 239 | 228, 232,
233, 238 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → ((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2))))‘𝑦) = (2 · (sin‘(𝑦 / 2)))) |
| 240 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → 2 ∈ ℂ) |
| 241 | 237 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (sin‘(𝑦 /
2)) ∈ ℂ) |
| 242 | 81 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → 2 ≠ 0) |
| 243 | | ioossicc 13473 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(-π(,)π) ⊆ (-π[,]π) |
| 244 | | eldifi 4131 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → 𝑦 ∈
(-π(,)π)) |
| 245 | 243, 244 | sselid 3981 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → 𝑦 ∈
(-π[,]π)) |
| 246 | | eldifsni 4790 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → 𝑦 ≠
0) |
| 247 | | fourierdlem44 46166 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 ∈ (-π[,]π) ∧
𝑦 ≠ 0) →
(sin‘(𝑦 / 2)) ≠
0) |
| 248 | 245, 246,
247 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (sin‘(𝑦 /
2)) ≠ 0) |
| 249 | 240, 241,
242, 248 | mulne0d 11915 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (2 · (sin‘(𝑦 / 2))) ≠ 0) |
| 250 | 239, 249 | eqnetrd 3008 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → ((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2))))‘𝑦) ≠ 0) |
| 251 | 250 | neneqd 2945 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → ¬ ((𝑥
∈ ((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2))))‘𝑦) = 0) |
| 252 | 251 | nrex 3074 |
. . . . . . . . . . . . . . . . 17
⊢ ¬
∃𝑦 ∈
((-π(,)π) ∖ {0})((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))‘𝑦) =
0 |
| 253 | 25 | fnmpt 6708 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∀𝑥 ∈
((-π(,)π) ∖ {0})(2 · (sin‘(𝑥 / 2))) ∈ ℝ → (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ (2 · (sin‘(𝑥 / 2)))) Fn ((-π(,)π) ∖
{0})) |
| 254 | 253, 30 | mprg 3067 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ (2 · (sin‘(𝑥 / 2)))) Fn ((-π(,)π) ∖
{0}) |
| 255 | | ssid 4006 |
. . . . . . . . . . . . . . . . . 18
⊢
((-π(,)π) ∖ {0}) ⊆ ((-π(,)π) ∖
{0}) |
| 256 | | fvelimab 6981 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ (2 · (sin‘(𝑥 / 2)))) Fn ((-π(,)π) ∖ {0})
∧ ((-π(,)π) ∖ {0}) ⊆ ((-π(,)π) ∖ {0})) →
(0 ∈ ((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2)))) “ ((-π(,)π) ∖
{0})) ↔ ∃𝑦
∈ ((-π(,)π) ∖ {0})((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))‘𝑦) =
0)) |
| 257 | 254, 255,
256 | mp2an 692 |
. . . . . . . . . . . . . . . . 17
⊢ (0 ∈
((𝑥 ∈ ((-π(,)π)
∖ {0}) ↦ (2 · (sin‘(𝑥 / 2)))) “ ((-π(,)π) ∖
{0})) ↔ ∃𝑦
∈ ((-π(,)π) ∖ {0})((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))‘𝑦) =
0) |
| 258 | 252, 257 | mtbir 323 |
. . . . . . . . . . . . . . . 16
⊢ ¬ 0
∈ ((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2)))) “ ((-π(,)π) ∖
{0})) |
| 259 | 258 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (⊤
→ ¬ 0 ∈ ((𝑥
∈ ((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2)))) “ ((-π(,)π)
∖ {0}))) |
| 260 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (cos‘(𝑥 / 2))) = (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(cos‘(𝑥 /
2)))) |
| 261 | 229 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 𝑦 → (cos‘(𝑥 / 2)) = (cos‘(𝑦 / 2))) |
| 262 | 261 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 ∈ ((-π(,)π) ∖
{0}) ∧ 𝑥 = 𝑦) → (cos‘(𝑥 / 2)) = (cos‘(𝑦 / 2))) |
| 263 | 235 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → 𝑦 ∈
ℂ) |
| 264 | 263 | halfcld 12511 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (𝑦 / 2) ∈
ℂ) |
| 265 | 264 | coscld 16167 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (cos‘(𝑦 /
2)) ∈ ℂ) |
| 266 | 260, 262,
233, 265 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → ((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (cos‘(𝑥 / 2)))‘𝑦) = (cos‘(𝑦 / 2))) |
| 267 | 236 | rered 15263 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (ℜ‘(𝑦 / 2)) = (𝑦 / 2)) |
| 268 | | halfpire 26506 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (π /
2) ∈ ℝ |
| 269 | 268 | renegcli 11570 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ -(π /
2) ∈ ℝ |
| 270 | 269 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → -(π / 2) ∈ ℝ) |
| 271 | 270 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → -(π / 2) ∈ ℝ*) |
| 272 | 268 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (π / 2) ∈ ℝ) |
| 273 | 272 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (π / 2) ∈ ℝ*) |
| 274 | | picn 26501 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ π
∈ ℂ |
| 275 | | divneg 11959 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((π
∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(π / 2) =
(-π / 2)) |
| 276 | 274, 102,
81, 275 | mp3an 1463 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ -(π /
2) = (-π / 2) |
| 277 | 39 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → -π ∈ ℝ) |
| 278 | | 2rp 13039 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ 2 ∈
ℝ+ |
| 279 | 278 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → 2 ∈ ℝ+) |
| 280 | 40 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → -π ∈ ℝ*) |
| 281 | 41 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → π ∈ ℝ*) |
| 282 | | ioogtlb 45508 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((-π
∈ ℝ* ∧ π ∈ ℝ* ∧ 𝑦 ∈ (-π(,)π)) →
-π < 𝑦) |
| 283 | 280, 281,
244, 282 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → -π < 𝑦) |
| 284 | 277, 235,
279, 283 | ltdiv1dd 13134 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (-π / 2) < (𝑦 / 2)) |
| 285 | 276, 284 | eqbrtrid 5178 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → -(π / 2) < (𝑦 / 2)) |
| 286 | 38 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → π ∈ ℝ) |
| 287 | | iooltub 45523 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((-π
∈ ℝ* ∧ π ∈ ℝ* ∧ 𝑦 ∈ (-π(,)π)) →
𝑦 <
π) |
| 288 | 280, 281,
244, 287 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → 𝑦 <
π) |
| 289 | 235, 286,
279, 288 | ltdiv1dd 13134 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (𝑦 / 2) <
(π / 2)) |
| 290 | 271, 273,
236, 285, 289 | eliood 45511 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (𝑦 / 2) ∈
(-(π / 2)(,)(π / 2))) |
| 291 | 267, 290 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (ℜ‘(𝑦 / 2)) ∈ (-(π / 2)(,)(π /
2))) |
| 292 | | cosne0 26571 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑦 / 2) ∈ ℂ ∧
(ℜ‘(𝑦 / 2))
∈ (-(π / 2)(,)(π / 2))) → (cos‘(𝑦 / 2)) ≠ 0) |
| 293 | 264, 291,
292 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → (cos‘(𝑦 /
2)) ≠ 0) |
| 294 | 266, 293 | eqnetrd 3008 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → ((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (cos‘(𝑥 / 2)))‘𝑦) ≠ 0) |
| 295 | 294 | neneqd 2945 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ ((-π(,)π) ∖
{0}) → ¬ ((𝑥
∈ ((-π(,)π) ∖ {0}) ↦ (cos‘(𝑥 / 2)))‘𝑦) = 0) |
| 296 | 295 | nrex 3074 |
. . . . . . . . . . . . . . . . . 18
⊢ ¬
∃𝑦 ∈
((-π(,)π) ∖ {0})((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(cos‘(𝑥 /
2)))‘𝑦) =
0 |
| 297 | 72, 73 | fnmpti 6711 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ (cos‘(𝑥
/ 2))) Fn ((-π(,)π) ∖ {0}) |
| 298 | | fvelimab 6981 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ (cos‘(𝑥
/ 2))) Fn ((-π(,)π) ∖ {0}) ∧ ((-π(,)π) ∖ {0})
⊆ ((-π(,)π) ∖ {0})) → (0 ∈ ((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(cos‘(𝑥 / 2)))
“ ((-π(,)π) ∖ {0})) ↔ ∃𝑦 ∈ ((-π(,)π) ∖ {0})((𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ (cos‘(𝑥
/ 2)))‘𝑦) =
0)) |
