Proof of Theorem dirkercncflem2
| Step | Hyp | Ref
| Expression |
| 1 | | difss 4136 |
. . . . 5
⊢ ((𝐴(,)𝐵) ∖ {𝑌}) ⊆ (𝐴(,)𝐵) |
| 2 | | ioossre 13448 |
. . . . 5
⊢ (𝐴(,)𝐵) ⊆ ℝ |
| 3 | 1, 2 | sstri 3993 |
. . . 4
⊢ ((𝐴(,)𝐵) ∖ {𝑌}) ⊆ ℝ |
| 4 | 3 | a1i 11 |
. . 3
⊢ (𝜑 → ((𝐴(,)𝐵) ∖ {𝑌}) ⊆ ℝ) |
| 5 | | dirkercncflem2.n |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 6 | 5 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → 𝑁 ∈ ℕ) |
| 7 | 6 | nnred 12281 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → 𝑁 ∈ ℝ) |
| 8 | | halfre 12480 |
. . . . . . . 8
⊢ (1 / 2)
∈ ℝ |
| 9 | 8 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (1 / 2) ∈
ℝ) |
| 10 | 7, 9 | readdcld 11290 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝑁 + (1 / 2)) ∈ ℝ) |
| 11 | 4 | sselda 3983 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → 𝑦 ∈ ℝ) |
| 12 | 10, 11 | remulcld 11291 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝑁 + (1 / 2)) · 𝑦) ∈ ℝ) |
| 13 | 12 | resincld 16179 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (sin‘((𝑁 + (1 / 2)) · 𝑦)) ∈ ℝ) |
| 14 | | dirkercncflem2.f |
. . . 4
⊢ 𝐹 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))) |
| 15 | 13, 14 | fmptd 7134 |
. . 3
⊢ (𝜑 → 𝐹:((𝐴(,)𝐵) ∖ {𝑌})⟶ℝ) |
| 16 | | 2re 12340 |
. . . . . . 7
⊢ 2 ∈
ℝ |
| 17 | | pire 26500 |
. . . . . . 7
⊢ π
∈ ℝ |
| 18 | 16, 17 | remulcli 11277 |
. . . . . 6
⊢ (2
· π) ∈ ℝ |
| 19 | 18 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (2 · π) ∈
ℝ) |
| 20 | 11 | rehalfcld 12513 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝑦 / 2) ∈ ℝ) |
| 21 | 20 | resincld 16179 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (sin‘(𝑦 / 2)) ∈ ℝ) |
| 22 | 19, 21 | remulcld 11291 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((2 · π) ·
(sin‘(𝑦 / 2))) ∈
ℝ) |
| 23 | | dirkercncflem2.g |
. . . 4
⊢ 𝐺 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((2 · π) ·
(sin‘(𝑦 /
2)))) |
| 24 | 22, 23 | fmptd 7134 |
. . 3
⊢ (𝜑 → 𝐺:((𝐴(,)𝐵) ∖ {𝑌})⟶ℝ) |
| 25 | | iooretop 24786 |
. . . 4
⊢ (𝐴(,)𝐵) ∈ (topGen‘ran
(,)) |
| 26 | 25 | a1i 11 |
. . 3
⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran
(,))) |
| 27 | | dirkercncflem2.y |
. . 3
⊢ (𝜑 → 𝑌 ∈ (𝐴(,)𝐵)) |
| 28 | | eqid 2737 |
. . 3
⊢ ((𝐴(,)𝐵) ∖ {𝑌}) = ((𝐴(,)𝐵) ∖ {𝑌}) |
| 29 | 14 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 𝐹 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦)))) |
| 30 | 29 | oveq2d 7447 |
. . . . . . 7
⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))))) |
| 31 | | resmpt 6055 |
. . . . . . . . . . . 12
⊢ (((𝐴(,)𝐵) ∖ {𝑌}) ⊆ ℝ → ((𝑦 ∈ ℝ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦))) ↾
((𝐴(,)𝐵) ∖ {𝑌})) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦)))) |
| 32 | 3, 31 | ax-mp 5 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ ℝ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦))) ↾
((𝐴(,)𝐵) ∖ {𝑌})) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))) |
| 33 | 32 | eqcomi 2746 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))) = ((𝑦 ∈ ℝ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))) ↾ ((𝐴(,)𝐵) ∖ {𝑌})) |
| 34 | 33 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))) = ((𝑦 ∈ ℝ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))) ↾ ((𝐴(,)𝐵) ∖ {𝑌}))) |
| 35 | 34 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝜑 → (ℝ D (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦)))) = (ℝ D ((𝑦 ∈ ℝ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))) ↾ ((𝐴(,)𝐵) ∖ {𝑌})))) |
| 36 | | ax-resscn 11212 |
. . . . . . . . . 10
⊢ ℝ
⊆ ℂ |
| 37 | 36 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ℝ ⊆
ℂ) |
| 38 | 5 | nncnd 12282 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 39 | | halfcn 12481 |
. . . . . . . . . . . . . . 15
⊢ (1 / 2)
∈ ℂ |
| 40 | 39 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1 / 2) ∈
ℂ) |
| 41 | 38, 40 | addcld 11280 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 + (1 / 2)) ∈ ℂ) |
| 42 | 41 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑁 + (1 / 2)) ∈ ℂ) |
| 43 | 37 | sselda 3983 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ) |
| 44 | 42, 43 | mulcld 11281 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑁 + (1 / 2)) · 𝑦) ∈ ℂ) |
| 45 | 44 | sincld 16166 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (sin‘((𝑁 + (1 / 2)) · 𝑦)) ∈
ℂ) |
| 46 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℝ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦))) = (𝑦 ∈ ℝ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦))) |
| 47 | 45, 46 | fmptd 7134 |
. . . . . . . . 9
⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))):ℝ⟶ℂ) |
| 48 | | ssid 4006 |
. . . . . . . . . . 11
⊢ ℝ
⊆ ℝ |
| 49 | 48, 3 | pm3.2i 470 |
. . . . . . . . . 10
⊢ (ℝ
⊆ ℝ ∧ ((𝐴(,)𝐵) ∖ {𝑌}) ⊆ ℝ) |
| 50 | 49 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (ℝ ⊆ ℝ
∧ ((𝐴(,)𝐵) ∖ {𝑌}) ⊆ ℝ)) |
| 51 | | eqid 2737 |
. . . . . . . . . 10
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
| 52 | | tgioo4 24826 |
. . . . . . . . . 10
⊢
(topGen‘ran (,)) = ((TopOpen‘ℂfld)
↾t ℝ) |
| 53 | 51, 52 | dvres 25946 |
. . . . . . . . 9
⊢
(((ℝ ⊆ ℂ ∧ (𝑦 ∈ ℝ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))):ℝ⟶ℂ)
∧ (ℝ ⊆ ℝ ∧ ((𝐴(,)𝐵) ∖ {𝑌}) ⊆ ℝ)) → (ℝ D
((𝑦 ∈ ℝ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦))) ↾
((𝐴(,)𝐵) ∖ {𝑌}))) = ((ℝ D (𝑦 ∈ ℝ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦)))) ↾
((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∖ {𝑌})))) |
| 54 | 37, 47, 50, 53 | syl21anc 838 |
. . . . . . . 8
⊢ (𝜑 → (ℝ D ((𝑦 ∈ ℝ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦))) ↾
((𝐴(,)𝐵) ∖ {𝑌}))) = ((ℝ D (𝑦 ∈ ℝ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦)))) ↾
((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∖ {𝑌})))) |
| 55 | | retop 24782 |
. . . . . . . . . . 11
⊢
(topGen‘ran (,)) ∈ Top |
| 56 | | rehaus 24820 |
. . . . . . . . . . . . 13
⊢
(topGen‘ran (,)) ∈ Haus |
| 57 | 27 | elioored 45562 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑌 ∈ ℝ) |
| 58 | | uniretop 24783 |
. . . . . . . . . . . . . 14
⊢ ℝ =
∪ (topGen‘ran (,)) |
| 59 | 58 | sncld 23379 |
. . . . . . . . . . . . 13
⊢
(((topGen‘ran (,)) ∈ Haus ∧ 𝑌 ∈ ℝ) → {𝑌} ∈ (Clsd‘(topGen‘ran
(,)))) |
| 60 | 56, 57, 59 | sylancr 587 |
. . . . . . . . . . . 12
⊢ (𝜑 → {𝑌} ∈ (Clsd‘(topGen‘ran
(,)))) |
| 61 | 58 | difopn 23042 |
. . . . . . . . . . . 12
⊢ (((𝐴(,)𝐵) ∈ (topGen‘ran (,)) ∧ {𝑌} ∈
(Clsd‘(topGen‘ran (,)))) → ((𝐴(,)𝐵) ∖ {𝑌}) ∈ (topGen‘ran
(,))) |
| 62 | 25, 60, 61 | sylancr 587 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐴(,)𝐵) ∖ {𝑌}) ∈ (topGen‘ran
(,))) |
| 63 | | isopn3i 23090 |
. . . . . . . . . . 11
⊢
(((topGen‘ran (,)) ∈ Top ∧ ((𝐴(,)𝐵) ∖ {𝑌}) ∈ (topGen‘ran (,))) →
((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∖ {𝑌})) = ((𝐴(,)𝐵) ∖ {𝑌})) |
| 64 | 55, 62, 63 | sylancr 587 |
. . . . . . . . . 10
⊢ (𝜑 →
((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∖ {𝑌})) = ((𝐴(,)𝐵) ∖ {𝑌})) |
| 65 | 64 | reseq2d 5997 |
. . . . . . . . 9
⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦)))) ↾
((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∖ {𝑌}))) = ((ℝ D (𝑦 ∈ ℝ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦)))) ↾ ((𝐴(,)𝐵) ∖ {𝑌}))) |
| 66 | | reelprrecn 11247 |
. . . . . . . . . . . . 13
⊢ ℝ
∈ {ℝ, ℂ} |
| 67 | 66 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → ℝ ∈ {ℝ,
ℂ}) |
| 68 | 41 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 + (1 / 2)) ∈ ℂ) |
| 69 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ) |
| 70 | 68, 69 | mulcld 11281 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑁 + (1 / 2)) · 𝑦) ∈ ℂ) |
| 71 | 70 | sincld 16166 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (sin‘((𝑁 + (1 / 2)) · 𝑦)) ∈
ℂ) |
| 72 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℂ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦))) = (𝑦 ∈ ℂ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦))) |
| 73 | 71, 72 | fmptd 7134 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))):ℂ⟶ℂ) |
| 74 | | ssid 4006 |
. . . . . . . . . . . . 13
⊢ ℂ
⊆ ℂ |
| 75 | 74 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → ℂ ⊆
ℂ) |
| 76 | | dvsinax 45928 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 + (1 / 2)) ∈ ℂ
→ (ℂ D (𝑦 ∈
ℂ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦)))) = (𝑦 ∈ ℂ ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))))) |
| 77 | 41, 76 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦)))) = (𝑦 ∈ ℂ ↦ ((𝑁 + (1 / 2)) ·
(cos‘((𝑁 + (1 / 2))
· 𝑦))))) |
| 78 | 77 | dmeqd 5916 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → dom (ℂ D (𝑦 ∈ ℂ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦)))) = dom (𝑦 ∈ ℂ ↦ ((𝑁 + (1 / 2)) ·
(cos‘((𝑁 + (1 / 2))
· 𝑦))))) |
| 79 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ℂ ↦ ((𝑁 + (1 / 2)) ·
(cos‘((𝑁 + (1 / 2))
· 𝑦)))) = (𝑦 ∈ ℂ ↦ ((𝑁 + (1 / 2)) ·
(cos‘((𝑁 + (1 / 2))
· 𝑦)))) |
| 80 | 70 | coscld 16167 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (cos‘((𝑁 + (1 / 2)) · 𝑦)) ∈
ℂ) |
| 81 | 68, 80 | mulcld 11281 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))) ∈
ℂ) |
| 82 | 79, 81 | dmmptd 6713 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → dom (𝑦 ∈ ℂ ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦)))) = ℂ) |
| 83 | 78, 82 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ (𝜑 → dom (ℂ D (𝑦 ∈ ℂ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦)))) =
ℂ) |
| 84 | 36, 83 | sseqtrrid 4027 |
. . . . . . . . . . . 12
⊢ (𝜑 → ℝ ⊆ dom
(ℂ D (𝑦 ∈
ℂ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))))) |
| 85 | | dvres3 25948 |
. . . . . . . . . . . 12
⊢
(((ℝ ∈ {ℝ, ℂ} ∧ (𝑦 ∈ ℂ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))):ℂ⟶ℂ)
∧ (ℂ ⊆ ℂ ∧ ℝ ⊆ dom (ℂ D (𝑦 ∈ ℂ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦)))))) →
(ℝ D ((𝑦 ∈
ℂ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))) ↾ ℝ)) = ((ℂ D (𝑦 ∈ ℂ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦)))) ↾
ℝ)) |
| 86 | 67, 73, 75, 84, 85 | syl22anc 839 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D ((𝑦 ∈ ℂ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦))) ↾
ℝ)) = ((ℂ D (𝑦
∈ ℂ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦)))) ↾ ℝ)) |
| 87 | | resmpt 6055 |
. . . . . . . . . . . . 13
⊢ (ℝ
⊆ ℂ → ((𝑦
∈ ℂ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))) ↾ ℝ) = (𝑦 ∈ ℝ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦)))) |
| 88 | 36, 87 | mp1i 13 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))) ↾ ℝ) = (𝑦 ∈ ℝ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦)))) |
| 89 | 88 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D ((𝑦 ∈ ℂ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦))) ↾
ℝ)) = (ℝ D (𝑦
∈ ℝ ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦))))) |
| 90 | 77 | reseq1d 5996 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((ℂ D (𝑦 ∈ ℂ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦)))) ↾
ℝ) = ((𝑦 ∈
ℂ ↦ ((𝑁 + (1 /
2)) · (cos‘((𝑁
+ (1 / 2)) · 𝑦))))
↾ ℝ)) |
| 91 | | resmpt 6055 |
. . . . . . . . . . . . 13
⊢ (ℝ
⊆ ℂ → ((𝑦
∈ ℂ ↦ ((𝑁
+ (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦)))) ↾ ℝ) = (𝑦 ∈ ℝ ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))))) |
| 92 | 36, 91 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ ℂ ↦ ((𝑁 + (1 / 2)) ·
(cos‘((𝑁 + (1 / 2))
· 𝑦)))) ↾
ℝ) = (𝑦 ∈
ℝ ↦ ((𝑁 + (1 /
2)) · (cos‘((𝑁
+ (1 / 2)) · 𝑦)))) |
| 93 | 90, 92 | eqtrdi 2793 |
. . . . . . . . . . 11
⊢ (𝜑 → ((ℂ D (𝑦 ∈ ℂ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦)))) ↾
ℝ) = (𝑦 ∈
ℝ ↦ ((𝑁 + (1 /
2)) · (cos‘((𝑁
+ (1 / 2)) · 𝑦))))) |
| 94 | 86, 89, 93 | 3eqtr3d 2785 |
. . . . . . . . . 10
⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦)))) = (𝑦 ∈ ℝ ↦ ((𝑁 + (1 / 2)) ·
(cos‘((𝑁 + (1 / 2))
· 𝑦))))) |
| 95 | 94 | reseq1d 5996 |
. . . . . . . . 9
⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦)))) ↾
((𝐴(,)𝐵) ∖ {𝑌})) = ((𝑦 ∈ ℝ ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦)))) ↾ ((𝐴(,)𝐵) ∖ {𝑌}))) |
| 96 | | resmpt 6055 |
. . . . . . . . . 10
⊢ (((𝐴(,)𝐵) ∖ {𝑌}) ⊆ ℝ → ((𝑦 ∈ ℝ ↦ ((𝑁 + (1 / 2)) ·
(cos‘((𝑁 + (1 / 2))
· 𝑦)))) ↾
((𝐴(,)𝐵) ∖ {𝑌})) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))))) |
| 97 | 3, 96 | mp1i 13 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦)))) ↾ ((𝐴(,)𝐵) ∖ {𝑌})) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))))) |
| 98 | 65, 95, 97 | 3eqtrd 2781 |
. . . . . . . 8
⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦
(sin‘((𝑁 + (1 / 2))
· 𝑦)))) ↾
((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∖ {𝑌}))) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))))) |
| 99 | 35, 54, 98 | 3eqtrd 2781 |
. . . . . . 7
⊢ (𝜑 → (ℝ D (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦)))) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))))) |
| 100 | | dirkercncflem2.h |
. . . . . . . . 9
⊢ 𝐻 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦)))) |
| 101 | 100 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 𝐻 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))))) |
| 102 | 101 | eqcomd 2743 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦)))) = 𝐻) |
| 103 | 30, 99, 102 | 3eqtrd 2781 |
. . . . . 6
⊢ (𝜑 → (ℝ D 𝐹) = 𝐻) |
