| Step | Hyp | Ref
| Expression |
| 1 | | 2re 12340 |
. . . . 5
⊢ 2 ∈
ℝ |
| 2 | 1 | a1i 11 |
. . . 4
⊢ (⊤
→ 2 ∈ ℝ) |
| 3 | | 4re 12350 |
. . . . 5
⊢ 4 ∈
ℝ |
| 4 | 3 | a1i 11 |
. . . 4
⊢ (⊤
→ 4 ∈ ℝ) |
| 5 | | 0red 11264 |
. . . 4
⊢ (⊤
→ 0 ∈ ℝ) |
| 6 | | 2lt4 12441 |
. . . . 5
⊢ 2 <
4 |
| 7 | 6 | a1i 11 |
. . . 4
⊢ (⊤
→ 2 < 4) |
| 8 | | iccssre 13469 |
. . . . . . 7
⊢ ((2
∈ ℝ ∧ 4 ∈ ℝ) → (2[,]4) ⊆
ℝ) |
| 9 | 1, 3, 8 | mp2an 692 |
. . . . . 6
⊢ (2[,]4)
⊆ ℝ |
| 10 | | ax-resscn 11212 |
. . . . . 6
⊢ ℝ
⊆ ℂ |
| 11 | 9, 10 | sstri 3993 |
. . . . 5
⊢ (2[,]4)
⊆ ℂ |
| 12 | 11 | a1i 11 |
. . . 4
⊢ (⊤
→ (2[,]4) ⊆ ℂ) |
| 13 | | sincn 26488 |
. . . . 5
⊢ sin
∈ (ℂ–cn→ℂ) |
| 14 | 13 | a1i 11 |
. . . 4
⊢ (⊤
→ sin ∈ (ℂ–cn→ℂ)) |
| 15 | 9 | sseli 3979 |
. . . . . 6
⊢ (𝑦 ∈ (2[,]4) → 𝑦 ∈
ℝ) |
| 16 | 15 | resincld 16179 |
. . . . 5
⊢ (𝑦 ∈ (2[,]4) →
(sin‘𝑦) ∈
ℝ) |
| 17 | 16 | adantl 481 |
. . . 4
⊢
((⊤ ∧ 𝑦
∈ (2[,]4)) → (sin‘𝑦) ∈ ℝ) |
| 18 | | sin4lt0 16231 |
. . . . . 6
⊢
(sin‘4) < 0 |
| 19 | | sincos2sgn 16230 |
. . . . . . 7
⊢ (0 <
(sin‘2) ∧ (cos‘2) < 0) |
| 20 | 19 | simpli 483 |
. . . . . 6
⊢ 0 <
(sin‘2) |
| 21 | 18, 20 | pm3.2i 470 |
. . . . 5
⊢
((sin‘4) < 0 ∧ 0 < (sin‘2)) |
| 22 | 21 | a1i 11 |
. . . 4
⊢ (⊤
→ ((sin‘4) < 0 ∧ 0 < (sin‘2))) |
| 23 | 2, 4, 5, 7, 12, 14, 17, 22 | ivth2 25490 |
. . 3
⊢ (⊤
→ ∃𝑥 ∈
(2(,)4)(sin‘𝑥) =
0) |
| 24 | 23 | mptru 1547 |
. 2
⊢
∃𝑥 ∈
(2(,)4)(sin‘𝑥) =
0 |
| 25 | | df-pi 16108 |
. . . . . . 7
⊢ π =
inf((ℝ+ ∩ (◡sin
“ {0})), ℝ, < ) |
| 26 | | inss1 4237 |
. . . . . . . . 9
⊢
(ℝ+ ∩ (◡sin
“ {0})) ⊆ ℝ+ |
| 27 | | rpssre 13042 |
. . . . . . . . 9
⊢
ℝ+ ⊆ ℝ |
| 28 | 26, 27 | sstri 3993 |
. . . . . . . 8
⊢
(ℝ+ ∩ (◡sin
“ {0})) ⊆ ℝ |
| 29 | | 0re 11263 |
. . . . . . . . 9
⊢ 0 ∈
ℝ |
| 30 | 26 | sseli 3979 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ (ℝ+
∩ (◡sin “ {0})) → 𝑧 ∈
ℝ+) |
| 31 | 30 | rpge0d 13081 |
. . . . . . . . . 10
⊢ (𝑧 ∈ (ℝ+
∩ (◡sin “ {0})) → 0 ≤
𝑧) |
| 32 | 31 | rgen 3063 |
. . . . . . . . 9
⊢
∀𝑧 ∈
(ℝ+ ∩ (◡sin
“ {0}))0 ≤ 𝑧 |
| 33 | | breq1 5146 |
. . . . . . . . . . 11
⊢ (𝑦 = 0 → (𝑦 ≤ 𝑧 ↔ 0 ≤ 𝑧)) |
| 34 | 33 | ralbidv 3178 |
. . . . . . . . . 10
⊢ (𝑦 = 0 → (∀𝑧 ∈ (ℝ+
∩ (◡sin “ {0}))𝑦 ≤ 𝑧 ↔ ∀𝑧 ∈ (ℝ+ ∩ (◡sin “ {0}))0 ≤ 𝑧)) |
| 35 | 34 | rspcev 3622 |
. . . . . . . . 9
⊢ ((0
∈ ℝ ∧ ∀𝑧 ∈ (ℝ+ ∩ (◡sin “ {0}))0 ≤ 𝑧) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ (ℝ+ ∩ (◡sin “ {0}))𝑦 ≤ 𝑧) |
| 36 | 29, 32, 35 | mp2an 692 |
. . . . . . . 8
⊢
∃𝑦 ∈
ℝ ∀𝑧 ∈
(ℝ+ ∩ (◡sin
“ {0}))𝑦 ≤ 𝑧 |
| 37 | | elioore 13417 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (2(,)4) → 𝑥 ∈
ℝ) |
| 38 | 37 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
𝑥 ∈