| 299 | 297, 255,
298 | mp2an 692 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 ∈
((𝑥 ∈ ((-π(,)π)
∖ {0}) ↦ (cos‘(𝑥 / 2))) “ ((-π(,)π) ∖ {0}))
↔ ∃𝑦 ∈
((-π(,)π) ∖ {0})((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(cos‘(𝑥 /
2)))‘𝑦) =
0) |
| 300 | 296, 299 | mtbir 323 |
. . . . . . . . . . . . . . . . 17
⊢ ¬ 0
∈ ((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (cos‘(𝑥 / 2))) “ ((-π(,)π) ∖
{0})) |
| 301 | 135 | imaeq1i 6075 |
. . . . . . . . . . . . . . . . . 18
⊢ ((ℝ
D (𝑥 ∈ ((-π(,)π)
∖ {0}) ↦ (2 · (sin‘(𝑥 / 2))))) “ ((-π(,)π) ∖
{0})) = ((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (cos‘(𝑥 / 2))) “ ((-π(,)π) ∖
{0})) |
| 302 | 301 | eleq2i 2833 |
. . . . . . . . . . . . . . . . 17
⊢ (0 ∈
((ℝ D (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2))))) “ ((-π(,)π) ∖
{0})) ↔ 0 ∈ ((𝑥
∈ ((-π(,)π) ∖ {0}) ↦ (cos‘(𝑥 / 2))) “ ((-π(,)π) ∖
{0}))) |
| 303 | 300, 302 | mtbir 323 |
. . . . . . . . . . . . . . . 16
⊢ ¬ 0
∈ ((ℝ D (𝑥
∈ ((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2))))) “
((-π(,)π) ∖ {0})) |
| 304 | 303 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (⊤
→ ¬ 0 ∈ ((ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))) “ ((-π(,)π) ∖ {0}))) |
| 305 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) ↦ (cos‘(𝑠
/ 2))) = (𝑠 ∈
((-π(,)π) ∖ {0}) ↦ (cos‘(𝑠 / 2))) |
| 306 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) ↦ (1 / (cos‘(𝑠 / 2)))) = (𝑠 ∈ ((-π(,)π) ∖ {0}) ↦
(1 / (cos‘(𝑠 /
2)))) |
| 307 | 19 | sseli 3979 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → 𝑠 ∈
ℝ) |
| 308 | 307 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → 𝑠 ∈
ℂ) |
| 309 | 308 | halfcld 12511 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (𝑠 / 2) ∈
ℂ) |
| 310 | 309 | coscld 16167 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (cos‘(𝑠 /
2)) ∈ ℂ) |
| 311 | 307 | rehalfcld 12513 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (𝑠 / 2) ∈
ℝ) |
| 312 | 311 | rered 15263 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (ℜ‘(𝑠 / 2)) = (𝑠 / 2)) |
| 313 | 269 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → -(π / 2) ∈ ℝ) |
| 314 | 313 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → -(π / 2) ∈ ℝ*) |
| 315 | 268 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (π / 2) ∈ ℝ) |
| 316 | 315 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (π / 2) ∈ ℝ*) |
| 317 | 38 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → π ∈ ℝ) |
| 318 | 317 | renegcld 11690 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → -π ∈ ℝ) |
| 319 | 278 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → 2 ∈ ℝ+) |
| 320 | 40 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → -π ∈ ℝ*) |
| 321 | 41 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → π ∈ ℝ*) |
| 322 | | eldifi 4131 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → 𝑠 ∈
(-π(,)π)) |
| 323 | | ioogtlb 45508 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((-π
∈ ℝ* ∧ π ∈ ℝ* ∧ 𝑠 ∈ (-π(,)π)) →
-π < 𝑠) |
| 324 | 320, 321,
322, 323 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → -π < 𝑠) |
| 325 | 318, 307,
319, 324 | ltdiv1dd 13134 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (-π / 2) < (𝑠 / 2)) |
| 326 | 276, 325 | eqbrtrid 5178 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → -(π / 2) < (𝑠 / 2)) |
| 327 | | iooltub 45523 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((-π
∈ ℝ* ∧ π ∈ ℝ* ∧ 𝑠 ∈ (-π(,)π)) →
𝑠 <
π) |
| 328 | 320, 321,
322, 327 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → 𝑠 <
π) |
| 329 | 307, 317,
319, 328 | ltdiv1dd 13134 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (𝑠 / 2) <
(π / 2)) |
| 330 | 314, 316,
311, 326, 329 | eliood 45511 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (𝑠 / 2) ∈
(-(π / 2)(,)(π / 2))) |
| 331 | 312, 330 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (ℜ‘(𝑠 / 2)) ∈ (-(π / 2)(,)(π /
2))) |
| 332 | | cosne0 26571 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑠 / 2) ∈ ℂ ∧
(ℜ‘(𝑠 / 2))
∈ (-(π / 2)(,)(π / 2))) → (cos‘(𝑠 / 2)) ≠ 0) |
| 333 | 309, 331,
332 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (cos‘(𝑠 /
2)) ≠ 0) |
| 334 | 333 | neneqd 2945 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → ¬ (cos‘(𝑠 / 2)) = 0) |
| 335 | 311 | recoscld 16180 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (cos‘(𝑠 /
2)) ∈ ℝ) |
| 336 | | elsng 4640 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((cos‘(𝑠 / 2))
∈ ℝ → ((cos‘(𝑠 / 2)) ∈ {0} ↔ (cos‘(𝑠 / 2)) = 0)) |
| 337 | 335, 336 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → ((cos‘(𝑠
/ 2)) ∈ {0} ↔ (cos‘(𝑠 / 2)) = 0)) |
| 338 | 334, 337 | mtbird 325 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → ¬ (cos‘(𝑠 / 2)) ∈ {0}) |
| 339 | 310, 338 | eldifd 3962 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (cos‘(𝑠 /
2)) ∈ (ℂ ∖ {0})) |
| 340 | 339 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢
((⊤ ∧ 𝑠
∈ ((-π(,)π) ∖ {0})) → (cos‘(𝑠 / 2)) ∈ (ℂ ∖
{0})) |
| 341 | 309 | ad2antrl 728 |
. . . . . . . . . . . . . . . . . 18
⊢
((⊤ ∧ (𝑠
∈ ((-π(,)π) ∖ {0}) ∧ (𝑠 / 2) ≠ 0)) → (𝑠 / 2) ∈ ℂ) |
| 342 | | cosf 16161 |
. . . . . . . . . . . . . . . . . . . 20
⊢
cos:ℂ⟶ℂ |
| 343 | 342 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (⊤
→ cos:ℂ⟶ℂ) |
| 344 | 343 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . . 18
⊢
((⊤ ∧ 𝑥
∈ ℂ) → (cos‘𝑥) ∈ ℂ) |
| 345 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ℂ ↦ (𝑠 / 2)) = (𝑠 ∈ ℂ ↦ (𝑠 / 2)) |
| 346 | 345 | divccncf 24932 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((2
∈ ℂ ∧ 2 ≠ 0) → (𝑠 ∈ ℂ ↦ (𝑠 / 2)) ∈ (ℂ–cn→ℂ)) |
| 347 | 102, 81, 346 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 ∈ ℂ ↦ (𝑠 / 2)) ∈
(ℂ–cn→ℂ) |
| 348 | 347 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (⊤
→ (𝑠 ∈ ℂ
↦ (𝑠 / 2)) ∈
(ℂ–cn→ℂ)) |
| 349 | 141 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((⊤ ∧ 𝑠
∈ (-π(,)π)) → 𝑠 ∈ ℂ) |
| 350 | 349 | halfcld 12511 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((⊤ ∧ 𝑠
∈ (-π(,)π)) → (𝑠 / 2) ∈ ℂ) |
| 351 | 345, 348,
143, 145, 350 | cncfmptssg 45886 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (⊤
→ (𝑠 ∈
(-π(,)π) ↦ (𝑠 /
2)) ∈ ((-π(,)π)–cn→ℂ)) |
| 352 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 = 0 → (𝑠 / 2) = (0 / 2)) |
| 353 | 352, 203 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 = 0 → (𝑠 / 2) = 0) |
| 354 | 351, 45, 353 | cnmptlimc 25925 |
. . . . . . . . . . . . . . . . . . 19
⊢ (⊤
→ 0 ∈ ((𝑠 ∈
(-π(,)π) ↦ (𝑠 /
2)) limℂ 0)) |
| 355 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈ (-π(,)π) ↦
(𝑠 / 2)) = (𝑠 ∈ (-π(,)π) ↦
(𝑠 / 2)) |
| 356 | 141 | halfcld 12511 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈ (-π(,)π) →
(𝑠 / 2) ∈
ℂ) |
| 357 | 355, 356 | fmpti 7132 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑠 ∈ (-π(,)π) ↦
(𝑠 /
2)):(-π(,)π)⟶ℂ |
| 358 | 357 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (⊤
→ (𝑠 ∈
(-π(,)π) ↦ (𝑠 /
2)):(-π(,)π)⟶ℂ) |
| 359 | 358 | limcdif 25911 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (⊤
→ ((𝑠 ∈
(-π(,)π) ↦ (𝑠 /
2)) limℂ 0) = (((𝑠 ∈ (-π(,)π) ↦ (𝑠 / 2)) ↾ ((-π(,)π)
∖ {0})) limℂ 0)) |
| 360 | 359 | mptru 1547 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑠 ∈ (-π(,)π) ↦
(𝑠 / 2))
limℂ 0) = (((𝑠 ∈ (-π(,)π) ↦ (𝑠 / 2)) ↾ ((-π(,)π)
∖ {0})) limℂ 0) |
| 361 | | resmpt 6055 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((-π(,)π) ∖ {0}) ⊆ (-π(,)π) → ((𝑠 ∈ (-π(,)π) ↦
(𝑠 / 2)) ↾
((-π(,)π) ∖ {0})) = (𝑠 ∈ ((-π(,)π) ∖ {0}) ↦
(𝑠 / 2))) |