| 104 | 103 | dmeqd 5916 |
. . . . 5
⊢ (𝜑 → dom (ℝ D 𝐹) = dom 𝐻) |
| 105 | 11 | recnd 11289 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → 𝑦 ∈ ℂ) |
| 106 | 105, 81 | syldan 591 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))) ∈
ℂ) |
| 107 | 100, 106 | dmmptd 6713 |
. . . . 5
⊢ (𝜑 → dom 𝐻 = ((𝐴(,)𝐵) ∖ {𝑌})) |
| 108 | 104, 107 | eqtr2d 2778 |
. . . 4
⊢ (𝜑 → ((𝐴(,)𝐵) ∖ {𝑌}) = dom (ℝ D 𝐹)) |
| 109 | | eqimss 4042 |
. . . 4
⊢ (((𝐴(,)𝐵) ∖ {𝑌}) = dom (ℝ D 𝐹) → ((𝐴(,)𝐵) ∖ {𝑌}) ⊆ dom (ℝ D 𝐹)) |
| 110 | 108, 109 | syl 17 |
. . 3
⊢ (𝜑 → ((𝐴(,)𝐵) ∖ {𝑌}) ⊆ dom (ℝ D 𝐹)) |
| 111 | | dirkercncflem2.i |
. . . . . . . 8
⊢ 𝐼 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (π · (cos‘(𝑦 / 2)))) |
| 112 | 111 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 𝐼 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (π · (cos‘(𝑦 / 2))))) |
| 113 | | resmpt 6055 |
. . . . . . . . . . . . 13
⊢ (((𝐴(,)𝐵) ∖ {𝑌}) ⊆ ℝ → ((𝑦 ∈ ℝ ↦ ((2
· π) · (sin‘(𝑦 / 2)))) ↾ ((𝐴(,)𝐵) ∖ {𝑌})) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((2 · π) ·
(sin‘(𝑦 /
2))))) |
| 114 | 3, 113 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ ℝ ↦ ((2
· π) · (sin‘(𝑦 / 2)))) ↾ ((𝐴(,)𝐵) ∖ {𝑌})) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((2 · π) ·
(sin‘(𝑦 /
2)))) |
| 115 | 114 | eqcomi 2746 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((2 · π) ·
(sin‘(𝑦 / 2)))) =
((𝑦 ∈ ℝ ↦
((2 · π) · (sin‘(𝑦 / 2)))) ↾ ((𝐴(,)𝐵) ∖ {𝑌})) |
| 116 | 115 | oveq2i 7442 |
. . . . . . . . . 10
⊢ (ℝ
D (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((2 · π) ·
(sin‘(𝑦 / 2))))) =
(ℝ D ((𝑦 ∈
ℝ ↦ ((2 · π) · (sin‘(𝑦 / 2)))) ↾ ((𝐴(,)𝐵) ∖ {𝑌}))) |
| 117 | 116 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (ℝ D (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((2 · π) ·
(sin‘(𝑦 / 2))))) =
(ℝ D ((𝑦 ∈
ℝ ↦ ((2 · π) · (sin‘(𝑦 / 2)))) ↾ ((𝐴(,)𝐵) ∖ {𝑌})))) |
| 118 | | 2cn 12341 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℂ |
| 119 | | picn 26501 |
. . . . . . . . . . . . . 14
⊢ π
∈ ℂ |
| 120 | 118, 119 | mulcli 11268 |
. . . . . . . . . . . . 13
⊢ (2
· π) ∈ ℂ |
| 121 | 120 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (2 · π)
∈ ℂ) |
| 122 | 43 | halfcld 12511 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 / 2) ∈ ℂ) |
| 123 | 122 | sincld 16166 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (sin‘(𝑦 / 2)) ∈
ℂ) |
| 124 | 121, 123 | mulcld 11281 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((2 · π)
· (sin‘(𝑦 /
2))) ∈ ℂ) |
| 125 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℝ ↦ ((2
· π) · (sin‘(𝑦 / 2)))) = (𝑦 ∈ ℝ ↦ ((2 · π)
· (sin‘(𝑦 /
2)))) |
| 126 | 124, 125 | fmptd 7134 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑦 ∈ ℝ ↦ ((2 · π)
· (sin‘(𝑦 /
2)))):ℝ⟶ℂ) |
| 127 | 51, 52 | dvres 25946 |
. . . . . . . . . 10
⊢
(((ℝ ⊆ ℂ ∧ (𝑦 ∈ ℝ ↦ ((2 · π)
· (sin‘(𝑦 /
2)))):ℝ⟶ℂ) ∧ (ℝ ⊆ ℝ ∧ ((𝐴(,)𝐵) ∖ {𝑌}) ⊆ ℝ)) → (ℝ D
((𝑦 ∈ ℝ ↦
((2 · π) · (sin‘(𝑦 / 2)))) ↾ ((𝐴(,)𝐵) ∖ {𝑌}))) = ((ℝ D (𝑦 ∈ ℝ ↦ ((2 · π)
· (sin‘(𝑦 /
2))))) ↾ ((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∖ {𝑌})))) |
| 128 | 37, 126, 50, 127 | syl21anc 838 |
. . . . . . . . 9
⊢ (𝜑 → (ℝ D ((𝑦 ∈ ℝ ↦ ((2
· π) · (sin‘(𝑦 / 2)))) ↾ ((𝐴(,)𝐵) ∖ {𝑌}))) = ((ℝ D (𝑦 ∈ ℝ ↦ ((2 · π)
· (sin‘(𝑦 /
2))))) ↾ ((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∖ {𝑌})))) |
| 129 | 64 | reseq2d 5997 |
. . . . . . . . . 10
⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ ((2
· π) · (sin‘(𝑦 / 2))))) ↾
((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∖ {𝑌}))) = ((ℝ D (𝑦 ∈ ℝ ↦ ((2 · π)
· (sin‘(𝑦 /
2))))) ↾ ((𝐴(,)𝐵) ∖ {𝑌}))) |
| 130 | 36 | sseli 3979 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ ℝ → 𝑦 ∈
ℂ) |
| 131 | | 1cnd 11256 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℂ → 1 ∈
ℂ) |
| 132 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℂ → 2 ∈
ℂ) |
| 133 | | id 22 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℂ → 𝑦 ∈
ℂ) |
| 134 | | 2ne0 12370 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ≠
0 |
| 135 | 134 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℂ → 2 ≠
0) |
| 136 | 131, 132,
133, 135 | div13d 12067 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ℂ → ((1 / 2)
· 𝑦) = ((𝑦 / 2) ·
1)) |
| 137 | | halfcl 12491 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℂ → (𝑦 / 2) ∈
ℂ) |
| 138 | 137 | mulridd 11278 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ℂ → ((𝑦 / 2) · 1) = (𝑦 / 2)) |
| 139 | 136, 138 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ ℂ → ((1 / 2)
· 𝑦) = (𝑦 / 2)) |
| 140 | 139 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ℂ →
(sin‘((1 / 2) · 𝑦)) = (sin‘(𝑦 / 2))) |
| 141 | 140 | oveq2d 7447 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ ℂ → ((2
· π) · (sin‘((1 / 2) · 𝑦))) = ((2 · π) ·
(sin‘(𝑦 /
2)))) |
| 142 | 141 | eqcomd 2743 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ ℂ → ((2
· π) · (sin‘(𝑦 / 2))) = ((2 · π) ·
(sin‘((1 / 2) · 𝑦)))) |
| 143 | 130, 142 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ℝ → ((2
· π) · (sin‘(𝑦 / 2))) = ((2 · π) ·
(sin‘((1 / 2) · 𝑦)))) |
| 144 | 143 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((2 · π)
· (sin‘(𝑦 /
2))) = ((2 · π) · (sin‘((1 / 2) · 𝑦)))) |
| 145 | 144 | mpteq2dva 5242 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑦 ∈ ℝ ↦ ((2 · π)
· (sin‘(𝑦 /
2)))) = (𝑦 ∈ ℝ
↦ ((2 · π) · (sin‘((1 / 2) · 𝑦))))) |
| 146 | 145 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ ((2
· π) · (sin‘(𝑦 / 2))))) = (ℝ D (𝑦 ∈ ℝ ↦ ((2 · π)
· (sin‘((1 / 2) · 𝑦)))))) |
| 147 | 120 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (2 · π)
∈ ℂ) |
| 148 | 39 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (1 / 2) ∈
ℂ) |
| 149 | 148, 69 | mulcld 11281 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((1 / 2) ·
𝑦) ∈
ℂ) |
| 150 | 149 | sincld 16166 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (sin‘((1 / 2)
· 𝑦)) ∈
ℂ) |
| 151 | 147, 150 | mulcld 11281 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((2 · π)
· (sin‘((1 / 2) · 𝑦))) ∈ ℂ) |
| 152 | | eqid 2737 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ ℂ ↦ ((2
· π) · (sin‘((1 / 2) · 𝑦)))) = (𝑦 ∈ ℂ ↦ ((2 · π)
· (sin‘((1 / 2) · 𝑦)))) |
| 153 | 151, 152 | fmptd 7134 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑦 ∈ ℂ ↦ ((2 · π)
· (sin‘((1 / 2) · 𝑦)))):ℂ⟶ℂ) |
| 154 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 2 ∈
ℂ) |
| 155 | 119 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → π ∈
ℂ) |
| 156 | 154, 155 | mulcld 11281 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (2 · π) ∈
ℂ) |
| 157 | | dvasinbx 45935 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((2
· π) ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (ℂ D
(𝑦 ∈ ℂ ↦
((2 · π) · (sin‘((1 / 2) · 𝑦))))) = (𝑦 ∈ ℂ ↦ (((2 · π)
· (1 / 2)) · (cos‘((1 / 2) · 𝑦))))) |
| 158 | 156, 39, 157 | sylancl 586 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ ((2
· π) · (sin‘((1 / 2) · 𝑦))))) = (𝑦 ∈ ℂ ↦ (((2 · π)
· (1 / 2)) · (cos‘((1 / 2) · 𝑦))))) |
| 159 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 2 ∈
ℂ) |
| 160 | 119 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → π ∈
ℂ) |
| 161 | 159, 160,
148 | mul32d 11471 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((2 · π)
· (1 / 2)) = ((2 · (1 / 2)) · π)) |
| 162 | 134 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 2 ≠
0) |
| 163 | 159, 162 | recidd 12038 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (2 · (1 / 2))
= 1) |
| 164 | 163 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((2 · (1 / 2))
· π) = (1 · π)) |
| 165 | 160 | mullidd 11279 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (1 · π) =
π) |
| 166 | 161, 164,
165 | 3eqtrd 2781 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((2 · π)
· (1 / 2)) = π) |
| 167 | 139 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ ℂ →
(cos‘((1 / 2) · 𝑦)) = (cos‘(𝑦 / 2))) |
| 168 | 167 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (cos‘((1 / 2)
· 𝑦)) =
(cos‘(𝑦 /
2))) |
| 169 | 166, 168 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((2 · π)
· (1 / 2)) · (cos‘((1 / 2) · 𝑦))) = (π · (cos‘(𝑦 / 2)))) |
| 170 | 169 | mpteq2dva 5242 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (((2 · π)
· (1 / 2)) · (cos‘((1 / 2) · 𝑦)))) = (𝑦 ∈ ℂ ↦ (π ·
(cos‘(𝑦 /
2))))) |
| 171 | 158, 170 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ ((2
· π) · (sin‘((1 / 2) · 𝑦))))) = (𝑦 ∈ ℂ ↦ (π ·
(cos‘(𝑦 /
2))))) |
| 172 | 171 | dmeqd 5916 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → dom (ℂ D (𝑦 ∈ ℂ ↦ ((2
· π) · (sin‘((1 / 2) · 𝑦))))) = dom (𝑦 ∈ ℂ ↦ (π ·
(cos‘(𝑦 /
2))))) |
| 173 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ℂ ↦ (π
· (cos‘(𝑦 /
2)))) = (𝑦 ∈ ℂ
↦ (π · (cos‘(𝑦 / 2)))) |
| 174 | 69 | halfcld 12511 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦 / 2) ∈ ℂ) |
| 175 | 174 | coscld 16167 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (cos‘(𝑦 / 2)) ∈
ℂ) |
| 176 | 160, 175 | mulcld 11281 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (π ·
(cos‘(𝑦 / 2))) ∈
ℂ) |
| 177 | 173, 176 | dmmptd 6713 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → dom (𝑦 ∈ ℂ ↦ (π ·
(cos‘(𝑦 / 2)))) =
ℂ) |
| 178 | 172, 177 | eqtrd 2777 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → dom (ℂ D (𝑦 ∈ ℂ ↦ ((2
· π) · (sin‘((1 / 2) · 𝑦))))) = ℂ) |
| 179 | 36, 178 | sseqtrrid 4027 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ℝ ⊆ dom
(ℂ D (𝑦 ∈
ℂ ↦ ((2 · π) · (sin‘((1 / 2) · 𝑦)))))) |
| 180 | | dvres3 25948 |
. . . . . . . . . . . . . . 15
⊢
(((ℝ ∈ {ℝ, ℂ} ∧ (𝑦 ∈ ℂ ↦ ((2 · π)
· (sin‘((1 / 2) · 𝑦)))):ℂ⟶ℂ) ∧ (ℂ
⊆ ℂ ∧ ℝ ⊆ dom (ℂ D (𝑦 ∈ ℂ ↦ ((2 · π)
· (sin‘((1 / 2) · 𝑦))))))) → (ℝ D ((𝑦 ∈ ℂ ↦ ((2
· π) · (sin‘((1 / 2) · 𝑦)))) ↾ ℝ)) = ((ℂ D (𝑦 ∈ ℂ ↦ ((2
· π) · (sin‘((1 / 2) · 𝑦))))) ↾ ℝ)) |
| 181 | 67, 153, 75, 179, 180 | syl22anc 839 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (ℝ D ((𝑦 ∈ ℂ ↦ ((2
· π) · (sin‘((1 / 2) · 𝑦)))) ↾ ℝ)) = ((ℂ D (𝑦 ∈ ℂ ↦ ((2
· π) · (sin‘((1 / 2) · 𝑦))))) ↾ ℝ)) |
| 182 | | resmpt 6055 |
. . . . . . . . . . . . . . . 16
⊢ (ℝ
⊆ ℂ → ((𝑦
∈ ℂ ↦ ((2 · π) · (sin‘((1 / 2) ·
𝑦)))) ↾ ℝ) =
(𝑦 ∈ ℝ ↦
((2 · π) · (sin‘((1 / 2) · 𝑦))))) |
| 183 | 36, 182 | mp1i 13 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ ((2 · π)
· (sin‘((1 / 2) · 𝑦)))) ↾ ℝ) = (𝑦 ∈ ℝ ↦ ((2 · π)
· (sin‘((1 / 2) · 𝑦))))) |
| 184 | 183 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (ℝ D ((𝑦 ∈ ℂ ↦ ((2
· π) · (sin‘((1 / 2) · 𝑦)))) ↾ ℝ)) = (ℝ D (𝑦 ∈ ℝ ↦ ((2
· π) · (sin‘((1 / 2) · 𝑦)))))) |
| 185 | 171 | reseq1d 5996 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((ℂ D (𝑦 ∈ ℂ ↦ ((2
· π) · (sin‘((1 / 2) · 𝑦))))) ↾ ℝ) = ((𝑦 ∈ ℂ ↦ (π ·
(cos‘(𝑦 / 2))))
↾ ℝ)) |
| 186 | 181, 184,
185 | 3eqtr3d 2785 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ ((2
· π) · (sin‘((1 / 2) · 𝑦))))) = ((𝑦 ∈ ℂ ↦ (π ·
(cos‘(𝑦 / 2))))
↾ ℝ)) |
| 187 | | resmpt 6055 |
. . . . . . . . . . . . . 14
⊢ (ℝ
⊆ ℂ → ((𝑦
∈ ℂ ↦ (π · (cos‘(𝑦 / 2)))) ↾ ℝ) = (𝑦 ∈ ℝ ↦ (π
· (cos‘(𝑦 /
2))))) |
| 188 | 36, 187 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ ℂ ↦ (π
· (cos‘(𝑦 /
2)))) ↾ ℝ) = (𝑦
∈ ℝ ↦ (π · (cos‘(𝑦 / 2)))) |
| 189 | 186, 188 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ ((2
· π) · (sin‘((1 / 2) · 𝑦))))) = (𝑦 ∈ ℝ ↦ (π ·
(cos‘(𝑦 /
2))))) |
| 190 | 146, 189 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ ((2
· π) · (sin‘(𝑦 / 2))))) = (𝑦 ∈ ℝ ↦ (π ·
(cos‘(𝑦 /
2))))) |
| 191 | 190 | reseq1d 5996 |
. . . . . . . . . 10
⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ ((2
· π) · (sin‘(𝑦 / 2))))) ↾ ((𝐴(,)𝐵) ∖ {𝑌})) = ((𝑦 ∈ ℝ ↦ (π ·
(cos‘(𝑦 / 2))))
↾ ((𝐴(,)𝐵) ∖ {𝑌}))) |
| 192 | 4 | resmptd 6058 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (π ·
(cos‘(𝑦 / 2))))
↾ ((𝐴(,)𝐵) ∖ {𝑌})) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (π · (cos‘(𝑦 / 2))))) |
| 193 | 129, 191,
192 | 3eqtrd 2781 |
. . . . . . . . 9
⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ ((2
· π) · (sin‘(𝑦 / 2))))) ↾
((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∖ {𝑌}))) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (π · (cos‘(𝑦 / 2))))) |
| 194 | 117, 128,
193 | 3eqtrd 2781 |
. . . . . . . 8
⊢ (𝜑 → (ℝ D (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((2 · π) ·
(sin‘(𝑦 / 2))))) =
(𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (π · (cos‘(𝑦 / 2))))) |
| 195 | 194 | eqcomd 2743 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (π · (cos‘(𝑦 / 2)))) = (ℝ D (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((2 · π) ·
(sin‘(𝑦 /
2)))))) |
| 196 | 23 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((2 · π) ·
(sin‘(𝑦 /
2))))) |
| 197 | 196 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝜑 → (ℝ D 𝐺) = (ℝ D (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((2 · π) ·
(sin‘(𝑦 /
2)))))) |
| 198 | 197 | eqcomd 2743 |
. . . . . . 7
⊢ (𝜑 → (ℝ D (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((2 · π) ·
(sin‘(𝑦 / 2))))) =
(ℝ D 𝐺)) |
| 199 | 112, 195,
198 | 3eqtrrd 2782 |
. . . . . 6
⊢ (𝜑 → (ℝ D 𝐺) = 𝐼) |
| 200 | 199 | dmeqd 5916 |
. . . . 5
⊢ (𝜑 → dom (ℝ D 𝐺) = dom 𝐼) |
| 201 | 105, 176 | syldan 591 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (π · (cos‘(𝑦 / 2))) ∈
ℂ) |
| 202 | 111, 201 | dmmptd 6713 |
. . . . 5