ℝ) |
| 39 | | 0red 11264 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) → 0
∈ ℝ) |
| 40 | 1 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) → 2
∈ ℝ) |
| 41 | | 2pos 12369 |
. . . . . . . . . . . 12
⊢ 0 <
2 |
| 42 | 41 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) → 0
< 2) |
| 43 | | eliooord 13446 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (2(,)4) → (2 <
𝑥 ∧ 𝑥 < 4)) |
| 44 | 43 | simpld 494 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (2(,)4) → 2 <
𝑥) |
| 45 | 44 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) → 2
< 𝑥) |
| 46 | 39, 40, 38, 42, 45 | lttrd 11422 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) → 0
< 𝑥) |
| 47 | 38, 46 | elrpd 13074 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
𝑥 ∈
ℝ+) |
| 48 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(sin‘𝑥) =
0) |
| 49 | | pilem1 26495 |
. . . . . . . . 9
⊢ (𝑥 ∈ (ℝ+
∩ (◡sin “ {0})) ↔ (𝑥 ∈ ℝ+
∧ (sin‘𝑥) =
0)) |
| 50 | 47, 48, 49 | sylanbrc 583 |
. . . . . . . 8
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
𝑥 ∈
(ℝ+ ∩ (◡sin
“ {0}))) |
| 51 | | infrelb 12253 |
. . . . . . . 8
⊢
(((ℝ+ ∩ (◡sin “ {0})) ⊆ ℝ ∧
∃𝑦 ∈ ℝ
∀𝑧 ∈
(ℝ+ ∩ (◡sin
“ {0}))𝑦 ≤ 𝑧 ∧ 𝑥 ∈ (ℝ+ ∩ (◡sin “ {0}))) →
inf((ℝ+ ∩ (◡sin
“ {0})), ℝ, < ) ≤ 𝑥) |
| 52 | 28, 36, 50, 51 | mp3an12i 1467 |
. . . . . . 7
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
inf((ℝ+ ∩ (◡sin
“ {0})), ℝ, < ) ≤ 𝑥) |
| 53 | 25, 52 | eqbrtrid 5178 |
. . . . . 6
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
π ≤ 𝑥) |
| 54 | | simpll 767 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) ∧
𝑦 ∈
(ℝ+ ∩ (◡sin
“ {0}))) → 𝑥
∈ (2(,)4)) |
| 55 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) ∧
𝑦 ∈
(ℝ+ ∩ (◡sin
“ {0}))) → 𝑦
∈ (ℝ+ ∩ (◡sin
“ {0}))) |
| 56 | | pilem1 26495 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (ℝ+
∩ (◡sin “ {0})) ↔ (𝑦 ∈ ℝ+
∧ (sin‘𝑦) =
0)) |
| 57 | 55, 56 | sylib 218 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) ∧
𝑦 ∈
(ℝ+ ∩ (◡sin
“ {0}))) → (𝑦
∈ ℝ+ ∧ (sin‘𝑦) = 0)) |
| 58 | 57 | simpld 494 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) ∧
𝑦 ∈
(ℝ+ ∩ (◡sin
“ {0}))) → 𝑦
∈ ℝ+) |
| 59 | | simplr 769 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) ∧
𝑦 ∈
(ℝ+ ∩ (◡sin
“ {0}))) → (sin‘𝑥) = 0) |
| 60 | 57 | simprd 495 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) ∧
𝑦 ∈
(ℝ+ ∩ (◡sin
“ {0}))) → (sin‘𝑦) = 0) |
| 61 | 54, 58, 59, 60 | pilem2 26496 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) ∧
𝑦 ∈
(ℝ+ ∩ (◡sin
“ {0}))) → ((π + 𝑥) / 2) ≤ 𝑦) |
| 62 | 61 | ralrimiva 3146 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
∀𝑦 ∈
(ℝ+ ∩ (◡sin