| 362 | 16, 361 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ∈ (-π(,)π) ↦
(𝑠 / 2)) ↾
((-π(,)π) ∖ {0})) = (𝑠 ∈ ((-π(,)π) ∖ {0}) ↦
(𝑠 / 2)) |
| 363 | 362 | oveq1i 7441 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑠 ∈ (-π(,)π) ↦
(𝑠 / 2)) ↾
((-π(,)π) ∖ {0})) limℂ 0) = ((𝑠 ∈ ((-π(,)π) ∖ {0}) ↦
(𝑠 / 2))
limℂ 0) |
| 364 | 360, 363 | eqtri 2765 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 ∈ (-π(,)π) ↦
(𝑠 / 2))
limℂ 0) = ((𝑠 ∈ ((-π(,)π) ∖ {0}) ↦
(𝑠 / 2))
limℂ 0) |
| 365 | 354, 364 | eleqtrdi 2851 |
. . . . . . . . . . . . . . . . . 18
⊢ (⊤
→ 0 ∈ ((𝑠 ∈
((-π(,)π) ∖ {0}) ↦ (𝑠 / 2)) limℂ
0)) |
| 366 | | ffn 6736 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(cos:ℂ⟶ℂ → cos Fn ℂ) |
| 367 | 342, 366 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ cos Fn
ℂ |
| 368 | | dffn5 6967 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (cos Fn
ℂ ↔ cos = (𝑥
∈ ℂ ↦ (cos‘𝑥))) |
| 369 | 367, 368 | mpbi 230 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ cos =
(𝑥 ∈ ℂ ↦
(cos‘𝑥)) |
| 370 | | coscn 26489 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ cos
∈ (ℂ–cn→ℂ) |
| 371 | 369, 370 | eqeltrri 2838 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ℂ ↦
(cos‘𝑥)) ∈
(ℂ–cn→ℂ) |
| 372 | 371 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (⊤
→ (𝑥 ∈ ℂ
↦ (cos‘𝑥))
∈ (ℂ–cn→ℂ)) |
| 373 | | 0cnd 11254 |
. . . . . . . . . . . . . . . . . . 19
⊢ (⊤
→ 0 ∈ ℂ) |
| 374 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 0 → (cos‘𝑥) =
(cos‘0)) |
| 375 | | cos0 16186 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(cos‘0) = 1 |
| 376 | 374, 375 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 0 → (cos‘𝑥) = 1) |
| 377 | 372, 373,
376 | cnmptlimc 25925 |
. . . . . . . . . . . . . . . . . 18
⊢ (⊤
→ 1 ∈ ((𝑥 ∈
ℂ ↦ (cos‘𝑥)) limℂ 0)) |
| 378 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = (𝑠 / 2) → (cos‘𝑥) = (cos‘(𝑠 / 2))) |
| 379 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑠 / 2) = 0 →
(cos‘(𝑠 / 2)) =
(cos‘0)) |
| 380 | 379, 375 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 / 2) = 0 →
(cos‘(𝑠 / 2)) =
1) |
| 381 | 380 | ad2antll 729 |
. . . . . . . . . . . . . . . . . 18
⊢
((⊤ ∧ (𝑠
∈ ((-π(,)π) ∖ {0}) ∧ (𝑠 / 2) = 0)) → (cos‘(𝑠 / 2)) = 1) |
| 382 | 341, 344,
365, 377, 378, 381 | limcco 25928 |
. . . . . . . . . . . . . . . . 17
⊢ (⊤
→ 1 ∈ ((𝑠 ∈
((-π(,)π) ∖ {0}) ↦ (cos‘(𝑠 / 2))) limℂ
0)) |
| 383 | | ax-1ne0 11224 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ≠
0 |
| 384 | 383 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (⊤
→ 1 ≠ 0) |
| 385 | 305, 306,
340, 382, 384 | reclimc 45668 |
. . . . . . . . . . . . . . . 16
⊢ (⊤
→ (1 / 1) ∈ ((𝑠
∈ ((-π(,)π) ∖ {0}) ↦ (1 / (cos‘(𝑠 / 2)))) limℂ
0)) |
| 386 | | 1div1e1 11958 |
. . . . . . . . . . . . . . . 16
⊢ (1 / 1) =
1 |
| 387 | 66 | fveq1i 6907 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((ℝ
D (𝑥 ∈ ((-π(,)π)
∖ {0}) ↦ 𝑥))‘𝑠) = ((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
1)‘𝑠) |
| 388 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 1) = (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
1)) |
| 389 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ∈ ((-π(,)π) ∖
{0}) ∧ 𝑥 = 𝑠) → 1 = 1) |
| 390 | | id 22 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → 𝑠 ∈
((-π(,)π) ∖ {0})) |
| 391 | | 1red 11262 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → 1 ∈ ℝ) |
| 392 | 388, 389,
390, 391 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → ((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 1)‘𝑠) = 1) |
| 393 | 387, 392 | eqtr2id 2790 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → 1 = ((ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
𝑥))‘𝑠)) |
| 394 | 135 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (ℝ D (𝑥
∈ ((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2))))) = (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(cos‘(𝑥 /
2)))) |
| 395 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 𝑠 → (𝑥 / 2) = (𝑠 / 2)) |
| 396 | 395 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑠 → (cos‘(𝑥 / 2)) = (cos‘(𝑠 / 2))) |
| 397 | 396 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ∈ ((-π(,)π) ∖
{0}) ∧ 𝑥 = 𝑠) → (cos‘(𝑥 / 2)) = (cos‘(𝑠 / 2))) |
| 398 | 394, 397,
390, 335 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → ((ℝ D (𝑥
∈ ((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2)))))‘𝑠) = (cos‘(𝑠 / 2))) |
| 399 | 398 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (cos‘(𝑠 /
2)) = ((ℝ D (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2)))))‘𝑠)) |
| 400 | 393, 399 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (1 / (cos‘(𝑠 / 2))) = (((ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
𝑥))‘𝑠) / ((ℝ D (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ (2 · (sin‘(𝑥 / 2)))))‘𝑠))) |
| 401 | 400 | mpteq2ia 5245 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) ↦ (1 / (cos‘(𝑠 / 2)))) = (𝑠 ∈ ((-π(,)π) ∖ {0}) ↦
(((ℝ D (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 𝑥))‘𝑠) / ((ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2)))))‘𝑠))) |
| 402 | 401 | oveq1i 7441 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑠 ∈ ((-π(,)π) ∖
{0}) ↦ (1 / (cos‘(𝑠 / 2)))) limℂ 0) = ((𝑠 ∈ ((-π(,)π) ∖
{0}) ↦ (((ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
𝑥))‘𝑠) / ((ℝ D (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ (2 · (sin‘(𝑥 / 2)))))‘𝑠))) limℂ 0) |
| 403 | 385, 386,
402 | 3eltr3g 2857 |
. . . . . . . . . . . . . . 15
⊢ (⊤
→ 1 ∈ ((𝑠 ∈
((-π(,)π) ∖ {0}) ↦ (((ℝ D (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
𝑥))‘𝑠) / ((ℝ D (𝑥 ∈ ((-π(,)π) ∖
{0}) ↦ (2 · (sin‘(𝑥 / 2)))))‘𝑠))) limℂ
0)) |
| 404 | 20, 24, 32, 34, 45, 46, 71, 140, 173, 227, 259, 304, 403 | lhop 26055 |
. . . . . . . . . . . . . 14
⊢ (⊤
→ 1 ∈ ((𝑠 ∈
((-π(,)π) ∖ {0}) ↦ (((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
𝑥)‘𝑠) / ((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))‘𝑠)))
limℂ 0)) |
| 405 | 404 | mptru 1547 |
. . . . . . . . . . . . 13
⊢ 1 ∈
((𝑠 ∈ ((-π(,)π)
∖ {0}) ↦ (((𝑥
∈ ((-π(,)π) ∖ {0}) ↦ 𝑥)‘𝑠) / ((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))‘𝑠)))
limℂ 0) |
| 406 | | eqidd 2738 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 𝑥) = (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
𝑥)) |
| 407 | | simpr 484 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∈ ((-π(,)π) ∖
{0}) ∧ 𝑥 = 𝑠) → 𝑥 = 𝑠) |
| 408 | 406, 407,
390, 307 | fvmptd 7023 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → ((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 𝑥)‘𝑠) = 𝑠) |
| 409 | | eqidd 2738 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2)))) = (𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))) |
| 410 | 407 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 ∈ ((-π(,)π) ∖
{0}) ∧ 𝑥 = 𝑠) → (𝑥 / 2) = (𝑠 / 2)) |
| 411 | 410 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑠 ∈ ((-π(,)π) ∖
{0}) ∧ 𝑥 = 𝑠) → (sin‘(𝑥 / 2)) = (sin‘(𝑠 / 2))) |
| 412 | 411 | oveq2d 7447 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∈ ((-π(,)π) ∖
{0}) ∧ 𝑥 = 𝑠) → (2 ·
(sin‘(𝑥 / 2))) = (2
· (sin‘(𝑠 /
2)))) |
| 413 | 26 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → 2 ∈ ℝ) |
| 414 | 311 | resincld 16179 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (sin‘(𝑠 /
2)) ∈ ℝ) |
| 415 | 413, 414 | remulcld 11291 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (2 · (sin‘(𝑠 / 2))) ∈ ℝ) |
| 416 | 409, 412,
390, 415 | fvmptd 7023 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → ((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ (2 · (sin‘(𝑥 / 2))))‘𝑠) = (2 · (sin‘(𝑠 / 2)))) |