⊢ (𝜑 → dom 𝐼 = ((𝐴(,)𝐵) ∖ {𝑌})) |
| 203 | 200, 202 | eqtr2d 2778 |
. . . 4
⊢ (𝜑 → ((𝐴(,)𝐵) ∖ {𝑌}) = dom (ℝ D 𝐺)) |
| 204 | | eqimss 4042 |
. . . 4
⊢ (((𝐴(,)𝐵) ∖ {𝑌}) = dom (ℝ D 𝐺) → ((𝐴(,)𝐵) ∖ {𝑌}) ⊆ dom (ℝ D 𝐺)) |
| 205 | 203, 204 | syl 17 |
. . 3
⊢ (𝜑 → ((𝐴(,)𝐵) ∖ {𝑌}) ⊆ dom (ℝ D 𝐺)) |
| 206 | 105, 70 | syldan 591 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝑁 + (1 / 2)) · 𝑦) ∈ ℂ) |
| 207 | 206 | ralrimiva 3146 |
. . . . . . 7
⊢ (𝜑 → ∀𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})((𝑁 + (1 / 2)) · 𝑦) ∈ ℂ) |
| 208 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦)) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦)) |
| 209 | 208 | fnmpt 6708 |
. . . . . . 7
⊢
(∀𝑦 ∈
((𝐴(,)𝐵) ∖ {𝑌})((𝑁 + (1 / 2)) · 𝑦) ∈ ℂ → (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦)) Fn ((𝐴(,)𝐵) ∖ {𝑌})) |
| 210 | 207, 209 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦)) Fn ((𝐴(,)𝐵) ∖ {𝑌})) |
| 211 | | eqidd 2738 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦)) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))) |
| 212 | | simpr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤) |
| 213 | 212 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ 𝑦 = 𝑤) → ((𝑁 + (1 / 2)) · 𝑦) = ((𝑁 + (1 / 2)) · 𝑤)) |
| 214 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) |
| 215 | 38 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → 𝑁 ∈ ℂ) |
| 216 | | 1cnd 11256 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → 1 ∈
ℂ) |
| 217 | 216 | halfcld 12511 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (1 / 2) ∈
ℂ) |
| 218 | 215, 217 | addcld 11280 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝑁 + (1 / 2)) ∈ ℂ) |
| 219 | | eldifi 4131 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → 𝑤 ∈ (𝐴(,)𝐵)) |
| 220 | 219 | elioored 45562 |
. . . . . . . . . . . . 13
⊢ (𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → 𝑤 ∈ ℝ) |
| 221 | 220 | recnd 11289 |
. . . . . . . . . . . 12
⊢ (𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → 𝑤 ∈ ℂ) |
| 222 | 221 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → 𝑤 ∈ ℂ) |
| 223 | 218, 222 | mulcld 11281 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝑁 + (1 / 2)) · 𝑤) ∈ ℂ) |
| 224 | 211, 213,
214, 223 | fvmptd 7023 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤) = ((𝑁 + (1 / 2)) · 𝑤)) |
| 225 | | eleq1w 2824 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑤 → (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↔ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌}))) |
| 226 | 225 | anbi2d 630 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑤 → ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ↔ (𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})))) |
| 227 | | oveq1 7438 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑤 → (𝑦 mod (2 · π)) = (𝑤 mod (2 · π))) |
| 228 | 227 | neeq1d 3000 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑤 → ((𝑦 mod (2 · π)) ≠ 0 ↔ (𝑤 mod (2 · π)) ≠
0)) |
| 229 | 226, 228 | imbi12d 344 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑤 → (((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝑦 mod (2 · π)) ≠ 0) ↔
((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝑤 mod (2 · π)) ≠
0))) |
| 230 | | eldifi 4131 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → 𝑦 ∈ (𝐴(,)𝐵)) |
| 231 | | elioore 13417 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (𝐴(,)𝐵) → 𝑦 ∈ ℝ) |
| 232 | 230, 231,
130 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → 𝑦 ∈ ℂ) |
| 233 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → 2 ∈ ℂ) |
| 234 | 119 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → π ∈
ℂ) |
| 235 | 134 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → 2 ≠ 0) |
| 236 | | 0re 11263 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 0 ∈
ℝ |
| 237 | | pipos 26502 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 0 <
π |
| 238 | 236, 237 | gtneii 11373 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ π ≠
0 |
| 239 | 238 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → π ≠ 0) |
| 240 | 232, 233,
234, 235, 239 | divdiv1d 12074 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → ((𝑦 / 2) / π) = (𝑦 / (2 · π))) |
| 241 | 240 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → (𝑦 / (2 · π)) = ((𝑦 / 2) / π)) |
| 242 | 241 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝑦 / (2 · π)) = ((𝑦 / 2) / π)) |
| 243 | | dirkercncflem2.yne0 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (sin‘(𝑦 / 2)) ≠ 0) |
| 244 | 243 | neneqd 2945 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ¬ (sin‘(𝑦 / 2)) = 0) |
| 245 | 105 | halfcld 12511 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝑦 / 2) ∈ ℂ) |
| 246 | | sineq0 26566 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑦 / 2) ∈ ℂ →
((sin‘(𝑦 / 2)) = 0
↔ ((𝑦 / 2) / π)
∈ ℤ)) |
| 247 | 245, 246 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((sin‘(𝑦 / 2)) = 0 ↔ ((𝑦 / 2) / π) ∈
ℤ)) |
| 248 | 244, 247 | mtbid 324 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ¬ ((𝑦 / 2) / π) ∈
ℤ) |
| 249 | 242, 248 | eqneltrd 2861 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ¬ (𝑦 / (2 · π)) ∈
ℤ) |
| 250 | | 2rp 13039 |
. . . . . . . . . . . . . . . . . 18
⊢ 2 ∈
ℝ+ |
| 251 | | pirp 26503 |
. . . . . . . . . . . . . . . . . 18
⊢ π
∈ ℝ+ |
| 252 | | rpmulcl 13058 |
. . . . . . . . . . . . . . . . . 18
⊢ ((2
∈ ℝ+ ∧ π ∈ ℝ+) → (2
· π) ∈ ℝ+) |
| 253 | 250, 251,
252 | mp2an 692 |
. . . . . . . . . . . . . . . . 17
⊢ (2
· π) ∈ ℝ+ |
| 254 | | mod0 13916 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑦 ∈ ℝ ∧ (2
· π) ∈ ℝ+) → ((𝑦 mod (2 · π)) = 0 ↔ (𝑦 / (2 · π)) ∈
ℤ)) |
| 255 | 11, 253, 254 | sylancl 586 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝑦 mod (2 · π)) = 0 ↔ (𝑦 / (2 · π)) ∈
ℤ)) |
| 256 | 249, 255 | mtbird 325 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ¬ (𝑦 mod (2 · π)) = 0) |
| 257 | 256 | neqned 2947 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝑦 mod (2 · π)) ≠
0) |
| 258 | 229, 257 | chvarvv 1998 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝑤 mod (2 · π)) ≠
0) |
| 259 | 258 | neneqd 2945 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ¬ (𝑤 mod (2 · π)) = 0) |
| 260 | | simpll 767 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ ((𝑁 + (1 / 2)) · 𝑤) = ((𝑁 + (1 / 2)) · 𝑌)) → 𝜑) |
| 261 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ ((𝑁 + (1 / 2)) · 𝑤) = ((𝑁 + (1 / 2)) · 𝑌)) → ((𝑁 + (1 / 2)) · 𝑤) = ((𝑁 + (1 / 2)) · 𝑌)) |
| 262 | 221 | ad2antlr 727 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ ((𝑁 + (1 / 2)) · 𝑤) = ((𝑁 + (1 / 2)) · 𝑌)) → 𝑤 ∈ ℂ) |
| 263 | 57 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑌 ∈ ℂ) |
| 264 | 263 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ ((𝑁 + (1 / 2)) · 𝑤) = ((𝑁 + (1 / 2)) · 𝑌)) → 𝑌 ∈ ℂ) |
| 265 | | 0red 11264 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 0 ∈
ℝ) |
| 266 | 5 | nnred 12281 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 267 | | 1red 11262 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 1 ∈
ℝ) |
| 268 | 267 | rehalfcld 12513 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (1 / 2) ∈
ℝ) |
| 269 | 266, 268 | readdcld 11290 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑁 + (1 / 2)) ∈ ℝ) |
| 270 | 5 | nngt0d 12315 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 0 < 𝑁) |
| 271 | 250 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 2 ∈
ℝ+) |
| 272 | 271 | rpreccld 13087 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (1 / 2) ∈
ℝ+) |
| 273 | 266, 272 | ltaddrpd 13110 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 < (𝑁 + (1 / 2))) |
| 274 | 265, 266,
269, 270, 273 | lttrd 11422 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 0 < (𝑁 + (1 / 2))) |
| 275 | 274 | gt0ne0d 11827 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑁 + (1 / 2)) ≠ 0) |
| 276 | 41, 275 | jca 511 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑁 + (1 / 2)) ∈ ℂ ∧ (𝑁 + (1 / 2)) ≠
0)) |
| 277 | 276 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ ((𝑁 + (1 / 2)) · 𝑤) = ((𝑁 + (1 / 2)) · 𝑌)) → ((𝑁 + (1 / 2)) ∈ ℂ ∧ (𝑁 + (1 / 2)) ≠
0)) |
| 278 | | mulcan 11900 |
. . . . . . . . . . . . . . 15
⊢ ((𝑤 ∈ ℂ ∧ 𝑌 ∈ ℂ ∧ ((𝑁 + (1 / 2)) ∈ ℂ ∧
(𝑁 + (1 / 2)) ≠ 0))
→ (((𝑁 + (1 / 2))
· 𝑤) = ((𝑁 + (1 / 2)) · 𝑌) ↔ 𝑤 = 𝑌)) |
| 279 | 262, 264,
277, 278 | syl3anc 1373 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ ((𝑁 + (1 / 2)) · 𝑤) = ((𝑁 + (1 / 2)) · 𝑌)) → (((𝑁 + (1 / 2)) · 𝑤) = ((𝑁 + (1 / 2)) · 𝑌) ↔ 𝑤 = 𝑌)) |
| 280 | 261, 279 | mpbid 232 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ ((𝑁 + (1 / 2)) · 𝑤) = ((𝑁 + (1 / 2)) · 𝑌)) → 𝑤 = 𝑌) |
| 281 | | oveq1 7438 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑌 → (𝑤 mod (2 · π)) = (𝑌 mod (2 · π))) |
| 282 | | dirkercncflem2.ymod |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑌 mod (2 · π)) =
0) |
| 283 | 281, 282 | sylan9eqr 2799 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑤 = 𝑌) → (𝑤 mod (2 · π)) = 0) |
| 284 | 260, 280,
283 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ ((𝑁 + (1 / 2)) · 𝑤) = ((𝑁 + (1 / 2)) · 𝑌)) → (𝑤 mod (2 · π)) = 0) |
| 285 | 259, 284 | mtand 816 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ¬ ((𝑁 + (1 / 2)) · 𝑤) = ((𝑁 + (1 / 2)) · 𝑌)) |
| 286 | 41, 263 | mulcld 11281 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑁 + (1 / 2)) · 𝑌) ∈ ℂ) |
| 287 | 286 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝑁 + (1 / 2)) · 𝑌) ∈ ℂ) |
| 288 | | elsn2g 4664 |
. . . . . . . . . . . 12
⊢ (((𝑁 + (1 / 2)) · 𝑌) ∈ ℂ → (((𝑁 + (1 / 2)) · 𝑤) ∈ {((𝑁 + (1 / 2)) · 𝑌)} ↔ ((𝑁 + (1 / 2)) · 𝑤) = ((𝑁 + (1 / 2)) · 𝑌))) |
| 289 | 287, 288 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (((𝑁 + (1 / 2)) · 𝑤) ∈ {((𝑁 + (1 / 2)) · 𝑌)} ↔ ((𝑁 + (1 / 2)) · 𝑤) = ((𝑁 + (1 / 2)) · 𝑌))) |
| 290 | 285, 289 | mtbird 325 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ¬ ((𝑁 + (1 / 2)) · 𝑤) ∈ {((𝑁 + (1 / 2)) · 𝑌)}) |
| 291 | 223, 290 | eldifd 3962 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝑁 + (1 / 2)) · 𝑤) ∈ (ℂ ∖ {((𝑁 + (1 / 2)) · 𝑌)})) |
| 292 | 224, 291 | eqeltrd 2841 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤) ∈ (ℂ ∖ {((𝑁 + (1 / 2)) · 𝑌)})) |
| 293 | | sinf 16160 |
. . . . . . . . . . . 12
⊢
sin:ℂ⟶ℂ |
| 294 | 293 | fdmi 6747 |
. . . . . . . . . . 11
⊢ dom sin =
ℂ |
| 295 | 294 | eqcomi 2746 |
. . . . . . . . . 10
⊢ ℂ =
dom sin |
| 296 | 295 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ℂ = dom
sin) |
| 297 | 296 | difeq1d 4125 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (ℂ ∖ {((𝑁 + (1 / 2)) · 𝑌)}) = (dom sin ∖ {((𝑁 + (1 / 2)) · 𝑌)})) |
| 298 | 292, 297 | eleqtrd 2843 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤) ∈ (dom sin ∖ {((𝑁 + (1 / 2)) · 𝑌)})) |
| 299 | 298 | ralrimiva 3146 |
. . . . . 6
⊢ (𝜑 → ∀𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤) ∈ (dom sin ∖ {((𝑁 + (1 / 2)) · 𝑌)})) |
| 300 | | fnfvrnss 7141 |
. . . . . 6
⊢ (((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦)) Fn ((𝐴(,)𝐵) ∖ {𝑌}) ∧ ∀𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤) ∈ (dom sin ∖ {((𝑁 + (1 / 2)) · 𝑌)})) → ran (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦)) ⊆ (dom sin ∖ {((𝑁 + (1 / 2)) · 𝑌)})) |
| 301 | 210, 299,
300 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → ran (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦)) ⊆ (dom sin ∖ {((𝑁 + (1 / 2)) · 𝑌)})) |
| 302 | | uncom 4158 |
. . . . . . . . . 10
⊢ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) = ({𝑌} ∪ ((𝐴(,)𝐵) ∖ {𝑌})) |
| 303 | 302 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) = ({𝑌} ∪ ((𝐴(,)𝐵) ∖ {𝑌}))) |
| 304 | 27 | snssd 4809 |
. . . . . . . . . 10
⊢ (𝜑 → {𝑌} ⊆ (𝐴(,)𝐵)) |
| 305 | | undif 4482 |
. . . . . . . . . 10
⊢ ({𝑌} ⊆ (𝐴(,)𝐵) ↔ ({𝑌} ∪ ((𝐴(,)𝐵) ∖ {𝑌})) = (𝐴(,)𝐵)) |
| 306 | 304, 305 | sylib 218 |
. . . . . . . . 9
⊢ (𝜑 → ({𝑌} ∪ ((𝐴(,)𝐵) ∖ {𝑌})) = (𝐴(,)𝐵)) |
| 307 | 303, 306 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) = (𝐴(,)𝐵)) |
| 308 | 307 | mpteq1d 5237 |
. . . . . . 7
⊢ (𝜑 → (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤))) = (𝑤 ∈ (𝐴(,)𝐵) ↦ if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)))) |
| 309 | | iftrue 4531 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑌 → if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)) = ((𝑁 + (1 / 2)) · 𝑌)) |
| 310 | | oveq2 7439 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑌 → ((𝑁 + (1 / 2)) · 𝑤) = ((𝑁 + (1 / 2)) · 𝑌)) |
| 311 | 309, 310 | eqtr4d 2780 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑌 → if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)) = ((𝑁 + (1 / 2)) · 𝑤)) |
| 312 | 311 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ 𝑤 = 𝑌) → if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)) = ((𝑁 + (1 / 2)) · 𝑤)) |
| 313 | | iffalse 4534 |
. . . . . . . . . . . . 13
⊢ (¬
𝑤 = 𝑌 → if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)) = ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)) |
| 314 | 313 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)) = ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)) |
| 315 | | eqidd 2738 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦)) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))) |
| 316 | | oveq2 7439 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑤 → ((𝑁 + (1 / 2)) · 𝑦) = ((𝑁 + (1 / 2)) · 𝑤)) |
| 317 | 316 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) ∧ 𝑦 = 𝑤) → ((𝑁 + (1 / 2)) · 𝑦) = ((𝑁 + (1 / 2)) · 𝑤)) |
| 318 | | simpl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝑤 ∈ (𝐴(,)𝐵) ∧ ¬ 𝑤 = 𝑌) → 𝑤 ∈ (𝐴(,)𝐵)) |
| 319 | | id 22 |
. . . . . . . . . . . . . . . . 17
⊢ (¬
𝑤 = 𝑌 → ¬ 𝑤 = 𝑌) |
| 320 | | velsn 4642 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 ∈ {𝑌} ↔ 𝑤 = 𝑌) |
| 321 | 319, 320 | sylnibr 329 |
. . . . . . . . . . . . . . . 16
⊢ (¬
𝑤 = 𝑌 → ¬ 𝑤 ∈ {𝑌}) |
| 322 | 321 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝑤 ∈ (𝐴(,)𝐵) ∧ ¬ 𝑤 = 𝑌) → ¬ 𝑤 ∈ {𝑌}) |
| 323 | 318, 322 | eldifd 3962 |
. . . . . . . . . . . . . 14
⊢ ((𝑤 ∈ (𝐴(,)𝐵) ∧ ¬ 𝑤 = 𝑌) → 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) |
| 324 | 323 | adantll 714 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) |
| 325 | 41 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝑁 + (1 / 2)) ∈ ℂ) |
| 326 | | elioore 13417 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 ∈ (𝐴(,)𝐵) → 𝑤 ∈ ℝ) |
| 327 | 326 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 ∈ (𝐴(,)𝐵) → 𝑤 ∈ ℂ) |
| 328 | 327 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → 𝑤 ∈ ℂ) |
| 329 | 325, 328 | mulcld 11281 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → ((𝑁 + (1 / 2)) · 𝑤) ∈ ℂ) |
| 330 | 329 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → ((𝑁 + (1 / 2)) · 𝑤) ∈ ℂ) |
| 331 | 315, 317,
324, 330 | fvmptd 7023 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤) = ((𝑁 + (1 / 2)) · 𝑤)) |
| 332 | 314, 331 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)) = ((𝑁 + (1 / 2)) · 𝑤)) |
| 333 | 312, 332 | pm2.61dan 813 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)) = ((𝑁 + (1 / 2)) · 𝑤)) |
| 334 | 333 | mpteq2dva 5242 |
. . . . . . . . 9
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤))) = (𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤))) |
| 335 | | ioosscn 13449 |
. . . . . . . . . . . . . 14
⊢ (𝐴(,)𝐵) ⊆ ℂ |
| 336 | | resmpt 6055 |
. . . . . . . . . . . . . 14
⊢ ((𝐴(,)𝐵) ⊆ ℂ → ((𝑤 ∈ ℂ ↦ ((𝑁 + (1 / 2)) · 𝑤)) ↾ (𝐴(,)𝐵)) = (𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤))) |
| 337 | 335, 336 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ ((𝑤 ∈ ℂ ↦ ((𝑁 + (1 / 2)) · 𝑤)) ↾ (𝐴(,)𝐵)) = (𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)) |
| 338 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 ∈ ℂ ↦ ((𝑁 + (1 / 2)) · 𝑤)) = (𝑤 ∈ ℂ ↦ ((𝑁 + (1 / 2)) · 𝑤)) |
| 339 | 338 | mulc1cncf 24931 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 + (1 / 2)) ∈ ℂ
→ (𝑤 ∈ ℂ
↦ ((𝑁 + (1 / 2))
· 𝑤)) ∈
(ℂ–cn→ℂ)) |
| 340 | 41, 339 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑤 ∈ ℂ ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈ (ℂ–cn→ℂ)) |
| 341 | 51 | cnfldtop 24804 |
. . . . . . . . . . . . . . . . . . 19
⊢
(TopOpen‘ℂfld) ∈ Top |
| 342 | | unicntop 24806 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ℂ =
∪
(TopOpen‘ℂfld) |
| 343 | 342 | restid 17478 |
. . . . . . . . . . . . . . . . . . 19
⊢
((TopOpen‘ℂfld) ∈ Top →
((TopOpen‘ℂfld) ↾t ℂ) =
(TopOpen‘ℂfld)) |
| 344 | 341, 343 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢
((TopOpen‘ℂfld) ↾t ℂ) =
(TopOpen‘ℂfld) |
| 345 | 344 | eqcomi 2746 |
. . . . . . . . . . . . . . . . 17
⊢
(TopOpen‘ℂfld) =
((TopOpen‘ℂfld) ↾t
ℂ) |
| 346 | 51, 345, 345 | cncfcn 24936 |
. . . . . . . . . . . . . . . 16
⊢ ((ℂ
⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℂ) =
((TopOpen‘ℂfld) Cn
(TopOpen‘ℂfld))) |
| 347 | 74, 75, 346 | sylancr 587 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (ℂ–cn→ℂ) =
((TopOpen‘ℂfld) Cn
(TopOpen‘ℂfld))) |
| 348 | 340, 347 | eleqtrd 2843 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑤 ∈ ℂ ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈
((TopOpen‘ℂfld) Cn
(TopOpen‘ℂfld))) |
| 349 | 2, 37 | sstrid 3995 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ) |
| 350 | 342 | cnrest 23293 |
. . . . . . . . . . . . . 14
⊢ (((𝑤 ∈ ℂ ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈
((TopOpen‘ℂfld) Cn
(TopOpen‘ℂfld)) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((𝑤 ∈ ℂ ↦ ((𝑁 + (1 / 2)) · 𝑤)) ↾ (𝐴(,)𝐵)) ∈
(((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn
(TopOpen‘ℂfld))) |
| 351 | 348, 349,
350 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑤 ∈ ℂ ↦ ((𝑁 + (1 / 2)) · 𝑤)) ↾ (𝐴(,)𝐵)) ∈
(((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn
(TopOpen‘ℂfld))) |
| 352 | 337, 351 | eqeltrrid 2846 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈
(((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn
(TopOpen‘ℂfld))) |
| 353 | 51 | cnfldtopon 24803 |
. . . . . . . . . . . . . 14
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
| 354 | | resttopon 23169 |
. . . . . . . . . . . . . 14
⊢
(((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
∧ (𝐴(,)𝐵) ⊆ ℂ) →
((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵))) |
| 355 | 353, 349,
354 | sylancr 587 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵))) |
| 356 | | cncnp 23288 |
. . . . . . . . . . . . 13
⊢
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)) ∧
(TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) →
((𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈
(((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld))
↔ ((𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)):(𝐴(,)𝐵)⟶ℂ ∧ ∀𝑦 ∈ (𝐴(,)𝐵)(𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦)))) |
| 357 | 355, 353,
356 | sylancl 586 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈
(((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld))
↔ ((𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)):(𝐴(,)𝐵)⟶ℂ ∧ ∀𝑦 ∈ (𝐴(,)𝐵)(𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦)))) |
| 358 | 352, 357 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)):(𝐴(,)𝐵)⟶ℂ ∧ ∀𝑦 ∈ (𝐴(,)𝐵)(𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦))) |
| 359 | 358 | simprd 495 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦)) |
| 360 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑌 →
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld)
↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 361 | 360 | eleq2d 2827 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑌 → ((𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦) ↔ (𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌))) |
| 362 | 361 | rspccva 3621 |
. . . . . . . . . 10
⊢
((∀𝑦 ∈
(𝐴(,)𝐵)(𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦) ∧ 𝑌 ∈ (𝐴(,)𝐵)) → (𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 363 | 359, 27, 362 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑤)) ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 364 | 334, 363 | eqeltrd 2841 |
. . . . . . . 8
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤))) ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 365 | 307 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴(,)𝐵) = (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) |
| 366 | 365 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) = ((TopOpen‘ℂfld)
↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}))) |
| 367 | 366 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝜑 →
(((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld)) =
(((TopOpen‘ℂfld) ↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) CnP
(TopOpen‘ℂfld))) |
| 368 | 367 | fveq1d 6908 |
. . . . . . . 8
⊢ (𝜑 →
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌) = ((((TopOpen‘ℂfld)
↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 369 | 364, 368 | eleqtrd 2843 |
. . . . . . 7
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤))) ∈
((((TopOpen‘ℂfld) ↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 370 | 308, 369 | eqeltrd 2841 |
. . . . . 6
⊢ (𝜑 → (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤))) ∈
((((TopOpen‘ℂfld) ↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 371 | | eqid 2737 |
. . . . . . 7
⊢
((TopOpen‘ℂfld) ↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) = ((TopOpen‘ℂfld)
↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) |
| 372 | | eqid 2737 |
. . . . . . 7
⊢ (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤))) = (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤))) |
| 373 | 206, 208 | fmptd 7134 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦)):((𝐴(,)𝐵) ∖ {𝑌})⟶ℂ) |
| 374 | 4, 36 | sstrdi 3996 |
. . . . . . 7
⊢ (𝜑 → ((𝐴(,)𝐵) ∖ {𝑌}) ⊆ ℂ) |
| 375 | 371, 51, 372, 373, 374, 263 | ellimc 25908 |
. . . . . 6
⊢ (𝜑 → (((𝑁 + (1 / 2)) · 𝑌) ∈ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦)) limℂ 𝑌) ↔ (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, ((𝑁 + (1 / 2)) · 𝑌), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤))) ∈
((((TopOpen‘ℂfld) ↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) CnP
(TopOpen‘ℂfld))‘𝑌))) |
| 376 | 370, 375 | mpbird 257 |
. . . . 5
⊢ (𝜑 → ((𝑁 + (1 / 2)) · 𝑌) ∈ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦)) limℂ 𝑌)) |
| 377 | 134 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 2 ≠ 0) |
| 378 | 238 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → π ≠
0) |
| 379 | 154, 155,
377, 378 | mulne0d 11915 |
. . . . . . . . . . 11
⊢ (𝜑 → (2 · π) ≠
0) |
| 380 | 263, 156,
379 | divcan1d 12044 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑌 / (2 · π)) · (2 ·
π)) = 𝑌) |
| 381 | 380 | eqcomd 2743 |
. . . . . . . . 9
⊢ (𝜑 → 𝑌 = ((𝑌 / (2 · π)) · (2 ·
π))) |
| 382 | 381 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁 + (1 / 2)) · 𝑌) = ((𝑁 + (1 / 2)) · ((𝑌 / (2 · π)) · (2 ·
π)))) |
| 383 | 382 | fveq2d 6910 |
. . . . . . 7
⊢ (𝜑 → (sin‘((𝑁 + (1 / 2)) · 𝑌)) = (sin‘((𝑁 + (1 / 2)) · ((𝑌 / (2 · π)) ·
(2 · π))))) |
| 384 | 263, 156,
379 | divcld 12043 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑌 / (2 · π)) ∈
ℂ) |
| 385 | 41, 384, 156 | mul12d 11470 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑁 + (1 / 2)) · ((𝑌 / (2 · π)) · (2 ·
π))) = ((𝑌 / (2 ·
π)) · ((𝑁 + (1 /
2)) · (2 · π)))) |
| 386 | 41, 154, 155 | mulassd 11284 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((𝑁 + (1 / 2)) · 2) · π) =
((𝑁 + (1 / 2)) · (2
· π))) |
| 387 | 386 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁 + (1 / 2)) · (2 · π)) =
(((𝑁 + (1 / 2)) · 2)
· π)) |
| 388 | 387 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑌 / (2 · π)) · ((𝑁 + (1 / 2)) · (2 ·
π))) = ((𝑌 / (2 ·
π)) · (((𝑁 + (1 /
2)) · 2) · π))) |
| 389 | 38, 40, 154 | adddird 11286 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑁 + (1 / 2)) · 2) = ((𝑁 · 2) + ((1 / 2) ·
2))) |
| 390 | 154, 377 | recid2d 12039 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((1 / 2) · 2) =
1) |
| 391 | 390 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑁 · 2) + ((1 / 2) · 2)) =
((𝑁 · 2) +
1)) |
| 392 | 389, 391 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑁 + (1 / 2)) · 2) = ((𝑁 · 2) +
1)) |
| 393 | 392 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (𝜑 → (((𝑁 + (1 / 2)) · 2) · π) =
(((𝑁 · 2) + 1)
· π)) |
| 394 | 393 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑌 / (2 · π)) · (((𝑁 + (1 / 2)) · 2) ·
π)) = ((𝑌 / (2 ·
π)) · (((𝑁
· 2) + 1) · π))) |
| 395 | 385, 388,
394 | 3eqtrd 2781 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑁 + (1 / 2)) · ((𝑌 / (2 · π)) · (2 ·
π))) = ((𝑌 / (2 ·
π)) · (((𝑁
· 2) + 1) · π))) |
| 396 | 38, 154 | mulcld 11281 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁 · 2) ∈
ℂ) |
| 397 | | 1cnd 11256 |
. . . . . . . . . . 11
⊢ (𝜑 → 1 ∈
ℂ) |
| 398 | 396, 397 | addcld 11280 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑁 · 2) + 1) ∈
ℂ) |
| 399 | 384, 398,
155 | mulassd 11284 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑌 / (2 · π)) · ((𝑁 · 2) + 1)) ·
π) = ((𝑌 / (2 ·
π)) · (((𝑁
· 2) + 1) · π))) |
| 400 | 395, 399 | eqtr4d 2780 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁 + (1 / 2)) · ((𝑌 / (2 · π)) · (2 ·
π))) = (((𝑌 / (2
· π)) · ((𝑁 · 2) + 1)) ·
π)) |
| 401 | 400 | fveq2d 6910 |
. . . . . . 7
⊢ (𝜑 → (sin‘((𝑁 + (1 / 2)) · ((𝑌 / (2 · π)) ·
(2 · π)))) = (sin‘(((𝑌 / (2 · π)) · ((𝑁 · 2) + 1)) ·
π))) |
| 402 | | mod0 13916 |
. . . . . . . . . . 11
⊢ ((𝑌 ∈ ℝ ∧ (2
· π) ∈ ℝ+) → ((𝑌 mod (2 · π)) = 0 ↔ (𝑌 / (2 · π)) ∈
ℤ)) |
| 403 | 57, 253, 402 | sylancl 586 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑌 mod (2 · π)) = 0 ↔ (𝑌 / (2 · π)) ∈
ℤ)) |
| 404 | 282, 403 | mpbid 232 |
. . . . . . . . 9
⊢ (𝜑 → (𝑌 / (2 · π)) ∈
ℤ) |
| 405 | 5 | nnzd 12640 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 406 | | 2z 12649 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℤ |
| 407 | 406 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → 2 ∈
ℤ) |
| 408 | 405, 407 | zmulcld 12728 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁 · 2) ∈
ℤ) |
| 409 | 408 | peano2zd 12725 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑁 · 2) + 1) ∈
ℤ) |
| 410 | 404, 409 | zmulcld 12728 |
. . . . . . . 8
⊢ (𝜑 → ((𝑌 / (2 · π)) · ((𝑁 · 2) + 1)) ∈