| 417 | 408, 416 | oveq12d 7449 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → (((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 𝑥)‘𝑠) / ((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))‘𝑠)) = (𝑠 / (2 · (sin‘(𝑠 / 2))))) |
| 418 | 417 | mpteq2ia 5245 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) ↦ (((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 𝑥)‘𝑠) / ((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))‘𝑠))) = (𝑠 ∈ ((-π(,)π) ∖
{0}) ↦ (𝑠 / (2
· (sin‘(𝑠 /
2))))) |
| 419 | 418 | oveq1i 7441 |
. . . . . . . . . . . . 13
⊢ ((𝑠 ∈ ((-π(,)π) ∖
{0}) ↦ (((𝑥 ∈
((-π(,)π) ∖ {0}) ↦ 𝑥)‘𝑠) / ((𝑥 ∈ ((-π(,)π) ∖ {0}) ↦
(2 · (sin‘(𝑥 /
2))))‘𝑠)))
limℂ 0) = ((𝑠 ∈ ((-π(,)π) ∖ {0}) ↦
(𝑠 / (2 ·
(sin‘(𝑠 / 2)))))
limℂ 0) |
| 420 | 405, 419 | eleqtri 2839 |
. . . . . . . . . . . 12
⊢ 1 ∈
((𝑠 ∈ ((-π(,)π)
∖ {0}) ↦ (𝑠 /
(2 · (sin‘(𝑠 /
2))))) limℂ 0) |
| 421 | 10 | oveq1i 7441 |
. . . . . . . . . . . . . 14
⊢ (𝐾 limℂ 0) =
((𝑠 ∈ (-π[,]π)
↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) limℂ
0) |
| 422 | 10 | feq1i 6727 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐾:(-π[,]π)⟶ℂ
↔ (𝑠 ∈
(-π[,]π) ↦ if(𝑠
= 0, 1, (𝑠 / (2 ·
(sin‘(𝑠 /
2)))))):(-π[,]π)⟶ℂ) |
| 423 | 14, 422 | mpbi 230 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ (-π[,]π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 /
2)))))):(-π[,]π)⟶ℂ |
| 424 | 423 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (⊤
→ (𝑠 ∈
(-π[,]π) ↦ if(𝑠
= 0, 1, (𝑠 / (2 ·
(sin‘(𝑠 /
2)))))):(-π[,]π)⟶ℂ) |
| 425 | 243 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (⊤
→ (-π(,)π) ⊆ (-π[,]π)) |
| 426 | | iccssre 13469 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((-π
∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆
ℝ) |
| 427 | 39, 38, 426 | mp2an 692 |
. . . . . . . . . . . . . . . . . . 19
⊢
(-π[,]π) ⊆ ℝ |
| 428 | 427 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (⊤
→ (-π[,]π) ⊆ ℝ) |
| 429 | 428, 12 | sstrdi 3996 |
. . . . . . . . . . . . . . . . 17
⊢ (⊤
→ (-π[,]π) ⊆ ℂ) |
| 430 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢
((TopOpen‘ℂfld) ↾t
((-π[,]π) ∪ {0})) = ((TopOpen‘ℂfld)
↾t ((-π[,]π) ∪ {0})) |
| 431 | 39, 35, 36 | ltleii 11384 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ -π
≤ 0 |
| 432 | 35, 38, 37 | ltleii 11384 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ 0 ≤
π |
| 433 | 39, 38 | elicc2i 13453 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (0 ∈
(-π[,]π) ↔ (0 ∈ ℝ ∧ -π ≤ 0 ∧ 0 ≤
π)) |
| 434 | 35, 431, 432, 433 | mpbir3an 1342 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 0 ∈
(-π[,]π) |
| 435 | 159 | snss 4785 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (0 ∈
(-π[,]π) ↔ {0} ⊆ (-π[,]π)) |
| 436 | 434, 435 | mpbi 230 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ {0}
⊆ (-π[,]π) |
| 437 | | ssequn2 4189 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ({0}
⊆ (-π[,]π) ↔ ((-π[,]π) ∪ {0}) =
(-π[,]π)) |
| 438 | 436, 437 | mpbi 230 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((-π[,]π) ∪ {0}) = (-π[,]π) |
| 439 | 438 | oveq2i 7442 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((TopOpen‘ℂfld) ↾t
((-π[,]π) ∪ {0})) = ((TopOpen‘ℂfld)
↾t (-π[,]π)) |
| 440 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(topGen‘ran (,)) = (topGen‘ran (,)) |
| 441 | 57, 440 | rerest 24825 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((-π[,]π) ⊆ ℝ →
((TopOpen‘ℂfld) ↾t (-π[,]π)) =
((topGen‘ran (,)) ↾t (-π[,]π))) |
| 442 | 427, 441 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((TopOpen‘ℂfld) ↾t
(-π[,]π)) = ((topGen‘ran (,)) ↾t
(-π[,]π)) |
| 443 | 439, 442 | eqtri 2765 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((TopOpen‘ℂfld) ↾t
((-π[,]π) ∪ {0})) = ((topGen‘ran (,)) ↾t
(-π[,]π)) |
| 444 | 443 | fveq2i 6909 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(int‘((TopOpen‘ℂfld) ↾t
((-π[,]π) ∪ {0}))) = (int‘((topGen‘ran (,))
↾t (-π[,]π))) |
| 445 | 159 | snss 4785 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (0 ∈
(-π(,)π) ↔ {0} ⊆ (-π(,)π)) |
| 446 | 44, 445 | mpbi 230 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ {0}
⊆ (-π(,)π) |
| 447 | | ssequn2 4189 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ({0}
⊆ (-π(,)π) ↔ ((-π(,)π) ∪ {0}) =
(-π(,)π)) |
| 448 | 446, 447 | mpbi 230 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((-π(,)π) ∪ {0}) = (-π(,)π) |
| 449 | 444, 448 | fveq12i 6912 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((int‘((TopOpen‘ℂfld) ↾t
((-π[,]π) ∪ {0})))‘((-π(,)π) ∪ {0})) =
((int‘((topGen‘ran (,)) ↾t
(-π[,]π)))‘(-π(,)π)) |
| 450 | | resttopon 23169 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧
(-π[,]π) ⊆ ℝ) → ((topGen‘ran (,))
↾t (-π[,]π)) ∈
(TopOn‘(-π[,]π))) |
| 451 | 60, 427, 450 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((topGen‘ran (,)) ↾t (-π[,]π)) ∈
(TopOn‘(-π[,]π)) |
| 452 | 451 | topontopi 22921 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((topGen‘ran (,)) ↾t (-π[,]π)) ∈
Top |
| 453 | | retop 24782 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(topGen‘ran (,)) ∈ Top |
| 454 | | ovex 7464 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(-π[,]π) ∈ V |
| 455 | 453, 454 | pm3.2i 470 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((topGen‘ran (,)) ∈ Top ∧ (-π[,]π) ∈
V) |
| 456 | | ssid 4006 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(-π(,)π) ⊆ (-π(,)π) |
| 457 | 33, 243, 456 | 3pm3.2i 1340 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((-π(,)π) ∈ (topGen‘ran (,)) ∧ (-π(,)π)
⊆ (-π[,]π) ∧ (-π(,)π) ⊆
(-π(,)π)) |
| 458 | | restopnb 23183 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((topGen‘ran (,)) ∈ Top ∧ (-π[,]π) ∈ V)
∧ ((-π(,)π) ∈ (topGen‘ran (,)) ∧ (-π(,)π)
⊆ (-π[,]π) ∧ (-π(,)π) ⊆ (-π(,)π))) →
((-π(,)π) ∈ (topGen‘ran (,)) ↔ (-π(,)π) ∈
((topGen‘ran (,)) ↾t (-π[,]π)))) |
| 459 | 455, 457,
458 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((-π(,)π) ∈ (topGen‘ran (,)) ↔ (-π(,)π)
∈ ((topGen‘ran (,)) ↾t
(-π[,]π))) |
| 460 | 33, 459 | mpbi 230 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(-π(,)π) ∈ ((topGen‘ran (,)) ↾t
(-π[,]π)) |
| 461 | | isopn3i 23090 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((topGen‘ran (,)) ↾t (-π[,]π)) ∈ Top
∧ (-π(,)π) ∈ ((topGen‘ran (,)) ↾t
(-π[,]π))) → ((int‘((topGen‘ran (,)) ↾t
(-π[,]π)))‘(-π(,)π)) = (-π(,)π)) |
| 462 | 452, 460,
461 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((int‘((topGen‘ran (,)) ↾t
(-π[,]π)))‘(-π(,)π)) = (-π(,)π) |
| 463 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(-π(,)π) = (-π(,)π) |
| 464 | 449, 462,
463 | 3eqtrri 2770 |
. . . . . . . . . . . . . . . . . . 19
⊢
(-π(,)π) = ((int‘((TopOpen‘ℂfld)
↾t ((-π[,]π) ∪ {0})))‘((-π(,)π) ∪
{0})) |
| 465 | 44, 464 | eleqtri 2839 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
((int‘((TopOpen‘ℂfld) ↾t
((-π[,]π) ∪ {0})))‘((-π(,)π) ∪ {0})) |
| 466 | 465 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (⊤
→ 0 ∈ ((int‘((TopOpen‘ℂfld)
↾t ((-π[,]π) ∪ {0})))‘((-π(,)π) ∪
{0}))) |
| 467 | 424, 425,
429, 57, 430, 466 | limcres 25921 |
. . . . . . . . . . . . . . . 16
⊢ (⊤
→ (((𝑠 ∈
(-π[,]π) ↦ if(𝑠
= 0, 1, (𝑠 / (2 ·
(sin‘(𝑠 / 2))))))
↾ (-π(,)π)) limℂ 0) = ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) limℂ
0)) |
| 468 | 467 | mptru 1547 |
. . . . . . . . . . . . . . 15
⊢ (((𝑠 ∈ (-π[,]π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) ↾
(-π(,)π)) limℂ 0) = ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) limℂ