ℤ) |
| 411 | | sinkpi 26564 |
. . . . . . . 8
⊢ (((𝑌 / (2 · π)) ·
((𝑁 · 2) + 1))
∈ ℤ → (sin‘(((𝑌 / (2 · π)) · ((𝑁 · 2) + 1)) ·
π)) = 0) |
| 412 | 410, 411 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (sin‘(((𝑌 / (2 · π)) ·
((𝑁 · 2) + 1))
· π)) = 0) |
| 413 | 383, 401,
412 | 3eqtrd 2781 |
. . . . . 6
⊢ (𝜑 → (sin‘((𝑁 + (1 / 2)) · 𝑌)) = 0) |
| 414 | | sincn 26488 |
. . . . . . . 8
⊢ sin
∈ (ℂ–cn→ℂ) |
| 415 | 414 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → sin ∈
(ℂ–cn→ℂ)) |
| 416 | 415, 286 | cnlimci 25924 |
. . . . . 6
⊢ (𝜑 → (sin‘((𝑁 + (1 / 2)) · 𝑌)) ∈ (sin
limℂ ((𝑁 +
(1 / 2)) · 𝑌))) |
| 417 | 413, 416 | eqeltrrd 2842 |
. . . . 5
⊢ (𝜑 → 0 ∈ (sin
limℂ ((𝑁 +
(1 / 2)) · 𝑌))) |
| 418 | 301, 376,
417 | limccog 45635 |
. . . 4
⊢ (𝜑 → 0 ∈ ((sin ∘
(𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))) limℂ 𝑌)) |
| 419 | 14 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → 𝐹 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦)))) |
| 420 | 213 | fveq2d 6910 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ 𝑦 = 𝑤) → (sin‘((𝑁 + (1 / 2)) · 𝑦)) = (sin‘((𝑁 + (1 / 2)) · 𝑤))) |
| 421 | 223 | sincld 16166 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (sin‘((𝑁 + (1 / 2)) · 𝑤)) ∈ ℂ) |
| 422 | 419, 420,
214, 421 | fvmptd 7023 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝐹‘𝑤) = (sin‘((𝑁 + (1 / 2)) · 𝑤))) |
| 423 | 224 | fveq2d 6910 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (sin‘((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)) = (sin‘((𝑁 + (1 / 2)) · 𝑤))) |
| 424 | 422, 423 | eqtr4d 2780 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝐹‘𝑤) = (sin‘((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤))) |
| 425 | 424 | mpteq2dva 5242 |
. . . . . 6
⊢ (𝜑 → (𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (𝐹‘𝑤)) = (𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (sin‘((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)))) |
| 426 | 15 | feqmptd 6977 |
. . . . . 6
⊢ (𝜑 → 𝐹 = (𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (𝐹‘𝑤))) |
| 427 | | fcompt 7153 |
. . . . . . 7
⊢
((sin:ℂ⟶ℂ ∧ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦)):((𝐴(,)𝐵) ∖ {𝑌})⟶ℂ) → (sin ∘
(𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))) = (𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (sin‘((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)))) |
| 428 | 293, 373,
427 | sylancr 587 |
. . . . . 6
⊢ (𝜑 → (sin ∘ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))) = (𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (sin‘((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)))) |
| 429 | 425, 426,
428 | 3eqtr4rd 2788 |
. . . . 5
⊢ (𝜑 → (sin ∘ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))) = 𝐹) |
| 430 | 429 | oveq1d 7446 |
. . . 4
⊢ (𝜑 → ((sin ∘ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · 𝑦))) limℂ 𝑌) = (𝐹 limℂ 𝑌)) |
| 431 | 418, 430 | eleqtrd 2843 |
. . 3
⊢ (𝜑 → 0 ∈ (𝐹 limℂ 𝑌)) |
| 432 | | simpr 484 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ 𝑤 = 𝑌) → 𝑤 = 𝑌) |
| 433 | 432 | iftrued 4533 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ 𝑤 = 𝑌) → if(𝑤 = 𝑌, 0, (𝐺‘𝑤)) = 0) |
| 434 | 263, 154,
156, 377, 379 | divdiv32d 12068 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑌 / 2) / (2 · π)) = ((𝑌 / (2 · π)) /
2)) |
| 435 | 434 | oveq1d 7446 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((𝑌 / 2) / (2 · π)) · (2
· π)) = (((𝑌 / (2
· π)) / 2) · (2 · π))) |
| 436 | 263 | halfcld 12511 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑌 / 2) ∈ ℂ) |
| 437 | 436, 156,
379 | divcan1d 12044 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((𝑌 / 2) / (2 · π)) · (2
· π)) = (𝑌 /
2)) |
| 438 | 384, 154,
156, 377 | div32d 12066 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (((𝑌 / (2 · π)) / 2) · (2
· π)) = ((𝑌 / (2
· π)) · ((2 · π) / 2))) |
| 439 | 155, 154,
377 | divcan3d 12048 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((2 · π) / 2) =
π) |
| 440 | 439 | oveq2d 7447 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑌 / (2 · π)) · ((2 ·
π) / 2)) = ((𝑌 / (2
· π)) · π)) |
| 441 | 438, 440 | eqtrd 2777 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((𝑌 / (2 · π)) / 2) · (2
· π)) = ((𝑌 / (2
· π)) · π)) |
| 442 | 435, 437,
441 | 3eqtr3d 2785 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑌 / 2) = ((𝑌 / (2 · π)) ·
π)) |
| 443 | 442 | fveq2d 6910 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (sin‘(𝑌 / 2)) = (sin‘((𝑌 / (2 · π)) ·
π))) |
| 444 | | sinkpi 26564 |
. . . . . . . . . . . . . . 15
⊢ ((𝑌 / (2 · π)) ∈
ℤ → (sin‘((𝑌 / (2 · π)) · π)) =
0) |
| 445 | 404, 444 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (sin‘((𝑌 / (2 · π)) ·
π)) = 0) |
| 446 | 443, 445 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (sin‘(𝑌 / 2)) = 0) |
| 447 | 446 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((2 · π)
· (sin‘(𝑌 /
2))) = ((2 · π) · 0)) |
| 448 | 156 | mul01d 11460 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((2 · π)
· 0) = 0) |
| 449 | 447, 448 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2 · π)
· (sin‘(𝑌 /
2))) = 0) |
| 450 | 449 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (𝜑 → 0 = ((2 · π)
· (sin‘(𝑌 /
2)))) |
| 451 | 450 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ 𝑤 = 𝑌) → 0 = ((2 · π) ·
(sin‘(𝑌 /
2)))) |
| 452 | | fvoveq1 7454 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑌 → (sin‘(𝑤 / 2)) = (sin‘(𝑌 / 2))) |
| 453 | 452 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑌 → ((2 · π) ·
(sin‘(𝑤 / 2))) = ((2
· π) · (sin‘(𝑌 / 2)))) |
| 454 | 453 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑌 → ((2 · π) ·
(sin‘(𝑌 / 2))) = ((2
· π) · (sin‘(𝑤 / 2)))) |
| 455 | 454 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ 𝑤 = 𝑌) → ((2 · π) ·
(sin‘(𝑌 / 2))) = ((2
· π) · (sin‘(𝑤 / 2)))) |
| 456 | 433, 451,
455 | 3eqtrd 2781 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ 𝑤 = 𝑌) → if(𝑤 = 𝑌, 0, (𝐺‘𝑤)) = ((2 · π) ·
(sin‘(𝑤 /
2)))) |
| 457 | | iffalse 4534 |
. . . . . . . . . 10
⊢ (¬
𝑤 = 𝑌 → if(𝑤 = 𝑌, 0, (𝐺‘𝑤)) = (𝐺‘𝑤)) |
| 458 | 457 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → if(𝑤 = 𝑌, 0, (𝐺‘𝑤)) = (𝐺‘𝑤)) |
| 459 | | fvoveq1 7454 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑤 → (sin‘(𝑦 / 2)) = (sin‘(𝑤 / 2))) |
| 460 | 459 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑤 → ((2 · π) ·
(sin‘(𝑦 / 2))) = ((2
· π) · (sin‘(𝑤 / 2)))) |
| 461 | 120 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (2 · π) ∈
ℂ) |
| 462 | 328 | halfcld 12511 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝑤 / 2) ∈ ℂ) |
| 463 | 462 | sincld 16166 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (sin‘(𝑤 / 2)) ∈ ℂ) |
| 464 | 461, 463 | mulcld 11281 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → ((2 · π) ·
(sin‘(𝑤 / 2))) ∈
ℂ) |
| 465 | 464 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → ((2 · π) ·
(sin‘(𝑤 / 2))) ∈
ℂ) |
| 466 | 23, 460, 324, 465 | fvmptd3 7039 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → (𝐺‘𝑤) = ((2 · π) ·
(sin‘(𝑤 /
2)))) |
| 467 | 458, 466 | eqtrd 2777 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → if(𝑤 = 𝑌, 0, (𝐺‘𝑤)) = ((2 · π) ·
(sin‘(𝑤 /
2)))) |
| 468 | 456, 467 | pm2.61dan 813 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → if(𝑤 = 𝑌, 0, (𝐺‘𝑤)) = ((2 · π) ·
(sin‘(𝑤 /
2)))) |
| 469 | 468 | mpteq2dva 5242 |
. . . . . 6
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ if(𝑤 = 𝑌, 0, (𝐺‘𝑤))) = (𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 /
2))))) |
| 470 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑤 ∈ ℂ ↦ ((2
· π) · (sin‘(𝑤 / 2)))) = (𝑤 ∈ ℂ ↦ ((2 · π)
· (sin‘(𝑤 /
2)))) |
| 471 | 75, 156, 75 | constcncfg 45887 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑤 ∈ ℂ ↦ (2 · π))
∈ (ℂ–cn→ℂ)) |
| 472 | | id 22 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 ∈ ℂ → 𝑤 ∈
ℂ) |
| 473 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 ∈ ℂ → 2 ∈
ℂ) |
| 474 | 134 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 ∈ ℂ → 2 ≠
0) |
| 475 | 472, 473,
474 | divrec2d 12047 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 ∈ ℂ → (𝑤 / 2) = ((1 / 2) · 𝑤)) |
| 476 | 475 | mpteq2ia 5245 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 ∈ ℂ ↦ (𝑤 / 2)) = (𝑤 ∈ ℂ ↦ ((1 / 2) ·
𝑤)) |
| 477 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 ∈ ℂ ↦ ((1 / 2)
· 𝑤)) = (𝑤 ∈ ℂ ↦ ((1 / 2)
· 𝑤)) |
| 478 | 477 | mulc1cncf 24931 |
. . . . . . . . . . . . . . . 16
⊢ ((1 / 2)
∈ ℂ → (𝑤
∈ ℂ ↦ ((1 / 2) · 𝑤)) ∈ (ℂ–cn→ℂ)) |
| 479 | 39, 478 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 ∈ ℂ ↦ ((1 / 2)
· 𝑤)) ∈
(ℂ–cn→ℂ) |
| 480 | 476, 479 | eqeltri 2837 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ∈ ℂ ↦ (𝑤 / 2)) ∈
(ℂ–cn→ℂ) |
| 481 | 480 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑤 ∈ ℂ ↦ (𝑤 / 2)) ∈ (ℂ–cn→ℂ)) |
| 482 | 415, 481 | cncfmpt1f 24940 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑤 ∈ ℂ ↦ (sin‘(𝑤 / 2))) ∈
(ℂ–cn→ℂ)) |
| 483 | 471, 482 | mulcncf 25480 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑤 ∈ ℂ ↦ ((2 · π)
· (sin‘(𝑤 /
2)))) ∈ (ℂ–cn→ℂ)) |
| 484 | 470, 483,
349, 75, 464 | cncfmptssg 45886 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 / 2))))
∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 485 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) = ((TopOpen‘ℂfld)
↾t (𝐴(,)𝐵)) |
| 486 | 51, 485, 345 | cncfcn 24936 |
. . . . . . . . . . 11
⊢ (((𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆
ℂ) → ((𝐴(,)𝐵)–cn→ℂ) =
(((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn
(TopOpen‘ℂfld))) |
| 487 | 349, 74, 486 | sylancl 586 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐴(,)𝐵)–cn→ℂ) =
(((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn
(TopOpen‘ℂfld))) |
| 488 | 484, 487 | eleqtrd 2843 |
. . . . . . . . 9
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 / 2))))
∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn
(TopOpen‘ℂfld))) |
| 489 | | cncnp 23288 |
. . . . . . . . . 10
⊢
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)) ∧
(TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) →
((𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 / 2))))
∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld))
↔ ((𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 /
2)))):(𝐴(,)𝐵)⟶ℂ ∧
∀𝑦 ∈ (𝐴(,)𝐵)(𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 / 2))))
∈ ((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦)))) |
| 490 | 355, 353,
489 | sylancl 586 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 / 2))))
∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld))
↔ ((𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 /
2)))):(𝐴(,)𝐵)⟶ℂ ∧
∀𝑦 ∈ (𝐴(,)𝐵)(𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 / 2))))
∈ ((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦)))) |
| 491 | 488, 490 | mpbid 232 |
. . . . . . . 8
⊢ (𝜑 → ((𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 /
2)))):(𝐴(,)𝐵)⟶ℂ ∧
∀𝑦 ∈ (𝐴(,)𝐵)(𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 / 2))))
∈ ((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦))) |
| 492 | 491 | simprd 495 |
. . . . . . 7
⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 / 2))))
∈ ((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦)) |
| 493 | 360 | eleq2d 2827 |
. . . . . . . 8
⊢ (𝑦 = 𝑌 → ((𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 / 2))))
∈ ((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦) ↔ (𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 / 2))))
∈ ((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌))) |
| 494 | 493 | rspccva 3621 |
. . . . . . 7
⊢
((∀𝑦 ∈
(𝐴(,)𝐵)(𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 / 2))))
∈ ((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦) ∧ 𝑌 ∈ (𝐴(,)𝐵)) → (𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 / 2))))
∈ ((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 495 | 492, 27, 494 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ ((2 · π) ·
(sin‘(𝑤 / 2))))
∈ ((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 496 | 469, 495 | eqeltrd 2841 |
. . . . 5
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ if(𝑤 = 𝑌, 0, (𝐺‘𝑤))) ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 497 | 307 | mpteq1d 5237 |
. . . . 5
⊢ (𝜑 → (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, 0, (𝐺‘𝑤))) = (𝑤 ∈ (𝐴(,)𝐵) ↦ if(𝑤 = 𝑌, 0, (𝐺‘𝑤)))) |
| 498 | 366 | eqcomd 2743 |
. . . . . . 7
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) = ((TopOpen‘ℂfld)
↾t (𝐴(,)𝐵))) |
| 499 | 498 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 →
(((TopOpen‘ℂfld) ↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) CnP
(TopOpen‘ℂfld)) =
(((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))) |
| 500 | 499 | fveq1d 6908 |
. . . . 5
⊢ (𝜑 →
((((TopOpen‘ℂfld) ↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) CnP
(TopOpen‘ℂfld))‘𝑌) = ((((TopOpen‘ℂfld)
↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 501 | 496, 497,
500 | 3eltr4d 2856 |
. . . 4
⊢ (𝜑 → (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, 0, (𝐺‘𝑤))) ∈
((((TopOpen‘ℂfld) ↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 502 | | eqid 2737 |
. . . . 5
⊢ (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, 0, (𝐺‘𝑤))) = (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, 0, (𝐺‘𝑤))) |
| 503 | 11, 124 | syldan 591 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((2 · π) ·
(sin‘(𝑦 / 2))) ∈
ℂ) |
| 504 | 503, 23 | fmptd 7134 |
. . . . 5
⊢ (𝜑 → 𝐺:((𝐴(,)𝐵) ∖ {𝑌})⟶ℂ) |
| 505 | 371, 51, 502, 504, 374, 263 | ellimc 25908 |
. . . 4
⊢ (𝜑 → (0 ∈ (𝐺 limℂ 𝑌) ↔ (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, 0, (𝐺‘𝑤))) ∈
((((TopOpen‘ℂfld) ↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) CnP
(TopOpen‘ℂfld))‘𝑌))) |
| 506 | 501, 505 | mpbird 257 |
. . 3
⊢ (𝜑 → 0 ∈ (𝐺 limℂ 𝑌)) |
| 507 | 256 | nrexdv 3149 |
. . . 4
⊢ (𝜑 → ¬ ∃𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})(𝑦 mod (2 · π)) = 0) |
| 508 | 504 | ffund 6740 |
. . . . . 6
⊢ (𝜑 → Fun 𝐺) |
| 509 | | fvelima 6974 |
. . . . . 6
⊢ ((Fun
𝐺 ∧ 0 ∈ (𝐺 “ ((𝐴(,)𝐵) ∖ {𝑌}))) → ∃𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})(𝐺‘𝑦) = 0) |
| 510 | 508, 509 | sylan 580 |
. . . . 5
⊢ ((𝜑 ∧ 0 ∈ (𝐺 “ ((𝐴(,)𝐵) ∖ {𝑌}))) → ∃𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})(𝐺‘𝑦) = 0) |
| 511 | | 2cnd 12344 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → 2 ∈
ℂ) |
| 512 | 119 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → π ∈
ℂ) |
| 513 | 134 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → 2 ≠ 0) |
| 514 | 238 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → π ≠ 0) |
| 515 | 105, 511,
512, 513, 514 | divdiv1d 12074 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝑦 / 2) / π) = (𝑦 / (2 · π))) |
| 516 | 515 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝑦 / (2 · π)) = ((𝑦 / 2) / π)) |
| 517 | 516 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ (𝐺‘𝑦) = 0) → (𝑦 / (2 · π)) = ((𝑦 / 2) / π)) |
| 518 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (𝐺‘𝑦) = 0) → 2 ∈
ℂ) |
| 519 | 119 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (𝐺‘𝑦) = 0) → π ∈
ℂ) |
| 520 | 518, 519 | mulcld 11281 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (𝐺‘𝑦) = 0) → (2 · π) ∈
ℂ) |
| 521 | 232 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (𝐺‘𝑦) = 0) → 𝑦 ∈ ℂ) |
| 522 | 521 | halfcld 12511 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (𝐺‘𝑦) = 0) → (𝑦 / 2) ∈ ℂ) |
| 523 | 522 | sincld 16166 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (𝐺‘𝑦) = 0) → (sin‘(𝑦 / 2)) ∈ ℂ) |
| 524 | 520, 523 | mulcld 11281 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (𝐺‘𝑦) = 0) → ((2 · π) ·
(sin‘(𝑦 / 2))) ∈
ℂ) |
| 525 | 23 | fvmpt2 7027 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ ((2 · π) ·
(sin‘(𝑦 / 2))) ∈
ℂ) → (𝐺‘𝑦) = ((2 · π) ·
(sin‘(𝑦 /
2)))) |
| 526 | 524, 525 | syldan 591 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (𝐺‘𝑦) = 0) → (𝐺‘𝑦) = ((2 · π) ·
(sin‘(𝑦 /
2)))) |
| 527 | 526 | eqcomd 2743 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (𝐺‘𝑦) = 0) → ((2 · π) ·
(sin‘(𝑦 / 2))) =
(𝐺‘𝑦)) |
| 528 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (𝐺‘𝑦) = 0) → (𝐺‘𝑦) = 0) |
| 529 | 527, 528 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (𝐺‘𝑦) = 0) → ((2 · π) ·
(sin‘(𝑦 / 2))) =
0) |
| 530 | 120 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → (2 · π) ∈
ℂ) |
| 531 | 232 | halfcld 12511 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → (𝑦 / 2) ∈ ℂ) |
| 532 | 531 | sincld 16166 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → (sin‘(𝑦 / 2)) ∈ ℂ) |
| 533 | 530, 532 | mul0ord 11913 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → (((2 · π) ·
(sin‘(𝑦 / 2))) = 0
↔ ((2 · π) = 0 ∨ (sin‘(𝑦 / 2)) = 0))) |
| 534 | 533 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (𝐺‘𝑦) = 0) → (((2 · π) ·
(sin‘(𝑦 / 2))) = 0
↔ ((2 · π) = 0 ∨ (sin‘(𝑦 / 2)) = 0))) |
| 535 | 529, 534 | mpbid 232 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (𝐺‘𝑦) = 0) → ((2 · π) = 0 ∨
(sin‘(𝑦 / 2)) =
0)) |
| 536 | | 2cnne0 12476 |
. . . . . . . . . . . . . . 15
⊢ (2 ∈
ℂ ∧ 2 ≠ 0) |
| 537 | 119, 238 | pm3.2i 470 |
. . . . . . . . . . . . . . 15
⊢ (π
∈ ℂ ∧ π ≠ 0) |
| 538 | | mulne0 11905 |
. . . . . . . . . . . . . . 15
⊢ (((2
∈ ℂ ∧ 2 ≠ 0) ∧ (π ∈ ℂ ∧ π ≠ 0))
→ (2 · π) ≠ 0) |
| 539 | 536, 537,
538 | mp2an 692 |
. . . . . . . . . . . . . 14
⊢ (2
· π) ≠ 0 |
| 540 | 539 | neii 2942 |
. . . . . . . . . . . . 13
⊢ ¬ (2
· π) = 0 |
| 541 | | pm2.53 852 |
. . . . . . . . . . . . 13
⊢ (((2
· π) = 0 ∨ (sin‘(𝑦 / 2)) = 0) → (¬ (2 · π) =
0 → (sin‘(𝑦 /
2)) = 0)) |
| 542 | 535, 540,
541 | mpisyl 21 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (𝐺‘𝑦) = 0) → (sin‘(𝑦 / 2)) = 0) |
| 543 | 542 | adantll 714 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ (𝐺‘𝑦) = 0) → (sin‘(𝑦 / 2)) = 0) |
| 544 | 105 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ (𝐺‘𝑦) = 0) → 𝑦 ∈ ℂ) |
| 545 | 544 | halfcld 12511 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ (𝐺‘𝑦) = 0) → (𝑦 / 2) ∈ ℂ) |
| 546 | 545, 246 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ (𝐺‘𝑦) = 0) → ((sin‘(𝑦 / 2)) = 0 ↔ ((𝑦 / 2) / π) ∈
ℤ)) |
| 547 | 543, 546 | mpbid 232 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ (𝐺‘𝑦) = 0) → ((𝑦 / 2) / π) ∈
ℤ) |
| 548 | 517, 547 | eqeltrd 2841 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ (𝐺‘𝑦) = 0) → (𝑦 / (2 · π)) ∈
ℤ) |
| 549 | 11 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ (𝐺‘𝑦) = 0) → 𝑦 ∈ ℝ) |
| 550 | 549, 253,
254 | sylancl 586 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ (𝐺‘𝑦) = 0) → ((𝑦 mod (2 · π)) = 0 ↔ (𝑦 / (2 · π)) ∈
ℤ)) |
| 551 | 548, 550 | mpbird 257 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) ∧ (𝐺‘𝑦) = 0) → (𝑦 mod (2 · π)) = 0) |
| 552 | 551 | ex 412 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝐺‘𝑦) = 0 → (𝑦 mod (2 · π)) =
0)) |
| 553 | 552 | reximdva 3168 |
. . . . . 6
⊢ (𝜑 → (∃𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})(𝐺‘𝑦) = 0 → ∃𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})(𝑦 mod (2 · π)) =
0)) |
| 554 | 553 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 0 ∈ (𝐺 “ ((𝐴(,)𝐵) ∖ {𝑌}))) → (∃𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})(𝐺‘𝑦) = 0 → ∃𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})(𝑦 mod (2 · π)) =
0)) |
| 555 | 510, 554 | mpd 15 |
. . . 4
⊢ ((𝜑 ∧ 0 ∈ (𝐺 “ ((𝐴(,)𝐵) ∖ {𝑌}))) → ∃𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})(𝑦 mod (2 · π)) = 0) |
| 556 | 507, 555 | mtand 816 |
. . 3
⊢ (𝜑 → ¬ 0 ∈ (𝐺 “ ((𝐴(,)𝐵) ∖ {𝑌}))) |
| 557 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) |
| 558 | 111 | fvmpt2 7027 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (π · (cos‘(𝑦 / 2))) ∈ ℂ) →
(𝐼‘𝑦) = (π · (cos‘(𝑦 / 2)))) |
| 559 | 557, 201,
558 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝐼‘𝑦) = (π · (cos‘(𝑦 / 2)))) |
| 560 | 531 | coscld 16167 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) → (cos‘(𝑦 / 2)) ∈ ℂ) |
| 561 | 560 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (cos‘(𝑦 / 2)) ∈ ℂ) |
| 562 | | dirkercncflem2.11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (cos‘(𝑦 / 2)) ≠ 0) |
| 563 | 512, 561,
514, 562 | mulne0d 11915 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (π · (cos‘(𝑦 / 2))) ≠ 0) |
| 564 | 559, 563 | eqnetrd 3008 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝐼‘𝑦) ≠ 0) |
| 565 | 564 | neneqd 2945 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ¬ (𝐼‘𝑦) = 0) |
| 566 | 565 | nrexdv 3149 |
. . . . 5
⊢ (𝜑 → ¬ ∃𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})(𝐼‘𝑦) = 0) |
| 567 | 201, 111 | fmptd 7134 |
. . . . . . 7
⊢ (𝜑 → 𝐼:((𝐴(,)𝐵) ∖ {𝑌})⟶ℂ) |
| 568 | 567 | ffund 6740 |
. . . . . 6
⊢ (𝜑 → Fun 𝐼) |
| 569 | | fvelima 6974 |
. . . . . 6
⊢ ((Fun
𝐼 ∧ 0 ∈ (𝐼 “ ((𝐴(,)𝐵) ∖ {𝑌}))) → ∃𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})(𝐼‘𝑦) = 0) |
| 570 | 568, 569 | sylan 580 |
. . . . 5
⊢ ((𝜑 ∧ 0 ∈ (𝐼 “ ((𝐴(,)𝐵) ∖ {𝑌}))) → ∃𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})(𝐼‘𝑦) = 0) |
| 571 | 566, 570 | mtand 816 |
. . . 4
⊢ (𝜑 → ¬ 0 ∈ (𝐼 “ ((𝐴(,)𝐵) ∖ {𝑌}))) |
| 572 | 199 | imaeq1d 6077 |
. . . 4
⊢ (𝜑 → ((ℝ D 𝐺) “ ((𝐴(,)𝐵) ∖ {𝑌})) = (𝐼 “ ((𝐴(,)𝐵) ∖ {𝑌}))) |
| 573 | 571, 572 | neleqtrrd 2864 |
. . 3
⊢ (𝜑 → ¬ 0 ∈ ((ℝ D
𝐺) “ ((𝐴(,)𝐵) ∖ {𝑌}))) |
| 574 | | dirkercncflem2.d |
. . . . . 6
⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0,
(((2 · 𝑛) + 1) / (2
· π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) ·
(sin‘(𝑦 /
2))))))) |
| 575 | 574 | dirkerval2 46109 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝑌 ∈ ℝ) → ((𝐷‘𝑁)‘𝑌) = if((𝑌 mod (2 · π)) = 0, (((2 ·
𝑁) + 1) / (2 ·
π)), ((sin‘((𝑁 +
(1 / 2)) · 𝑌)) / ((2
· π) · (sin‘(𝑌 / 2)))))) |
| 576 | 5, 57, 575 | syl2anc 584 |
. . . 4
⊢ (𝜑 → ((𝐷‘𝑁)‘𝑌) = if((𝑌 mod (2 · π)) = 0, (((2 ·
𝑁) + 1) / (2 ·
π)), ((sin‘((𝑁 +
(1 / 2)) · 𝑌)) / ((2
· π) · (sin‘(𝑌 / 2)))))) |
| 577 | 282 | iftrued 4533 |
. . . . 5
⊢ (𝜑 → if((𝑌 mod (2 · π)) = 0, (((2 ·
𝑁) + 1) / (2 ·
π)), ((sin‘((𝑁 +
(1 / 2)) · 𝑌)) / ((2
· π) · (sin‘(𝑌 / 2))))) = (((2 · 𝑁) + 1) / (2 ·
π))) |
| 578 | | dirkercncflem2.l |
. . . . . . . . . . 11
⊢ 𝐿 = (𝑤 ∈ (𝐴(,)𝐵) ↦ (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) / (π ·
(cos‘(𝑤 /
2))))) |
| 579 | 578 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐿 = (𝑤 ∈ (𝐴(,)𝐵) ↦ (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) / (π ·
(cos‘(𝑤 /
2)))))) |
| 580 | | iftrue 4531 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑌 → if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤)) = (((2 · 𝑁) + 1) / (2 ·
π))) |
| 581 | 580 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ 𝑤 = 𝑌) → if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤)) = (((2 · 𝑁) + 1) / (2 ·
π))) |
| 582 | 154, 38 | mulcld 11281 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (2 · 𝑁) ∈
ℂ) |
| 583 | 582, 397 | addcld 11280 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2 · 𝑁) + 1) ∈
ℂ) |
| 584 | 583, 154,
155, 377, 378 | divdiv1d 12074 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((((2 · 𝑁) + 1) / 2) / π) = (((2
· 𝑁) + 1) / (2
· π))) |
| 585 | 584 | eqcomd 2743 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2 · 𝑁) + 1) / (2 · π)) =
((((2 · 𝑁) + 1) / 2)
/ π)) |
| 586 | 582, 397,
154, 377 | divdird 12081 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (((2 · 𝑁) + 1) / 2) = (((2 ·
𝑁) / 2) + (1 /
2))) |
| 587 | 38, 154, 377 | divcan3d 12048 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((2 · 𝑁) / 2) = 𝑁) |
| 588 | 587 | oveq1d 7446 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (((2 · 𝑁) / 2) + (1 / 2)) = (𝑁 + (1 / 2))) |
| 589 | 586, 588 | eqtrd 2777 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((2 · 𝑁) + 1) / 2) = (𝑁 + (1 / 2))) |
| 590 | 589 | oveq1d 7446 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((((2 · 𝑁) + 1) / 2) / π) = ((𝑁 + (1 / 2)) /
π)) |
| 591 | 585, 590 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((2 · 𝑁) + 1) / (2 · π)) =
((𝑁 + (1 / 2)) /
π)) |
| 592 | 591 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ 𝑤 = 𝑌) → (((2 · 𝑁) + 1) / (2 · π)) = ((𝑁 + (1 / 2)) /
π)) |
| 593 | 310 | fveq2d 6910 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 = 𝑌 → (cos‘((𝑁 + (1 / 2)) · 𝑤)) = (cos‘((𝑁 + (1 / 2)) · 𝑌))) |
| 594 | 593 | oveq2d 7447 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 𝑌 → ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) = ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑌)))) |
| 595 | | fvoveq1 7454 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 = 𝑌 → (cos‘(𝑤 / 2)) = (cos‘(𝑌 / 2))) |
| 596 | 595 | oveq2d 7447 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 𝑌 → (π · (cos‘(𝑤 / 2))) = (π ·
(cos‘(𝑌 /
2)))) |
| 597 | 594, 596 | oveq12d 7449 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑌 → (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) / (π ·
(cos‘(𝑤 / 2)))) =
(((𝑁 + (1 / 2)) ·
(cos‘((𝑁 + (1 / 2))
· 𝑌))) / (π
· (cos‘(𝑌 /
2))))) |
| 598 | 597 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ 𝑤 = 𝑌) → (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) / (π ·
(cos‘(𝑤 / 2)))) =
(((𝑁 + (1 / 2)) ·
(cos‘((𝑁 + (1 / 2))
· 𝑌))) / (π
· (cos‘(𝑌 /
2))))) |
| 599 | 38, 40, 263 | adddird 11286 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑁 + (1 / 2)) · 𝑌) = ((𝑁 · 𝑌) + ((1 / 2) · 𝑌))) |
| 600 | 397, 154,
263, 377 | div32d 12066 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((1 / 2) · 𝑌) = (1 · (𝑌 / 2))) |
| 601 | 436 | mullidd 11279 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (1 · (𝑌 / 2)) = (𝑌 / 2)) |
| 602 | 600, 601 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1 / 2) · 𝑌) = (𝑌 / 2)) |
| 603 | 602 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑁 · 𝑌) + ((1 / 2) · 𝑌)) = ((𝑁 · 𝑌) + (𝑌 / 2))) |
| 604 | 38, 263 | mulcld 11281 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑁 · 𝑌) ∈ ℂ) |
| 605 | 604, 436 | addcomd 11463 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑁 · 𝑌) + (𝑌 / 2)) = ((𝑌 / 2) + (𝑁 · 𝑌))) |
| 606 | 599, 603,
605 | 3eqtrd 2781 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑁 + (1 / 2)) · 𝑌) = ((𝑌 / 2) + (𝑁 · 𝑌))) |
| 607 | 606 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (cos‘((𝑁 + (1 / 2)) · 𝑌)) = (cos‘((𝑌 / 2) + (𝑁 · 𝑌)))) |
| 608 | 604, 156,