0) |
| 469 | 468 | eqcomi 2746 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 ∈ (-π[,]π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) limℂ
0) = (((𝑠 ∈
(-π[,]π) ↦ if(𝑠
= 0, 1, (𝑠 / (2 ·
(sin‘(𝑠 / 2))))))
↾ (-π(,)π)) limℂ 0) |
| 470 | | resmpt 6055 |
. . . . . . . . . . . . . . . 16
⊢
((-π(,)π) ⊆ (-π[,]π) → ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) ↾ (-π(,)π)) = (𝑠 ∈ (-π(,)π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))) |
| 471 | 243, 470 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ((𝑠 ∈ (-π[,]π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) ↾
(-π(,)π)) = (𝑠 ∈
(-π(,)π) ↦ if(𝑠
= 0, 1, (𝑠 / (2 ·
(sin‘(𝑠 /
2)))))) |
| 472 | 471 | oveq1i 7441 |
. . . . . . . . . . . . . 14
⊢ (((𝑠 ∈ (-π[,]π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) ↾
(-π(,)π)) limℂ 0) = ((𝑠 ∈ (-π(,)π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) limℂ
0) |
| 473 | 421, 469,
472 | 3eqtri 2769 |
. . . . . . . . . . . . 13
⊢ (𝐾 limℂ 0) =
((𝑠 ∈ (-π(,)π)
↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) limℂ
0) |
| 474 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ (-π(,)π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) = (𝑠 ∈ (-π(,)π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) |
| 475 | | iftrue 4531 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 = 0 → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = 1) |
| 476 | | 1cnd 11256 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 = 0 → 1 ∈
ℂ) |
| 477 | 475, 476 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 = 0 → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ ℂ) |
| 478 | 477 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑠 ∈ (-π(,)π) ∧
𝑠 = 0) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ ℂ) |
| 479 | | iffalse 4534 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬
𝑠 = 0 → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = (𝑠 / (2 · (sin‘(𝑠 / 2))))) |
| 480 | 479 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 ∈ (-π(,)π) ∧
¬ 𝑠 = 0) →
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = (𝑠 / (2 · (sin‘(𝑠 / 2))))) |
| 481 | 141 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑠 ∈ (-π(,)π) ∧
¬ 𝑠 = 0) → 𝑠 ∈
ℂ) |
| 482 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ∈ (-π(,)π) ∧
¬ 𝑠 = 0) → 2
∈ ℂ) |
| 483 | 481 | halfcld 12511 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ (-π(,)π) ∧
¬ 𝑠 = 0) → (𝑠 / 2) ∈
ℂ) |
| 484 | 483 | sincld 16166 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ∈ (-π(,)π) ∧
¬ 𝑠 = 0) →
(sin‘(𝑠 / 2)) ∈
ℂ) |
| 485 | 482, 484 | mulcld 11281 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑠 ∈ (-π(,)π) ∧
¬ 𝑠 = 0) → (2
· (sin‘(𝑠 /
2))) ∈ ℂ) |
| 486 | 81 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ∈ (-π(,)π) ∧
¬ 𝑠 = 0) → 2 ≠
0) |
| 487 | 243 | sseli 3979 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 ∈ (-π(,)π) →
𝑠 ∈
(-π[,]π)) |
| 488 | | neqne 2948 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (¬
𝑠 = 0 → 𝑠 ≠ 0) |
| 489 | | fourierdlem44 46166 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ (-π[,]π) ∧
𝑠 ≠ 0) →
(sin‘(𝑠 / 2)) ≠
0) |
| 490 | 487, 488,
489 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ∈ (-π(,)π) ∧
¬ 𝑠 = 0) →
(sin‘(𝑠 / 2)) ≠
0) |
| 491 | 482, 484,
486, 490 | mulne0d 11915 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑠 ∈ (-π(,)π) ∧
¬ 𝑠 = 0) → (2
· (sin‘(𝑠 /
2))) ≠ 0) |
| 492 | 481, 485,
491 | divcld 12043 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 ∈ (-π(,)π) ∧
¬ 𝑠 = 0) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈
ℂ) |
| 493 | 480, 492 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑠 ∈ (-π(,)π) ∧
¬ 𝑠 = 0) →
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈
ℂ) |
| 494 | 478, 493 | pm2.61dan 813 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ (-π(,)π) →
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈
ℂ) |
| 495 | 474, 494 | fmpti 7132 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ (-π(,)π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 /
2)))))):(-π(,)π)⟶ℂ |
| 496 | 495 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (⊤
→ (𝑠 ∈
(-π(,)π) ↦ if(𝑠
= 0, 1, (𝑠 / (2 ·
(sin‘(𝑠 /
2)))))):(-π(,)π)⟶ℂ) |
| 497 | 496 | limcdif 25911 |
. . . . . . . . . . . . . 14
⊢ (⊤
→ ((𝑠 ∈
(-π(,)π) ↦ if(𝑠
= 0, 1, (𝑠 / (2 ·
(sin‘(𝑠 / 2))))))
limℂ 0) = (((𝑠 ∈ (-π(,)π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) ↾ ((-π(,)π) ∖
{0})) limℂ 0)) |
| 498 | 497 | mptru 1547 |
. . . . . . . . . . . . 13
⊢ ((𝑠 ∈ (-π(,)π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) limℂ
0) = (((𝑠 ∈
(-π(,)π) ↦ if(𝑠
= 0, 1, (𝑠 / (2 ·
(sin‘(𝑠 / 2))))))
↾ ((-π(,)π) ∖ {0})) limℂ 0) |
| 499 | | resmpt 6055 |
. . . . . . . . . . . . . . . 16
⊢
(((-π(,)π) ∖ {0}) ⊆ (-π(,)π) → ((𝑠 ∈ (-π(,)π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) ↾
((-π(,)π) ∖ {0})) = (𝑠 ∈ ((-π(,)π) ∖ {0}) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))) |
| 500 | 16, 499 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ((𝑠 ∈ (-π(,)π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) ↾
((-π(,)π) ∖ {0})) = (𝑠 ∈ ((-π(,)π) ∖ {0}) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) |
| 501 | | eldifn 4132 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → ¬ 𝑠 ∈
{0}) |
| 502 | | velsn 4642 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ {0} ↔ 𝑠 = 0) |
| 503 | 501, 502 | sylnib 328 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → ¬ 𝑠 =
0) |
| 504 | 503, 479 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) → if(𝑠 = 0, 1,
(𝑠 / (2 ·
(sin‘(𝑠 / 2))))) =
(𝑠 / (2 ·
(sin‘(𝑠 /
2))))) |
| 505 | 504 | mpteq2ia 5245 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈ ((-π(,)π) ∖
{0}) ↦ if(𝑠 = 0, 1,
(𝑠 / (2 ·
(sin‘(𝑠 / 2)))))) =
(𝑠 ∈ ((-π(,)π)
∖ {0}) ↦ (𝑠 /
(2 · (sin‘(𝑠 /
2))))) |
| 506 | 500, 505 | eqtri 2765 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 ∈ (-π(,)π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) ↾
((-π(,)π) ∖ {0})) = (𝑠 ∈ ((-π(,)π) ∖ {0}) ↦
(𝑠 / (2 ·
(sin‘(𝑠 /
2))))) |
| 507 | 506 | oveq1i 7441 |
. . . . . . . . . . . . 13
⊢ (((𝑠 ∈ (-π(,)π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) ↾
((-π(,)π) ∖ {0})) limℂ 0) = ((𝑠 ∈ ((-π(,)π) ∖ {0}) ↦
(𝑠 / (2 ·
(sin‘(𝑠 / 2)))))
limℂ 0) |
| 508 | 473, 498,
507 | 3eqtrri 2770 |
. . . . . . . . . . . 12
⊢ ((𝑠 ∈ ((-π(,)π) ∖
{0}) ↦ (𝑠 / (2
· (sin‘(𝑠 /
2))))) limℂ 0) = (𝐾 limℂ 0) |
| 509 | 420, 508 | eleqtri 2839 |
. . . . . . . . . . 11
⊢ 1 ∈
(𝐾 limℂ
0) |
| 510 | 509 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑠 = 0 → 1 ∈ (𝐾 limℂ
0)) |
| 511 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑠 = 0 → (𝐾‘𝑠) = (𝐾‘0)) |
| 512 | 475, 10, 47 | fvmpt 7016 |
. . . . . . . . . . . 12
⊢ (0 ∈
(-π[,]π) → (𝐾‘0) = 1) |
| 513 | 434, 512 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (𝐾‘0) = 1 |
| 514 | 511, 513 | eqtrdi 2793 |
. . . . . . . . . 10
⊢ (𝑠 = 0 → (𝐾‘𝑠) = 1) |
| 515 | | oveq2 7439 |
. . . . . . . . . 10
⊢ (𝑠 = 0 → (𝐾 limℂ 𝑠) = (𝐾 limℂ 0)) |
| 516 | 510, 514,
515 | 3eltr4d 2856 |
. . . . . . . . 9
⊢ (𝑠 = 0 → (𝐾‘𝑠) ∈ (𝐾 limℂ 𝑠)) |
| 517 | 427, 12 | sstri 3993 |
. . . . . . . . . . 11
⊢
(-π[,]π) ⊆ ℂ |
| 518 | 517 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑠 = 0 → (-π[,]π)
⊆ ℂ) |
| 519 | 38 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑠 = 0 → π ∈
ℝ) |
| 520 | 519 | renegcld 11690 |
. . . . . . . . . . 11
⊢ (𝑠 = 0 → -π ∈
ℝ) |
| 521 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑠 = 0 → 𝑠 = 0) |
| 522 | 35 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑠 = 0 → 0 ∈
ℝ) |
| 523 | 521, 522 | eqeltrd 2841 |
. . . . . . . . . . 11
⊢ (𝑠 = 0 → 𝑠 ∈ ℝ) |
| 524 | 431, 521 | breqtrrid 5181 |
. . . . . . . . . . 11
⊢ (𝑠 = 0 → -π ≤ 𝑠) |
| 525 | 521, 432 | eqbrtrdi 5182 |
. . . . . . . . . . 11
⊢ (𝑠 = 0 → 𝑠 ≤ π) |
| 526 | 520, 519,
523, 524, 525 | eliccd 45517 |
. . . . . . . . . 10
⊢ (𝑠 = 0 → 𝑠 ∈ (-π[,]π)) |
| 527 | 56 | oveq1i 7441 |
. . . . . . . . . . . 12
⊢
((topGen‘ran (,)) ↾t (-π[,]π)) =
(((TopOpen‘ℂfld) ↾t ℝ)
↾t (-π[,]π)) |
| 528 | 57 | cnfldtop 24804 |
. . . . . . . . . . . . 13
⊢
(TopOpen‘ℂfld) ∈ Top |
| 529 | | reex 11246 |
. . . . . . . . . . . . 13
⊢ ℝ
∈ V |
| 530 | | restabs 23173 |
. . . . . . . . . . . . 13
⊢
(((TopOpen‘ℂfld) ∈ Top ∧ (-π[,]π)
⊆ ℝ ∧ ℝ ∈ V) →
(((TopOpen‘ℂfld) ↾t ℝ)
↾t (-π[,]π)) = ((TopOpen‘ℂfld)
↾t (-π[,]π))) |
| 531 | 528, 427,
529, 530 | mp3an 1463 |
. . . . . . . . . . . 12
⊢
(((TopOpen‘ℂfld) ↾t ℝ)
↾t (-π[,]π)) = ((TopOpen‘ℂfld)
↾t (-π[,]π)) |
| 532 | 527, 531 | eqtri 2765 |
. . . . . . . . . . 11
⊢
((topGen‘ran (,)) ↾t (-π[,]π)) =
((TopOpen‘ℂfld) ↾t
(-π[,]π)) |
| 533 | 57, 532 | cnplimc 25922 |
. . . . . . . . . 10
⊢
(((-π[,]π) ⊆ ℂ ∧ 𝑠 ∈ (-π[,]π)) → (𝐾 ∈ ((((topGen‘ran
(,)) ↾t (-π[,]π)) CnP
(TopOpen‘ℂfld))‘𝑠) ↔ (𝐾:(-π[,]π)⟶ℂ ∧ (𝐾‘𝑠) ∈ (𝐾 limℂ 𝑠)))) |
| 534 | 518, 526,
533 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝑠 = 0 → (𝐾 ∈ ((((topGen‘ran (,))
↾t (-π[,]π)) CnP
(TopOpen‘ℂfld))‘𝑠) ↔ (𝐾:(-π[,]π)⟶ℂ ∧ (𝐾‘𝑠) ∈ (𝐾 limℂ 𝑠)))) |
| 535 | 15, 516, 534 | mpbir2and 713 |
. . . . . . . 8
⊢ (𝑠 = 0 → 𝐾 ∈ ((((topGen‘ran (,))
↾t (-π[,]π)) CnP
(TopOpen‘ℂfld))‘𝑠)) |
| 536 | 535 | adantl 481 |
. . . . . . 7
⊢ ((𝑠 ∈ (-π[,]π) ∧
𝑠 = 0) → 𝐾 ∈ ((((topGen‘ran
(,)) ↾t (-π[,]π)) CnP
(TopOpen‘ℂfld))‘𝑠)) |
| 537 | | simpl 482 |
. . . . . . . . . . 11
⊢ ((𝑠 ∈ (-π[,]π) ∧
¬ 𝑠 = 0) → 𝑠 ∈
(-π[,]π)) |
| 538 | 502 | notbii 320 |
. . . . . . . . . . . . 13
⊢ (¬
𝑠 ∈ {0} ↔ ¬
𝑠 = 0) |
| 539 | 538 | biimpri 228 |
. . . . . . . . . . . 12
⊢ (¬
𝑠 = 0 → ¬ 𝑠 ∈ {0}) |
| 540 | 539 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑠 ∈ (-π[,]π) ∧
¬ 𝑠 = 0) → ¬
𝑠 ∈
{0}) |
| 541 | 537, 540 | eldifd 3962 |
. . . . . . . . . 10
⊢ ((𝑠 ∈ (-π[,]π) ∧
¬ 𝑠 = 0) → 𝑠 ∈ ((-π[,]π) ∖
{0})) |
| 542 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑠 → ((((topGen‘ran (,))
↾t ((-π[,]π) ∖ {0})) CnP
(TopOpen‘ℂfld))‘𝑥) = ((((topGen‘ran (,))
↾t ((-π[,]π) ∖ {0})) CnP
(TopOpen‘ℂfld))‘𝑠)) |
| 543 | 542 | eleq2d 2827 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑠 → ((𝑠 ∈ ((-π[,]π) ∖ {0}) ↦
(𝑠 / (2 ·
(sin‘(𝑠 / 2)))))
∈ ((((topGen‘ran (,)) ↾t ((-π[,]π) ∖
{0})) CnP (TopOpen‘ℂfld))‘𝑥) ↔ (𝑠 ∈ ((-π[,]π) ∖ {0}) ↦
(𝑠 / (2 ·
(sin‘(𝑠 / 2)))))
∈ ((((topGen‘ran (,)) ↾t ((-π[,]π) ∖
{0})) CnP (TopOpen‘ℂfld))‘𝑠))) |
| 544 | 429 | ssdifssd 4147 |
. . . . . . . . . . . . . . . . 17
⊢ (⊤
→ ((-π[,]π) ∖ {0}) ⊆ ℂ) |
| 545 | 544, 145 | idcncfg 45888 |
. . . . . . . . . . . . . . . 16
⊢ (⊤
→ (𝑠 ∈
((-π[,]π) ∖ {0}) ↦ 𝑠) ∈ (((-π[,]π) ∖
{0})–cn→ℂ)) |
| 546 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ (2 · (sin‘(𝑠 / 2)))) = (𝑠 ∈ ((-π[,]π) ∖ {0}) ↦
(2 · (sin‘(𝑠 /
2)))) |
| 547 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → 2 ∈ ℂ) |
| 548 | | eldifi 4131 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → 𝑠 ∈
(-π[,]π)) |
| 549 | 517, 548 | sselid 3981 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → 𝑠 ∈
ℂ) |
| 550 | 549 | halfcld 12511 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → (𝑠 / 2) ∈
ℂ) |
| 551 | 550 | sincld 16166 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → (sin‘(𝑠 /
2)) ∈ ℂ) |
| 552 | 547, 551 | mulcld 11281 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ) |
| 553 | 81 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → 2 ≠ 0) |
| 554 | | eldifsni 4790 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → 𝑠 ≠
0) |
| 555 | 548, 554,
489 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → (sin‘(𝑠 /
2)) ≠ 0) |
| 556 | 547, 551,
553, 555 | mulne0d 11915 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → (2 · (sin‘(𝑠 / 2))) ≠ 0) |
| 557 | 556 | neneqd 2945 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → ¬ (2 · (sin‘(𝑠 / 2))) = 0) |
| 558 | | elsng 4640 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((2
· (sin‘(𝑠 /
2))) ∈ ℂ → ((2 · (sin‘(𝑠 / 2))) ∈ {0} ↔ (2 ·
(sin‘(𝑠 / 2))) =
0)) |
| 559 | 552, 558 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → ((2 · (sin‘(𝑠 / 2))) ∈ {0} ↔ (2 ·
(sin‘(𝑠 / 2))) =
0)) |
| 560 | 557, 559 | mtbird 325 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → ¬ (2 · (sin‘(𝑠 / 2))) ∈ {0}) |
| 561 | 552, 560 | eldifd 3962 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → (2 · (sin‘(𝑠 / 2))) ∈ (ℂ ∖
{0})) |
| 562 | 546, 561 | fmpti 7132 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ (2 · (sin‘(𝑠 / 2)))):((-π[,]π) ∖
{0})⟶(ℂ ∖ {0}) |
| 563 | | difss 4136 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℂ
∖ {0}) ⊆ ℂ |
| 564 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 ∈ ℂ ↦ 2) =
(𝑠 ∈ ℂ ↦
2) |
| 565 | 175, 176,
175 | constcncfg 45887 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (2 ∈
ℂ → (𝑠 ∈
ℂ ↦ 2) ∈ (ℂ–cn→ℂ)) |
| 566 | 102, 565 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (⊤
→ (𝑠 ∈ ℂ
↦ 2) ∈ (ℂ–cn→ℂ)) |
| 567 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((⊤ ∧ 𝑠
∈ ((-π[,]π) ∖ {0})) → 2 ∈ ℂ) |
| 568 | 564, 566,
544, 145, 567 | cncfmptssg 45886 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (⊤
→ (𝑠 ∈
((-π[,]π) ∖ {0}) ↦ 2) ∈ (((-π[,]π) ∖
{0})–cn→ℂ)) |
| 569 | 549, 547,
553 | divrecd 12046 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → (𝑠 / 2) =
(𝑠 · (1 /
2))) |
| 570 | 569 | mpteq2ia 5245 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ (𝑠 / 2)) =
(𝑠 ∈ ((-π[,]π)
∖ {0}) ↦ (𝑠
· (1 / 2))) |
| 571 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑠 ∈ ℂ ↦ (1 / 2))
= (𝑠 ∈ ℂ ↦
(1 / 2)) |
| 572 | 144 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((1 / 2)
∈ ℂ → ℂ ⊆ ℂ) |
| 573 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((1 / 2)
∈ ℂ → (1 / 2) ∈ ℂ) |
| 574 | 572, 573,
572 | constcncfg 45887 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((1 / 2)
∈ ℂ → (𝑠
∈ ℂ ↦ (1 / 2)) ∈ (ℂ–cn→ℂ)) |
| 575 | 94, 574 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (⊤
→ (𝑠 ∈ ℂ
↦ (1 / 2)) ∈ (ℂ–cn→ℂ)) |
| 576 | 94 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((⊤ ∧ 𝑠
∈ ((-π[,]π) ∖ {0})) → (1 / 2) ∈
ℂ) |
| 577 | 571, 575,
544, 145, 576 | cncfmptssg 45886 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (⊤
→ (𝑠 ∈
((-π[,]π) ∖ {0}) ↦ (1 / 2)) ∈ (((-π[,]π) ∖
{0})–cn→ℂ)) |
| 578 | 545, 577 | mulcncf 25480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (⊤
→ (𝑠 ∈
((-π[,]π) ∖ {0}) ↦ (𝑠 · (1 / 2))) ∈ (((-π[,]π)
∖ {0})–cn→ℂ)) |
| 579 | 570, 578 | eqeltrid 2845 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (⊤
→ (𝑠 ∈
((-π[,]π) ∖ {0}) ↦ (𝑠 / 2)) ∈ (((-π[,]π) ∖
{0})–cn→ℂ)) |
| 580 | 182, 579 | cncfmpt1f 24940 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (⊤
→ (𝑠 ∈
((-π[,]π) ∖ {0}) ↦ (sin‘(𝑠 / 2))) ∈ (((-π[,]π) ∖
{0})–cn→ℂ)) |
| 581 | 568, 580 | mulcncf 25480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (⊤
→ (𝑠 ∈
((-π[,]π) ∖ {0}) ↦ (2 · (sin‘(𝑠 / 2)))) ∈ (((-π[,]π) ∖
{0})–cn→ℂ)) |
| 582 | 581 | mptru 1547 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ (2 · (sin‘(𝑠 / 2)))) ∈ (((-π[,]π) ∖
{0})–cn→ℂ) |
| 583 | | cncfcdm 24924 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((ℂ ∖ {0}) ⊆ ℂ ∧ (𝑠 ∈ ((-π[,]π) ∖ {0}) ↦
(2 · (sin‘(𝑠 /
2)))) ∈ (((-π[,]π) ∖ {0})–cn→ℂ)) → ((𝑠 ∈ ((-π[,]π) ∖ {0}) ↦