379 | divcan1d 12044 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (((𝑁 · 𝑌) / (2 · π)) · (2 ·
π)) = (𝑁 · 𝑌)) |
| 609 | 608 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑁 · 𝑌) = (((𝑁 · 𝑌) / (2 · π)) · (2 ·
π))) |
| 610 | 609 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑌 / 2) + (𝑁 · 𝑌)) = ((𝑌 / 2) + (((𝑁 · 𝑌) / (2 · π)) · (2 ·
π)))) |
| 611 | 610 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (cos‘((𝑌 / 2) + (𝑁 · 𝑌))) = (cos‘((𝑌 / 2) + (((𝑁 · 𝑌) / (2 · π)) · (2 ·
π))))) |
| 612 | 38, 263, 156, 379 | divassd 12078 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑁 · 𝑌) / (2 · π)) = (𝑁 · (𝑌 / (2 · π)))) |
| 613 | 405, 404 | zmulcld 12728 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑁 · (𝑌 / (2 · π))) ∈
ℤ) |
| 614 | 612, 613 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑁 · 𝑌) / (2 · π)) ∈
ℤ) |
| 615 | | cosper 26524 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑌 / 2) ∈ ℂ ∧
((𝑁 · 𝑌) / (2 · π)) ∈
ℤ) → (cos‘((𝑌 / 2) + (((𝑁 · 𝑌) / (2 · π)) · (2 ·
π)))) = (cos‘(𝑌 /
2))) |
| 616 | 436, 614,
615 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (cos‘((𝑌 / 2) + (((𝑁 · 𝑌) / (2 · π)) · (2 ·
π)))) = (cos‘(𝑌 /
2))) |
| 617 | 607, 611,
616 | 3eqtrd 2781 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (cos‘((𝑁 + (1 / 2)) · 𝑌)) = (cos‘(𝑌 / 2))) |
| 618 | 617 | oveq2d 7447 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑌))) = ((𝑁 + (1 / 2)) · (cos‘(𝑌 / 2)))) |
| 619 | 618 | oveq1d 7446 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑌))) / (π ·
(cos‘(𝑌 / 2)))) =
(((𝑁 + (1 / 2)) ·
(cos‘(𝑌 / 2))) /
(π · (cos‘(𝑌 / 2))))) |
| 620 | 436 | coscld 16167 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (cos‘(𝑌 / 2)) ∈
ℂ) |
| 621 | 263, 154,
155, 377, 378 | divdiv1d 12074 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑌 / 2) / π) = (𝑌 / (2 · π))) |
| 622 | 621, 404 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑌 / 2) / π) ∈
ℤ) |
| 623 | 622 | zred 12722 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑌 / 2) / π) ∈
ℝ) |
| 624 | 623, 272 | ltaddrpd 13110 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑌 / 2) / π) < (((𝑌 / 2) / π) + (1 / 2))) |
| 625 | | halflt1 12484 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (1 / 2)
< 1 |
| 626 | 625 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (1 / 2) <
1) |
| 627 | 268, 267,
623, 626 | ltadd2dd 11420 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((𝑌 / 2) / π) + (1 / 2)) < (((𝑌 / 2) / π) +
1)) |
| 628 | | btwnnz 12694 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑌 / 2) / π) ∈ ℤ
∧ ((𝑌 / 2) / π) <
(((𝑌 / 2) / π) + (1 /
2)) ∧ (((𝑌 / 2) / π)
+ (1 / 2)) < (((𝑌 / 2) /
π) + 1)) → ¬ (((𝑌 / 2) / π) + (1 / 2)) ∈
ℤ) |
| 629 | 622, 624,
627, 628 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ¬ (((𝑌 / 2) / π) + (1 / 2)) ∈
ℤ) |
| 630 | | coseq0 45879 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑌 / 2) ∈ ℂ →
((cos‘(𝑌 / 2)) = 0
↔ (((𝑌 / 2) / π) +
(1 / 2)) ∈ ℤ)) |
| 631 | 436, 630 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((cos‘(𝑌 / 2)) = 0 ↔ (((𝑌 / 2) / π) + (1 / 2)) ∈
ℤ)) |
| 632 | 629, 631 | mtbird 325 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ¬ (cos‘(𝑌 / 2)) = 0) |
| 633 | 632 | neqned 2947 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (cos‘(𝑌 / 2)) ≠ 0) |
| 634 | 41, 155, 620, 378, 633 | divcan5rd 12070 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((𝑁 + (1 / 2)) · (cos‘(𝑌 / 2))) / (π ·
(cos‘(𝑌 / 2)))) =
((𝑁 + (1 / 2)) /
π)) |
| 635 | 619, 634 | eqtrd 2777 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑌))) / (π ·
(cos‘(𝑌 / 2)))) =
((𝑁 + (1 / 2)) /
π)) |
| 636 | 635 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ 𝑤 = 𝑌) → (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑌))) / (π ·
(cos‘(𝑌 / 2)))) =
((𝑁 + (1 / 2)) /
π)) |
| 637 | 598, 636 | eqtr2d 2778 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ 𝑤 = 𝑌) → ((𝑁 + (1 / 2)) / π) = (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) / (π ·
(cos‘(𝑤 /
2))))) |
| 638 | 581, 592,
637 | 3eqtrrd 2782 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ 𝑤 = 𝑌) → (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) / (π ·
(cos‘(𝑤 / 2)))) =
if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤))) |
| 639 | | iffalse 4534 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑤 = 𝑌 → if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤)) = ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤)) |
| 640 | 639 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤)) = ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤)) |
| 641 | | eqidd 2738 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦))) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))) |
| 642 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑤 → (𝐻‘𝑦) = (𝐻‘𝑤)) |
| 643 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑤 → (𝐼‘𝑦) = (𝐼‘𝑤)) |
| 644 | 642, 643 | oveq12d 7449 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑤 → ((𝐻‘𝑦) / (𝐼‘𝑦)) = ((𝐻‘𝑤) / (𝐼‘𝑤))) |
| 645 | 644 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) ∧ 𝑦 = 𝑤) → ((𝐻‘𝑦) / (𝐼‘𝑦)) = ((𝐻‘𝑤) / (𝐼‘𝑤))) |
| 646 | 106, 100 | fmptd 7134 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐻:((𝐴(,)𝐵) ∖ {𝑌})⟶ℂ) |
| 647 | 646 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → 𝐻:((𝐴(,)𝐵) ∖ {𝑌})⟶ℂ) |
| 648 | 647, 324 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → (𝐻‘𝑤) ∈ ℂ) |
| 649 | 567 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → 𝐼:((𝐴(,)𝐵) ∖ {𝑌})⟶ℂ) |
| 650 | 649, 324 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → (𝐼‘𝑤) ∈ ℂ) |
| 651 | 111 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → 𝐼 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (π · (cos‘(𝑦 / 2))))) |
| 652 | | simpr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤) |
| 653 | 652 | fvoveq1d 7453 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) ∧ 𝑦 = 𝑤) → (cos‘(𝑦 / 2)) = (cos‘(𝑤 / 2))) |
| 654 | 653 | oveq2d 7447 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) ∧ 𝑦 = 𝑤) → (π · (cos‘(𝑦 / 2))) = (π ·
(cos‘(𝑤 /
2)))) |
| 655 | 119 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 ∈ (𝐴(,)𝐵) → π ∈
ℂ) |
| 656 | 327 | halfcld 12511 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 ∈ (𝐴(,)𝐵) → (𝑤 / 2) ∈ ℂ) |
| 657 | 656 | coscld 16167 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 ∈ (𝐴(,)𝐵) → (cos‘(𝑤 / 2)) ∈ ℂ) |
| 658 | 655, 657 | mulcld 11281 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 ∈ (𝐴(,)𝐵) → (π · (cos‘(𝑤 / 2))) ∈
ℂ) |
| 659 | 658 | ad2antlr 727 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → (π · (cos‘(𝑤 / 2))) ∈
ℂ) |
| 660 | 651, 654,
324, 659 | fvmptd 7023 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → (𝐼‘𝑤) = (π · (cos‘(𝑤 / 2)))) |
| 661 | 537 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → (π ∈ ℂ ∧ π
≠ 0)) |
| 662 | 657 | ad2antlr 727 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → (cos‘(𝑤 / 2)) ∈ ℂ) |
| 663 | | simpll 767 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → 𝜑) |
| 664 | | fvoveq1 7454 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑤 → (cos‘(𝑦 / 2)) = (cos‘(𝑤 / 2))) |
| 665 | 664 | neeq1d 3000 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑤 → ((cos‘(𝑦 / 2)) ≠ 0 ↔ (cos‘(𝑤 / 2)) ≠ 0)) |
| 666 | 226, 665 | imbi12d 344 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑤 → (((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (cos‘(𝑦 / 2)) ≠ 0) ↔ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (cos‘(𝑤 / 2)) ≠ 0))) |
| 667 | 666, 562 | chvarvv 1998 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (cos‘(𝑤 / 2)) ≠ 0) |
| 668 | 663, 324,
667 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → (cos‘(𝑤 / 2)) ≠ 0) |
| 669 | | mulne0 11905 |
. . . . . . . . . . . . . . . . 17
⊢ (((π
∈ ℂ ∧ π ≠ 0) ∧ ((cos‘(𝑤 / 2)) ∈ ℂ ∧ (cos‘(𝑤 / 2)) ≠ 0)) → (π
· (cos‘(𝑤 /
2))) ≠ 0) |
| 670 | 661, 662,
668, 669 | syl12anc 837 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → (π · (cos‘(𝑤 / 2))) ≠ 0) |
| 671 | 660, 670 | eqnetrd 3008 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → (𝐼‘𝑤) ≠ 0) |
| 672 | 648, 650,
671 | divcld 12043 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → ((𝐻‘𝑤) / (𝐼‘𝑤)) ∈ ℂ) |
| 673 | 641, 645,
324, 672 | fvmptd 7023 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤) = ((𝐻‘𝑤) / (𝐼‘𝑤))) |
| 674 | 100 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → 𝐻 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))))) |
| 675 | 317 | fveq2d 6910 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) ∧ 𝑦 = 𝑤) → (cos‘((𝑁 + (1 / 2)) · 𝑦)) = (cos‘((𝑁 + (1 / 2)) · 𝑤))) |
| 676 | 675 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) ∧ 𝑦 = 𝑤) → ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))) = ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤)))) |
| 677 | 329 | coscld 16167 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (cos‘((𝑁 + (1 / 2)) · 𝑤)) ∈ ℂ) |
| 678 | 325, 677 | mulcld 11281 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) ∈
ℂ) |
| 679 | 678 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) ∈
ℂ) |
| 680 | 674, 676,
324, 679 | fvmptd 7023 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → (𝐻‘𝑤) = ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤)))) |
| 681 | 680, 660 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → ((𝐻‘𝑤) / (𝐼‘𝑤)) = (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) / (π ·
(cos‘(𝑤 /
2))))) |
| 682 | 640, 673,
681 | 3eqtrrd 2782 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑤 = 𝑌) → (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) / (π ·
(cos‘(𝑤 / 2)))) =
if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤))) |
| 683 | 638, 682 | pm2.61dan 813 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) / (π ·
(cos‘(𝑤 / 2)))) =
if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤))) |
| 684 | 683 | mpteq2dva 5242 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) / (π ·
(cos‘(𝑤 / 2))))) =
(𝑤 ∈ (𝐴(,)𝐵) ↦ if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤)))) |
| 685 | 579, 684 | eqtr2d 2778 |
. . . . . . . . 9
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤))) = 𝐿) |
| 686 | 349, 41, 75 | constcncfg 45887 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ (𝑁 + (1 / 2))) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 687 | | cosf 16161 |
. . . . . . . . . . . . . . . . . . 19
⊢
cos:ℂ⟶ℂ |
| 688 | 231, 44 | sylan2 593 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((𝑁 + (1 / 2)) · 𝑦) ∈ ℂ) |
| 689 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦)) |
| 690 | 688, 689 | fmptd 7134 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦)):(𝐴(,)𝐵)⟶ℂ) |
| 691 | | fcompt 7153 |
. . . . . . . . . . . . . . . . . . 19
⊢
((cos:ℂ⟶ℂ ∧ (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦)):(𝐴(,)𝐵)⟶ℂ) → (cos ∘ (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦))) = (𝑤 ∈ (𝐴(,)𝐵) ↦ (cos‘((𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)))) |
| 692 | 687, 690,
691 | sylancr 587 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (cos ∘ (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦))) = (𝑤 ∈ (𝐴(,)𝐵) ↦ (cos‘((𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)))) |
| 693 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦))) |
| 694 | 316 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ 𝑦 = 𝑤) → ((𝑁 + (1 / 2)) · 𝑦) = ((𝑁 + (1 / 2)) · 𝑤)) |
| 695 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → 𝑤 ∈ (𝐴(,)𝐵)) |
| 696 | 693, 694,
695, 329 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → ((𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤) = ((𝑁 + (1 / 2)) · 𝑤)) |
| 697 | 696 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (cos‘((𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤)) = (cos‘((𝑁 + (1 / 2)) · 𝑤))) |
| 698 | 697 | mpteq2dva 5242 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ (cos‘((𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦))‘𝑤))) = (𝑤 ∈ (𝐴(,)𝐵) ↦ (cos‘((𝑁 + (1 / 2)) · 𝑤)))) |
| 699 | 692, 698 | eqtr2d 2778 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ (cos‘((𝑁 + (1 / 2)) · 𝑤))) = (cos ∘ (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦)))) |
| 700 | 349, 41, 75 | constcncfg 45887 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝑁 + (1 / 2))) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 701 | 349, 75 | idcncfg 45888 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 𝑦) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 702 | 700, 701 | mulcncf 25480 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦)) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 703 | | coscn 26489 |
. . . . . . . . . . . . . . . . . . 19
⊢ cos
∈ (ℂ–cn→ℂ) |
| 704 | 703 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → cos ∈
(ℂ–cn→ℂ)) |
| 705 | 702, 704 | cncfco 24933 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (cos ∘ (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · 𝑦))) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 706 | 699, 705 | eqeltrd 2841 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ (cos‘((𝑁 + (1 / 2)) · 𝑤))) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 707 | 686, 706 | mulcncf 25480 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤)))) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 708 | | eqid 2737 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 ∈ (𝐴(,)𝐵) ↦ (π · (cos‘(𝑤 / 2)))) = (𝑤 ∈ (𝐴(,)𝐵) ↦ (π · (cos‘(𝑤 / 2)))) |