(2 · (sin‘(𝑠 /
2)))) ∈ (((-π[,]π) ∖ {0})–cn→(ℂ ∖ {0})) ↔ (𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ (2 · (sin‘(𝑠 / 2)))):((-π[,]π) ∖
{0})⟶(ℂ ∖ {0}))) |
| 584 | 563, 582,
583 | mp2an 692 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ (2 · (sin‘(𝑠 / 2)))) ∈ (((-π[,]π) ∖
{0})–cn→(ℂ ∖ {0}))
↔ (𝑠 ∈
((-π[,]π) ∖ {0}) ↦ (2 · (sin‘(𝑠 / 2)))):((-π[,]π) ∖
{0})⟶(ℂ ∖ {0})) |
| 585 | 562, 584 | mpbir 231 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ (2 · (sin‘(𝑠 / 2)))) ∈ (((-π[,]π) ∖
{0})–cn→(ℂ ∖
{0})) |
| 586 | 585 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (⊤
→ (𝑠 ∈
((-π[,]π) ∖ {0}) ↦ (2 · (sin‘(𝑠 / 2)))) ∈ (((-π[,]π) ∖
{0})–cn→(ℂ ∖
{0}))) |
| 587 | 545, 586 | divcncf 25482 |
. . . . . . . . . . . . . . 15
⊢ (⊤
→ (𝑠 ∈
((-π[,]π) ∖ {0}) ↦ (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ (((-π[,]π) ∖
{0})–cn→ℂ)) |
| 588 | 587 | mptru 1547 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ (𝑠 / (2
· (sin‘(𝑠 /
2))))) ∈ (((-π[,]π) ∖ {0})–cn→ℂ) |
| 589 | 428 | ssdifssd 4147 |
. . . . . . . . . . . . . . . . 17
⊢ (⊤
→ ((-π[,]π) ∖ {0}) ⊆ ℝ) |
| 590 | 589 | mptru 1547 |
. . . . . . . . . . . . . . . 16
⊢
((-π[,]π) ∖ {0}) ⊆ ℝ |
| 591 | 590, 12 | sstri 3993 |
. . . . . . . . . . . . . . 15
⊢
((-π[,]π) ∖ {0}) ⊆ ℂ |
| 592 | 56 | oveq1i 7441 |
. . . . . . . . . . . . . . . . 17
⊢
((topGen‘ran (,)) ↾t ((-π[,]π) ∖
{0})) = (((TopOpen‘ℂfld) ↾t ℝ)
↾t ((-π[,]π) ∖ {0})) |
| 593 | | restabs 23173 |
. . . . . . . . . . . . . . . . . 18
⊢
(((TopOpen‘ℂfld) ∈ Top ∧ ((-π[,]π)
∖ {0}) ⊆ ℝ ∧ ℝ ∈ V) →
(((TopOpen‘ℂfld) ↾t ℝ)
↾t ((-π[,]π) ∖ {0})) =
((TopOpen‘ℂfld) ↾t ((-π[,]π)
∖ {0}))) |
| 594 | 528, 590,
529, 593 | mp3an 1463 |
. . . . . . . . . . . . . . . . 17
⊢
(((TopOpen‘ℂfld) ↾t ℝ)
↾t ((-π[,]π) ∖ {0})) =
((TopOpen‘ℂfld) ↾t ((-π[,]π)
∖ {0})) |
| 595 | 592, 594 | eqtri 2765 |
. . . . . . . . . . . . . . . 16
⊢
((topGen‘ran (,)) ↾t ((-π[,]π) ∖
{0})) = ((TopOpen‘ℂfld) ↾t
((-π[,]π) ∖ {0})) |
| 596 | | unicntop 24806 |
. . . . . . . . . . . . . . . . . . 19
⊢ ℂ =
∪
(TopOpen‘ℂfld) |
| 597 | 596 | restid 17478 |
. . . . . . . . . . . . . . . . . 18
⊢
((TopOpen‘ℂfld) ∈ Top →
((TopOpen‘ℂfld) ↾t ℂ) =
(TopOpen‘ℂfld)) |
| 598 | 528, 597 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
((TopOpen‘ℂfld) ↾t ℂ) =
(TopOpen‘ℂfld) |
| 599 | 598 | eqcomi 2746 |
. . . . . . . . . . . . . . . 16
⊢
(TopOpen‘ℂfld) =
((TopOpen‘ℂfld) ↾t
ℂ) |
| 600 | 57, 595, 599 | cncfcn 24936 |
. . . . . . . . . . . . . . 15
⊢
((((-π[,]π) ∖ {0}) ⊆ ℂ ∧ ℂ ⊆
ℂ) → (((-π[,]π) ∖ {0})–cn→ℂ) = (((topGen‘ran (,))
↾t ((-π[,]π) ∖ {0})) Cn
(TopOpen‘ℂfld))) |
| 601 | 591, 144,
600 | mp2an 692 |
. . . . . . . . . . . . . 14
⊢
(((-π[,]π) ∖ {0})–cn→ℂ) = (((topGen‘ran (,))
↾t ((-π[,]π) ∖ {0})) Cn
(TopOpen‘ℂfld)) |
| 602 | 588, 601 | eleqtri 2839 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ (𝑠 / (2
· (sin‘(𝑠 /
2))))) ∈ (((topGen‘ran (,)) ↾t ((-π[,]π)
∖ {0})) Cn (TopOpen‘ℂfld)) |
| 603 | | resttopon 23169 |
. . . . . . . . . . . . . . 15
⊢
(((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧
((-π[,]π) ∖ {0}) ⊆ ℝ) → ((topGen‘ran (,))
↾t ((-π[,]π) ∖ {0})) ∈
(TopOn‘((-π[,]π) ∖ {0}))) |
| 604 | 60, 590, 603 | mp2an 692 |
. . . . . . . . . . . . . 14
⊢
((topGen‘ran (,)) ↾t ((-π[,]π) ∖
{0})) ∈ (TopOn‘((-π[,]π) ∖ {0})) |
| 605 | 57 | cnfldtopon 24803 |
. . . . . . . . . . . . . 14
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
| 606 | | cncnp 23288 |
. . . . . . . . . . . . . 14
⊢
((((topGen‘ran (,)) ↾t ((-π[,]π) ∖
{0})) ∈ (TopOn‘((-π[,]π) ∖ {0})) ∧
(TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) →
((𝑠 ∈ ((-π[,]π)
∖ {0}) ↦ (𝑠 /
(2 · (sin‘(𝑠 /
2))))) ∈ (((topGen‘ran (,)) ↾t ((-π[,]π)
∖ {0})) Cn (TopOpen‘ℂfld)) ↔ ((𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ (𝑠 / (2
· (sin‘(𝑠 /
2))))):((-π[,]π) ∖ {0})⟶ℂ ∧ ∀𝑥 ∈ ((-π[,]π) ∖
{0})(𝑠 ∈
((-π[,]π) ∖ {0}) ↦ (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ ((((topGen‘ran (,))
↾t ((-π[,]π) ∖ {0})) CnP
(TopOpen‘ℂfld))‘𝑥)))) |
| 607 | 604, 605,
606 | mp2an 692 |
. . . . . . . . . . . . 13
⊢ ((𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ (𝑠 / (2
· (sin‘(𝑠 /
2))))) ∈ (((topGen‘ran (,)) ↾t ((-π[,]π)
∖ {0})) Cn (TopOpen‘ℂfld)) ↔ ((𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ (𝑠 / (2
· (sin‘(𝑠 /
2))))):((-π[,]π) ∖ {0})⟶ℂ ∧ ∀𝑥 ∈ ((-π[,]π) ∖
{0})(𝑠 ∈
((-π[,]π) ∖ {0}) ↦ (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ ((((topGen‘ran (,))
↾t ((-π[,]π) ∖ {0})) CnP
(TopOpen‘ℂfld))‘𝑥))) |
| 608 | 602, 607 | mpbi 230 |
. . . . . . . . . . . 12
⊢ ((𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ (𝑠 / (2
· (sin‘(𝑠 /
2))))):((-π[,]π) ∖ {0})⟶ℂ ∧ ∀𝑥 ∈ ((-π[,]π) ∖
{0})(𝑠 ∈
((-π[,]π) ∖ {0}) ↦ (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ ((((topGen‘ran (,))
↾t ((-π[,]π) ∖ {0})) CnP
(TopOpen‘ℂfld))‘𝑥)) |
| 609 | 608 | simpri 485 |
. . . . . . . . . . 11
⊢
∀𝑥 ∈
((-π[,]π) ∖ {0})(𝑠 ∈ ((-π[,]π) ∖ {0}) ↦
(𝑠 / (2 ·
(sin‘(𝑠 / 2)))))
∈ ((((topGen‘ran (,)) ↾t ((-π[,]π) ∖
{0})) CnP (TopOpen‘ℂfld))‘𝑥) |
| 610 | 543, 609 | vtoclri 3590 |
. . . . . . . . . 10
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → (𝑠 ∈
((-π[,]π) ∖ {0}) ↦ (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ ((((topGen‘ran (,))
↾t ((-π[,]π) ∖ {0})) CnP
(TopOpen‘ℂfld))‘𝑠)) |
| 611 | 541, 610 | syl 17 |
. . . . . . . . 9
⊢ ((𝑠 ∈ (-π[,]π) ∧
¬ 𝑠 = 0) → (𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ (𝑠 / (2
· (sin‘(𝑠 /
2))))) ∈ ((((topGen‘ran (,)) ↾t ((-π[,]π)
∖ {0})) CnP (TopOpen‘ℂfld))‘𝑠)) |
| 612 | 10 | reseq1i 5993 |
. . . . . . . . . 10
⊢ (𝐾 ↾ ((-π[,]π) ∖
{0})) = ((𝑠 ∈
(-π[,]π) ↦ if(𝑠
= 0, 1, (𝑠 / (2 ·
(sin‘(𝑠 / 2))))))
↾ ((-π[,]π) ∖ {0})) |
| 613 | | difss 4136 |
. . . . . . . . . . 11
⊢
((-π[,]π) ∖ {0}) ⊆ (-π[,]π) |
| 614 | | resmpt 6055 |
. . . . . . . . . . 11
⊢
(((-π[,]π) ∖ {0}) ⊆ (-π[,]π) → ((𝑠 ∈ (-π[,]π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) ↾
((-π[,]π) ∖ {0})) = (𝑠 ∈ ((-π[,]π) ∖ {0}) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))) |
| 615 | 613, 614 | ax-mp 5 |
. . . . . . . . . 10
⊢ ((𝑠 ∈ (-π[,]π) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) ↾
((-π[,]π) ∖ {0})) = (𝑠 ∈ ((-π[,]π) ∖ {0}) ↦
if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) |
| 616 | | eldifn 4132 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → ¬ 𝑠 ∈
{0}) |
| 617 | 616, 502 | sylnib 328 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → ¬ 𝑠 =
0) |
| 618 | 617, 479 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → if(𝑠 = 0, 1,
(𝑠 / (2 ·
(sin‘(𝑠 / 2))))) =
(𝑠 / (2 ·
(sin‘(𝑠 /
2))))) |
| 619 | 618 | mpteq2ia 5245 |
. . . . . . . . . 10
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) ↦ if(𝑠 = 0, 1,
(𝑠 / (2 ·
(sin‘(𝑠 / 2)))))) =
(𝑠 ∈ ((-π[,]π)
∖ {0}) ↦ (𝑠 /
(2 · (sin‘(𝑠 /
2))))) |
| 620 | 612, 615,
619 | 3eqtri 2769 |
. . . . . . . . 9
⊢ (𝐾 ↾ ((-π[,]π) ∖
{0})) = (𝑠 ∈
((-π[,]π) ∖ {0}) ↦ (𝑠 / (2 · (sin‘(𝑠 / 2))))) |
| 621 | | restabs 23173 |
. . . . . . . . . . . 12
⊢
(((topGen‘ran (,)) ∈ Top ∧ ((-π[,]π) ∖ {0})
⊆ (-π[,]π) ∧ (-π[,]π) ∈ V) →
(((topGen‘ran (,)) ↾t (-π[,]π))
↾t ((-π[,]π) ∖ {0})) = ((topGen‘ran (,))
↾t ((-π[,]π) ∖ {0}))) |
| 622 | 453, 613,
454, 621 | mp3an 1463 |
. . . . . . . . . . 11
⊢
(((topGen‘ran (,)) ↾t (-π[,]π))
↾t ((-π[,]π) ∖ {0})) = ((topGen‘ran (,))
↾t ((-π[,]π) ∖ {0})) |
| 623 | 622 | oveq1i 7441 |
. . . . . . . . . 10
⊢
((((topGen‘ran (,)) ↾t (-π[,]π))
↾t ((-π[,]π) ∖ {0})) CnP
(TopOpen‘ℂfld)) = (((topGen‘ran (,))
↾t ((-π[,]π) ∖ {0})) CnP
(TopOpen‘ℂfld)) |
| 624 | 623 | fveq1i 6907 |
. . . . . . . . 9
⊢