| 709 | 349, 155,
75 | constcncfg 45887 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ π) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 710 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ) |
| 711 | 134 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → 2 ≠ 0) |
| 712 | 328, 710,
711 | divrecd 12046 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝑤 / 2) = (𝑤 · (1 / 2))) |
| 713 | 712 | mpteq2dva 5242 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ (𝑤 / 2)) = (𝑤 ∈ (𝐴(,)𝐵) ↦ (𝑤 · (1 / 2)))) |
| 714 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 ∈ ℂ ↦ (𝑤 · (1 / 2))) = (𝑤 ∈ ℂ ↦ (𝑤 · (1 /
2))) |
| 715 | | cncfmptid 24939 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ℂ
⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑤 ∈ ℂ ↦ 𝑤) ∈ (ℂ–cn→ℂ)) |
| 716 | 74, 74, 715 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑤 ∈ ℂ ↦ 𝑤) ∈ (ℂ–cn→ℂ) |
| 717 | 716 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑤 ∈ ℂ ↦ 𝑤) ∈ (ℂ–cn→ℂ)) |
| 718 | 74 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((1 / 2)
∈ ℂ → ℂ ⊆ ℂ) |
| 719 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((1 / 2)
∈ ℂ → (1 / 2) ∈ ℂ) |
| 720 | 718, 719,
718 | constcncfg 45887 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((1 / 2)
∈ ℂ → (𝑤
∈ ℂ ↦ (1 / 2)) ∈ (ℂ–cn→ℂ)) |
| 721 | 39, 720 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑤 ∈ ℂ ↦ (1 / 2)) ∈
(ℂ–cn→ℂ)) |
| 722 | 717, 721 | mulcncf 25480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑤 ∈ ℂ ↦ (𝑤 · (1 / 2))) ∈
(ℂ–cn→ℂ)) |
| 723 | 712, 462 | eqeltrrd 2842 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝑤 · (1 / 2)) ∈
ℂ) |
| 724 | 714, 722,
349, 75, 723 | cncfmptssg 45886 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ (𝑤 · (1 / 2))) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 725 | 713, 724 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ (𝑤 / 2)) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 726 | 704, 725 | cncfmpt1f 24940 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ (cos‘(𝑤 / 2))) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 727 | 709, 726 | mulcncf 25480 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ (π · (cos‘(𝑤 / 2)))) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 728 | | ssid 4006 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵) |
| 729 | 728 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵)) |
| 730 | | difssd 4137 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (ℂ ∖ {0})
⊆ ℂ) |
| 731 | 658 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (π · (cos‘(𝑤 / 2))) ∈
ℂ) |
| 732 | 119 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → π ∈
ℂ) |
| 733 | 657 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (cos‘(𝑤 / 2)) ∈ ℂ) |
| 734 | 238 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → π ≠ 0) |
| 735 | 595 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑤 = 𝑌) → (cos‘(𝑤 / 2)) = (cos‘(𝑌 / 2))) |
| 736 | 633 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑤 = 𝑌) → (cos‘(𝑌 / 2)) ≠ 0) |
| 737 | 735, 736 | eqnetrd 3008 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑤 = 𝑌) → (cos‘(𝑤 / 2)) ≠ 0) |
| 738 | 737 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) ∧ 𝑤 = 𝑌) → (cos‘(𝑤 / 2)) ≠ 0) |
| 739 | 738, 668 | pm2.61dan 813 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (cos‘(𝑤 / 2)) ≠ 0) |
| 740 | 732, 733,
734, 739 | mulne0d 11915 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (π · (cos‘(𝑤 / 2))) ≠ 0) |
| 741 | 740 | neneqd 2945 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → ¬ (π ·
(cos‘(𝑤 / 2))) =
0) |
| 742 | | elsng 4640 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((π
· (cos‘(𝑤 /
2))) ∈ ℂ → ((π · (cos‘(𝑤 / 2))) ∈ {0} ↔ (π ·
(cos‘(𝑤 / 2))) =
0)) |
| 743 | 731, 742 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → ((π · (cos‘(𝑤 / 2))) ∈ {0} ↔ (π
· (cos‘(𝑤 /
2))) = 0)) |
| 744 | 741, 743 | mtbird 325 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → ¬ (π ·
(cos‘(𝑤 / 2))) ∈
{0}) |
| 745 | 731, 744 | eldifd 3962 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (π · (cos‘(𝑤 / 2))) ∈ (ℂ ∖
{0})) |
| 746 | 708, 727,
729, 730, 745 | cncfmptssg 45886 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ (π · (cos‘(𝑤 / 2)))) ∈ ((𝐴(,)𝐵)–cn→(ℂ ∖ {0}))) |
| 747 | 707, 746 | divcncf 25482 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) / (π ·
(cos‘(𝑤 / 2)))))
∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 748 | 747, 487 | eleqtrd 2843 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) / (π ·
(cos‘(𝑤 / 2)))))
∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn
(TopOpen‘ℂfld))) |
| 749 | 579, 748 | eqeltrd 2841 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐿 ∈
(((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn
(TopOpen‘ℂfld))) |
| 750 | | cncnp 23288 |
. . . . . . . . . . . . 13
⊢
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)) ∧
(TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) →
(𝐿 ∈
(((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld))
↔ (𝐿:(𝐴(,)𝐵)⟶ℂ ∧ ∀𝑦 ∈ (𝐴(,)𝐵)𝐿 ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦)))) |
| 751 | 355, 353,
750 | sylancl 586 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐿 ∈
(((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld))
↔ (𝐿:(𝐴(,)𝐵)⟶ℂ ∧ ∀𝑦 ∈ (𝐴(,)𝐵)𝐿 ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦)))) |
| 752 | 749, 751 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐿:(𝐴(,)𝐵)⟶ℂ ∧ ∀𝑦 ∈ (𝐴(,)𝐵)𝐿 ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦))) |
| 753 | 752 | simprd 495 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)𝐿 ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦)) |
| 754 | 360 | eleq2d 2827 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑌 → (𝐿 ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦) ↔ 𝐿 ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌))) |
| 755 | 754 | rspccva 3621 |
. . . . . . . . . 10
⊢
((∀𝑦 ∈
(𝐴(,)𝐵)𝐿 ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑦) ∧ 𝑌 ∈ (𝐴(,)𝐵)) → 𝐿 ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 756 | 753, 27, 755 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → 𝐿 ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 757 | 685, 756 | eqeltrd 2841 |
. . . . . . . 8
⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤))) ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 758 | 307 | mpteq1d 5237 |
. . . . . . . 8
⊢ (𝜑 → (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤))) = (𝑤 ∈ (𝐴(,)𝐵) ↦ if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤)))) |
| 759 | 757, 758,
500 | 3eltr4d 2856 |
. . . . . . 7
⊢ (𝜑 → (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤))) ∈
((((TopOpen‘ℂfld) ↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) CnP
(TopOpen‘ℂfld))‘𝑌)) |
| 760 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤))) = (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤))) |
| 761 | 100 | fvmpt2 7027 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))) ∈ ℂ) →
(𝐻‘𝑦) = ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦)))) |
| 762 | 557, 106,
761 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝐻‘𝑦) = ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦)))) |
| 763 | 762, 559 | oveq12d 7449 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝐻‘𝑦) / (𝐼‘𝑦)) = (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))) / (π ·
(cos‘(𝑦 /
2))))) |
| 764 | 106, 201,
563 | divcld 12043 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))) / (π ·
(cos‘(𝑦 / 2))))
∈ ℂ) |
| 765 | 763, 764 | eqeltrd 2841 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝐻‘𝑦) / (𝐼‘𝑦)) ∈ ℂ) |
| 766 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦))) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦))) |
| 767 | 765, 766 | fmptd 7134 |
. . . . . . . 8
⊢ (𝜑 → (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦))):((𝐴(,)𝐵) ∖ {𝑌})⟶ℂ) |
| 768 | 371, 51, 760, 767, 374, 263 | ellimc 25908 |
. . . . . . 7
⊢ (𝜑 → ((((2 · 𝑁) + 1) / (2 · π))
∈ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦))) limℂ 𝑌) ↔ (𝑤 ∈ (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌}) ↦ if(𝑤 = 𝑌, (((2 · 𝑁) + 1) / (2 · π)), ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦)))‘𝑤))) ∈
((((TopOpen‘ℂfld) ↾t (((𝐴(,)𝐵) ∖ {𝑌}) ∪ {𝑌})) CnP
(TopOpen‘ℂfld))‘𝑌))) |
| 769 | 759, 768 | mpbird 257 |
. . . . . 6
⊢ (𝜑 → (((2 · 𝑁) + 1) / (2 · π))
∈ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦))) limℂ 𝑌)) |
| 770 | 103 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐻 = (ℝ D 𝐹)) |
| 771 | 770 | fveq1d 6908 |
. . . . . . . . 9
⊢ (𝜑 → (𝐻‘𝑦) = ((ℝ D 𝐹)‘𝑦)) |
| 772 | 199 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 = (ℝ D 𝐺)) |
| 773 | 772 | fveq1d 6908 |
. . . . . . . . 9
⊢ (𝜑 → (𝐼‘𝑦) = ((ℝ D 𝐺)‘𝑦)) |
| 774 | 771, 773 | oveq12d 7449 |
. . . . . . . 8
⊢ (𝜑 → ((𝐻‘𝑦) / (𝐼‘𝑦)) = (((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦))) |
| 775 | 774 | mpteq2dv 5244 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦))) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)))) |
| 776 | 775 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐻‘𝑦) / (𝐼‘𝑦))) limℂ 𝑌) = ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦))) limℂ 𝑌)) |
| 777 | 769, 776 | eleqtrd 2843 |
. . . . 5
⊢ (𝜑 → (((2 · 𝑁) + 1) / (2 · π))
∈ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦))) limℂ 𝑌)) |
| 778 | 577, 777 | eqeltrd 2841 |
. . . 4
⊢ (𝜑 → if((𝑌 mod (2 · π)) = 0, (((2 ·
𝑁) + 1) / (2 ·
π)), ((sin‘((𝑁 +
(1 / 2)) · 𝑌)) / ((2
· π) · (sin‘(𝑌 / 2))))) ∈ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦))) limℂ 𝑌)) |
| 779 | 576, 778 | eqeltrd 2841 |
. . 3
⊢ (𝜑 → ((𝐷‘𝑁)‘𝑌) ∈ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦))) limℂ 𝑌)) |
| 780 | 4, 15, 24, 26, 27, 28, 110, 205, 431, 506, 556, 573, 779 | lhop 26055 |
. 2
⊢ (𝜑 → ((𝐷‘𝑁)‘𝑌) ∈ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐹‘𝑦) / (𝐺‘𝑦))) limℂ 𝑌)) |
| 781 | 574 | dirkerval 46106 |
. . . . . 6
⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁) = (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0,
(((2 · 𝑁) + 1) / (2
· π)), ((sin‘((𝑁 + (1 / 2)) · 𝑦)) / ((2 · π) ·
(sin‘(𝑦 /
2))))))) |
| 782 | 5, 781 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐷‘𝑁) = (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0,
(((2 · 𝑁) + 1) / (2
· π)), ((sin‘((𝑁 + (1 / 2)) · 𝑦)) / ((2 · π) ·
(sin‘(𝑦 /
2))))))) |
| 783 | 782 | reseq1d 5996 |
. . . 4
⊢ (𝜑 → ((𝐷‘𝑁) ↾ ((𝐴(,)𝐵) ∖ {𝑌})) = ((𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0,
(((2 · 𝑁) + 1) / (2
· π)), ((sin‘((𝑁 + (1 / 2)) · 𝑦)) / ((2 · π) ·
(sin‘(𝑦 / 2))))))
↾ ((𝐴(,)𝐵) ∖ {𝑌}))) |
| 784 | 4 | resmptd 6058 |
. . . 4
⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0,
(((2 · 𝑁) + 1) / (2
· π)), ((sin‘((𝑁 + (1 / 2)) · 𝑦)) / ((2 · π) ·
(sin‘(𝑦 / 2))))))
↾ ((𝐴(,)𝐵) ∖ {𝑌})) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ if((𝑦 mod (2 · π)) = 0, (((2 ·
𝑁) + 1) / (2 ·
π)), ((sin‘((𝑁 +
(1 / 2)) · 𝑦)) / ((2
· π) · (sin‘(𝑦 / 2))))))) |
| 785 | 256 | iffalsed 4536 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → if((𝑦 mod (2 · π)) = 0, (((2 ·
𝑁) + 1) / (2 ·
π)), ((sin‘((𝑁 +
(1 / 2)) · 𝑦)) / ((2
· π) · (sin‘(𝑦 / 2))))) = ((sin‘((𝑁 + (1 / 2)) · 𝑦)) / ((2 · π) ·
(sin‘(𝑦 /
2))))) |
| 786 | 13 | recnd 11289 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (sin‘((𝑁 + (1 / 2)) · 𝑦)) ∈ ℂ) |
| 787 | 14 | fvmpt2 7027 |
. . . . . . . 8
⊢ ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ∧ (sin‘((𝑁 + (1 / 2)) · 𝑦)) ∈ ℂ) → (𝐹‘𝑦) = (sin‘((𝑁 + (1 / 2)) · 𝑦))) |
| 788 | 557, 786,
787 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝐹‘𝑦) = (sin‘((𝑁 + (1 / 2)) · 𝑦))) |
| 789 | 557, 503,
525 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (𝐺‘𝑦) = ((2 · π) ·
(sin‘(𝑦 /
2)))) |
| 790 | 788, 789 | oveq12d 7449 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → ((𝐹‘𝑦) / (𝐺‘𝑦)) = ((sin‘((𝑁 + (1 / 2)) · 𝑦)) / ((2 · π) ·
(sin‘(𝑦 /
2))))) |
| 791 | 785, 790 | eqtr4d 2780 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → if((𝑦 mod (2 · π)) = 0, (((2 ·
𝑁) + 1) / (2 ·
π)), ((sin‘((𝑁 +
(1 / 2)) · 𝑦)) / ((2
· π) · (sin‘(𝑦 / 2))))) = ((𝐹‘𝑦) / (𝐺‘𝑦))) |
| 792 | 791 | mpteq2dva 5242 |
. . . 4
⊢ (𝜑 → (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ if((𝑦 mod (2 · π)) = 0, (((2 ·
𝑁) + 1) / (2 ·
π)), ((sin‘((𝑁 +
(1 / 2)) · 𝑦)) / ((2
· π) · (sin‘(𝑦 / 2)))))) = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐹‘𝑦) / (𝐺‘𝑦)))) |
| 793 | 783, 784,
792 | 3eqtrrd 2782 |
. . 3
⊢ (𝜑 → (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐹‘𝑦) / (𝐺‘𝑦))) = ((𝐷‘𝑁) ↾ ((𝐴(,)𝐵) ∖ {𝑌}))) |
| 794 | 793 | oveq1d 7446 |
. 2
⊢ (𝜑 → ((𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝐹‘𝑦) / (𝐺‘𝑦))) limℂ 𝑌) = (((𝐷‘𝑁) ↾ ((𝐴(,)𝐵) ∖ {𝑌})) limℂ 𝑌)) |
| 795 | 780, 794 | eleqtrd 2843 |
1
⊢ (𝜑 → ((𝐷‘𝑁)‘𝑌) ∈ (((𝐷‘𝑁) ↾ ((𝐴(,)𝐵) ∖ {𝑌})) limℂ 𝑌)) |