(((((topGen‘ran (,)) ↾t (-π[,]π))
↾t ((-π[,]π) ∖ {0})) CnP
(TopOpen‘ℂfld))‘𝑠) = ((((topGen‘ran (,))
↾t ((-π[,]π) ∖ {0})) CnP
(TopOpen‘ℂfld))‘𝑠) |
| 625 | 611, 620,
624 | 3eltr4g 2858 |
. . . . . . . 8
⊢ ((𝑠 ∈ (-π[,]π) ∧
¬ 𝑠 = 0) → (𝐾 ↾ ((-π[,]π) ∖
{0})) ∈ (((((topGen‘ran (,)) ↾t (-π[,]π))
↾t ((-π[,]π) ∖ {0})) CnP
(TopOpen‘ℂfld))‘𝑠)) |
| 626 | 452, 613 | pm3.2i 470 |
. . . . . . . . . 10
⊢
(((topGen‘ran (,)) ↾t (-π[,]π)) ∈ Top
∧ ((-π[,]π) ∖ {0}) ⊆ (-π[,]π)) |
| 627 | 626 | a1i 11 |
. . . . . . . . 9
⊢ ((𝑠 ∈ (-π[,]π) ∧
¬ 𝑠 = 0) →
(((topGen‘ran (,)) ↾t (-π[,]π)) ∈ Top ∧
((-π[,]π) ∖ {0}) ⊆ (-π[,]π))) |
| 628 | | ssdif 4144 |
. . . . . . . . . . . . . 14
⊢
((-π[,]π) ⊆ ℝ → ((-π[,]π) ∖ {0})
⊆ (ℝ ∖ {0})) |
| 629 | 427, 628 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
((-π[,]π) ∖ {0}) ⊆ (ℝ ∖
{0}) |
| 630 | 629, 541 | sselid 3981 |
. . . . . . . . . . . 12
⊢ ((𝑠 ∈ (-π[,]π) ∧
¬ 𝑠 = 0) → 𝑠 ∈ (ℝ ∖
{0})) |
| 631 | | sscon 4143 |
. . . . . . . . . . . . . . . . 17
⊢ ({0}
⊆ (-π[,]π) → (ℝ ∖ (-π[,]π)) ⊆ (ℝ
∖ {0})) |
| 632 | 436, 631 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ (ℝ
∖ (-π[,]π)) ⊆ (ℝ ∖ {0}) |
| 633 | 629, 632 | unssi 4191 |
. . . . . . . . . . . . . . 15
⊢
(((-π[,]π) ∖ {0}) ∪ (ℝ ∖ (-π[,]π)))
⊆ (ℝ ∖ {0}) |
| 634 | | simpr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 ∈ (ℝ ∖ {0})
∧ 𝑠 ∈
(-π[,]π)) → 𝑠
∈ (-π[,]π)) |
| 635 | | eldifn 4132 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ∈ (ℝ ∖ {0})
→ ¬ 𝑠 ∈
{0}) |
| 636 | 635 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 ∈ (ℝ ∖ {0})
∧ 𝑠 ∈
(-π[,]π)) → ¬ 𝑠 ∈ {0}) |
| 637 | 634, 636 | eldifd 3962 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑠 ∈ (ℝ ∖ {0})
∧ 𝑠 ∈
(-π[,]π)) → 𝑠
∈ ((-π[,]π) ∖ {0})) |
| 638 | | elun1 4182 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ ((-π[,]π) ∖
{0}) → 𝑠 ∈
(((-π[,]π) ∖ {0}) ∪ (ℝ ∖
(-π[,]π)))) |
| 639 | 637, 638 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∈ (ℝ ∖ {0})
∧ 𝑠 ∈
(-π[,]π)) → 𝑠
∈ (((-π[,]π) ∖ {0}) ∪ (ℝ ∖
(-π[,]π)))) |
| 640 | | eldifi 4131 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ∈ (ℝ ∖ {0})
→ 𝑠 ∈
ℝ) |
| 641 | 640 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 ∈ (ℝ ∖ {0})
∧ ¬ 𝑠 ∈
(-π[,]π)) → 𝑠
∈ ℝ) |
| 642 | | simpr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 ∈ (ℝ ∖ {0})
∧ ¬ 𝑠 ∈
(-π[,]π)) → ¬ 𝑠 ∈ (-π[,]π)) |
| 643 | 641, 642 | eldifd 3962 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑠 ∈ (ℝ ∖ {0})
∧ ¬ 𝑠 ∈
(-π[,]π)) → 𝑠
∈ (ℝ ∖ (-π[,]π))) |
| 644 | | elun2 4183 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ (ℝ ∖
(-π[,]π)) → 𝑠
∈ (((-π[,]π) ∖ {0}) ∪ (ℝ ∖
(-π[,]π)))) |
| 645 | 643, 644 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∈ (ℝ ∖ {0})
∧ ¬ 𝑠 ∈
(-π[,]π)) → 𝑠
∈ (((-π[,]π) ∖ {0}) ∪ (ℝ ∖
(-π[,]π)))) |
| 646 | 639, 645 | pm2.61dan 813 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ (ℝ ∖ {0})
→ 𝑠 ∈
(((-π[,]π) ∖ {0}) ∪ (ℝ ∖
(-π[,]π)))) |
| 647 | 646 | ssriv 3987 |
. . . . . . . . . . . . . . 15
⊢ (ℝ
∖ {0}) ⊆ (((-π[,]π) ∖ {0}) ∪ (ℝ ∖
(-π[,]π))) |
| 648 | 633, 647 | eqssi 4000 |
. . . . . . . . . . . . . 14
⊢
(((-π[,]π) ∖ {0}) ∪ (ℝ ∖ (-π[,]π))) =
(ℝ ∖ {0}) |
| 649 | 648 | fveq2i 6909 |
. . . . . . . . . . . . 13
⊢
((int‘(topGen‘ran (,)))‘(((-π[,]π) ∖ {0})
∪ (ℝ ∖ (-π[,]π)))) = ((int‘(topGen‘ran
(,)))‘(ℝ ∖ {0})) |
| 650 | 61 | cldopn 23039 |
. . . . . . . . . . . . . . 15
⊢ ({0}
∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ {0}) ∈
(topGen‘ran (,))) |
| 651 | 59, 650 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ (ℝ
∖ {0}) ∈ (topGen‘ran (,)) |
| 652 | | isopn3i 23090 |
. . . . . . . . . . . . . 14
⊢
(((topGen‘ran (,)) ∈ Top ∧ (ℝ ∖ {0}) ∈
(topGen‘ran (,))) → ((int‘(topGen‘ran
(,)))‘(ℝ ∖ {0})) = (ℝ ∖ {0})) |
| 653 | 453, 651,
652 | mp2an 692 |
. . . . . . . . . . . . 13
⊢
((int‘(topGen‘ran (,)))‘(ℝ ∖ {0})) =
(ℝ ∖ {0}) |
| 654 | 649, 653 | eqtri 2765 |
. . . . . . . . . . . 12
⊢
((int‘(topGen‘ran (,)))‘(((-π[,]π) ∖ {0})
∪ (ℝ ∖ (-π[,]π)))) = (ℝ ∖
{0}) |
| 655 | 630, 654 | eleqtrrdi 2852 |
. . . . . . . . . . 11
⊢ ((𝑠 ∈ (-π[,]π) ∧
¬ 𝑠 = 0) → 𝑠 ∈
((int‘(topGen‘ran (,)))‘(((-π[,]π) ∖ {0}) ∪
(ℝ ∖ (-π[,]π))))) |
| 656 | 655, 537 | elind 4200 |
. . . . . . . . . 10
⊢ ((𝑠 ∈ (-π[,]π) ∧
¬ 𝑠 = 0) → 𝑠 ∈
(((int‘(topGen‘ran (,)))‘(((-π[,]π) ∖ {0}) ∪
(ℝ ∖ (-π[,]π)))) ∩ (-π[,]π))) |
| 657 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
((topGen‘ran (,)) ↾t (-π[,]π)) =
((topGen‘ran (,)) ↾t (-π[,]π)) |
| 658 | 61, 657 | restntr 23190 |
. . . . . . . . . . 11
⊢
(((topGen‘ran (,)) ∈ Top ∧ (-π[,]π) ⊆ ℝ
∧ ((-π[,]π) ∖ {0}) ⊆ (-π[,]π)) →
((int‘((topGen‘ran (,)) ↾t
(-π[,]π)))‘((-π[,]π) ∖ {0})) =
(((int‘(topGen‘ran (,)))‘(((-π[,]π) ∖ {0}) ∪
(ℝ ∖ (-π[,]π)))) ∩ (-π[,]π))) |
| 659 | 453, 427,
613, 658 | mp3an 1463 |
. . . . . . . . . 10
⊢
((int‘((topGen‘ran (,)) ↾t
(-π[,]π)))‘((-π[,]π) ∖ {0})) =
(((int‘(topGen‘ran (,)))‘(((-π[,]π) ∖ {0}) ∪
(ℝ ∖ (-π[,]π)))) ∩ (-π[,]π)) |
| 660 | 656, 659 | eleqtrrdi 2852 |
. . . . . . . . 9
⊢ ((𝑠 ∈ (-π[,]π) ∧
¬ 𝑠 = 0) → 𝑠 ∈
((int‘((topGen‘ran (,)) ↾t
(-π[,]π)))‘((-π[,]π) ∖ {0}))) |
| 661 | 14 | a1i 11 |
. . . . . . . . 9
⊢ ((𝑠 ∈ (-π[,]π) ∧
¬ 𝑠 = 0) → 𝐾:(-π[,]π)⟶ℂ) |
| 662 | 451 | toponunii 22922 |
. . . . . . . . . 10
⊢
(-π[,]π) = ∪ ((topGen‘ran (,))
↾t (-π[,]π)) |
| 663 | 662, 596 | cnprest 23297 |
. . . . . . . . 9
⊢
(((((topGen‘ran (,)) ↾t (-π[,]π)) ∈
Top ∧ ((-π[,]π) ∖ {0}) ⊆ (-π[,]π)) ∧ (𝑠 ∈
((int‘((topGen‘ran (,)) ↾t
(-π[,]π)))‘((-π[,]π) ∖ {0})) ∧ 𝐾:(-π[,]π)⟶ℂ)) →
(𝐾 ∈
((((topGen‘ran (,)) ↾t (-π[,]π)) CnP
(TopOpen‘ℂfld))‘𝑠) ↔ (𝐾 ↾ ((-π[,]π) ∖ {0}))
∈ (((((topGen‘ran (,)) ↾t (-π[,]π))
↾t ((-π[,]π) ∖ {0})) CnP
(TopOpen‘ℂfld))‘𝑠))) |
| 664 | 627, 660,
661, 663 | syl12anc 837 |
. . . . . . . 8
⊢ ((𝑠 ∈ (-π[,]π) ∧
¬ 𝑠 = 0) → (𝐾 ∈ ((((topGen‘ran
(,)) ↾t (-π[,]π)) CnP
(TopOpen‘ℂfld))‘𝑠) ↔ (𝐾 ↾ ((-π[,]π) ∖ {0}))
∈ (((((topGen‘ran (,)) ↾t (-π[,]π))
↾t ((-π[,]π) ∖ {0})) CnP
(TopOpen‘ℂfld))‘𝑠))) |
| 665 | 625, 664 | mpbird 257 |
. . . . . . 7
⊢ ((𝑠 ∈ (-π[,]π) ∧
¬ 𝑠 = 0) → 𝐾 ∈ ((((topGen‘ran
(,)) ↾t (-π[,]π)) CnP
(TopOpen‘ℂfld))‘𝑠)) |
| 666 | 536, 665 | pm2.61dan 813 |
. . . . . 6
⊢ (𝑠 ∈ (-π[,]π) →
𝐾 ∈
((((topGen‘ran (,)) ↾t (-π[,]π)) CnP
(TopOpen‘ℂfld))‘𝑠)) |
| 667 | 666 | rgen 3063 |
. . . . 5
⊢
∀𝑠 ∈
(-π[,]π)𝐾 ∈
((((topGen‘ran (,)) ↾t (-π[,]π)) CnP
(TopOpen‘ℂfld))‘𝑠) |
| 668 | | cncnp 23288 |
. . . . . 6
⊢
((((topGen‘ran (,)) ↾t (-π[,]π)) ∈
(TopOn‘(-π[,]π)) ∧ (TopOpen‘ℂfld) ∈
(TopOn‘ℂ)) → (𝐾 ∈ (((topGen‘ran (,))
↾t (-π[,]π)) Cn (TopOpen‘ℂfld))
↔ (𝐾:(-π[,]π)⟶ℂ ∧
∀𝑠 ∈
(-π[,]π)𝐾 ∈
((((topGen‘ran (,)) ↾t (-π[,]π)) CnP
(TopOpen‘ℂfld))‘𝑠)))) |
| 669 | 451, 605,
668 | mp2an 692 |
. . . . 5
⊢ (𝐾 ∈ (((topGen‘ran (,))
↾t (-π[,]π)) Cn (TopOpen‘ℂfld))
↔ (𝐾:(-π[,]π)⟶ℂ ∧
∀𝑠 ∈
(-π[,]π)𝐾 ∈
((((topGen‘ran (,)) ↾t (-π[,]π)) CnP
(TopOpen‘ℂfld))‘𝑠))) |
| 670 | 14, 667, 669 | mpbir2an 711 |
. . . 4
⊢ 𝐾 ∈ (((topGen‘ran (,))
↾t (-π[,]π)) Cn
(TopOpen‘ℂfld)) |
| 671 | 57, 532, 599 | cncfcn 24936 |
. . . . . 6
⊢
(((-π[,]π) ⊆ ℂ ∧ ℂ ⊆ ℂ) →
((-π[,]π)–cn→ℂ) =
(((topGen‘ran (,)) ↾t (-π[,]π)) Cn
(TopOpen‘ℂfld))) |
| 672 | 517, 144,
671 | mp2an 692 |
. . . . 5
⊢
((-π[,]π)–cn→ℂ) = (((topGen‘ran (,))
↾t (-π[,]π)) Cn
(TopOpen‘ℂfld)) |
| 673 | 672 | eqcomi 2746 |
. . . 4
⊢
(((topGen‘ran (,)) ↾t (-π[,]π)) Cn
(TopOpen‘ℂfld)) = ((-π[,]π)–cn→ℂ) |
| 674 | 670, 673 | eleqtri 2839 |
. . 3
⊢ 𝐾 ∈
((-π[,]π)–cn→ℂ) |
| 675 | | cncfcdm 24924 |
. . 3
⊢ ((ℝ
⊆ ℂ ∧ 𝐾
∈ ((-π[,]π)–cn→ℂ)) → (𝐾 ∈ ((-π[,]π)–cn→ℝ) ↔ 𝐾:(-π[,]π)⟶ℝ)) |
| 676 | 12, 674, 675 | mp2an 692 |
. 2
⊢ (𝐾 ∈
((-π[,]π)–cn→ℝ)
↔ 𝐾:(-π[,]π)⟶ℝ) |
| 677 | 11, 676 | mpbir 231 |
1
⊢ 𝐾 ∈
((-π[,]π)–cn→ℝ) |