“ {0}))((π + 𝑥) /
2) ≤ 𝑦) |
| 63 | 28 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(ℝ+ ∩ (◡sin
“ {0})) ⊆ ℝ) |
| 64 | 50 | ne0d 4342 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(ℝ+ ∩ (◡sin
“ {0})) ≠ ∅) |
| 65 | 36 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
∃𝑦 ∈ ℝ
∀𝑧 ∈
(ℝ+ ∩ (◡sin
“ {0}))𝑦 ≤ 𝑧) |
| 66 | | infrecl 12250 |
. . . . . . . . . . . . . . 15
⊢
(((ℝ+ ∩ (◡sin “ {0})) ⊆ ℝ ∧
(ℝ+ ∩ (◡sin
“ {0})) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ (ℝ+ ∩ (◡sin “ {0}))𝑦 ≤ 𝑧) → inf((ℝ+ ∩
(◡sin “ {0})), ℝ, < )
∈ ℝ) |
| 67 | 28, 36, 66 | mp3an13 1454 |
. . . . . . . . . . . . . 14
⊢
((ℝ+ ∩ (◡sin “ {0})) ≠ ∅ →
inf((ℝ+ ∩ (◡sin
“ {0})), ℝ, < ) ∈ ℝ) |
| 68 | 64, 67 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
inf((ℝ+ ∩ (◡sin
“ {0})), ℝ, < ) ∈ ℝ) |
| 69 | 25, 68 | eqeltrid 2845 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
π ∈ ℝ) |
| 70 | 69, 38 | readdcld 11290 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(π + 𝑥) ∈
ℝ) |
| 71 | 70 | rehalfcld 12513 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
((π + 𝑥) / 2) ∈
ℝ) |
| 72 | | infregelb 12252 |
. . . . . . . . . 10
⊢
((((ℝ+ ∩ (◡sin “ {0})) ⊆ ℝ ∧
(ℝ+ ∩ (◡sin
“ {0})) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ (ℝ+ ∩ (◡sin “ {0}))𝑦 ≤ 𝑧) ∧ ((π + 𝑥) / 2) ∈ ℝ) → (((π + 𝑥) / 2) ≤
inf((ℝ+ ∩ (◡sin
“ {0})), ℝ, < ) ↔ ∀𝑦 ∈ (ℝ+ ∩ (◡sin “ {0}))((π + 𝑥) / 2) ≤ 𝑦)) |
| 73 | 63, 64, 65, 71, 72 | syl31anc 1375 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(((π + 𝑥) / 2) ≤
inf((ℝ+ ∩ (◡sin
“ {0})), ℝ, < ) ↔ ∀𝑦 ∈ (ℝ+ ∩ (◡sin “ {0}))((π + 𝑥) / 2) ≤ 𝑦)) |
| 74 | 62, 73 | mpbird 257 |
. . . . . . . 8
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
((π + 𝑥) / 2) ≤
inf((ℝ+ ∩ (◡sin
“ {0})), ℝ, < )) |
| 75 | 74, 25 | breqtrrdi 5185 |
. . . . . . 7
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
((π + 𝑥) / 2) ≤
π) |
| 76 | 69 | recnd 11289 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
π ∈ ℂ) |
| 77 | 38 | recnd 11289 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
𝑥 ∈
ℂ) |
| 78 | 76, 77 | addcomd 11463 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(π + 𝑥) = (𝑥 + π)) |
| 79 | 78 | oveq1d 7446 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
((π + 𝑥) / 2) = ((𝑥 + π) / 2)) |
| 80 | 79 | breq1d 5153 |
. . . . . . . 8
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(((π + 𝑥) / 2) ≤ π
↔ ((𝑥 + π) / 2)
≤ π)) |
| 81 | | avgle2 12507 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧ π
∈ ℝ) → (𝑥
≤ π ↔ ((𝑥 +
π) / 2) ≤ π)) |
| 82 | 38, 69, 81 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(𝑥 ≤ π ↔ ((𝑥 + π) / 2) ≤
π)) |
| 83 | 80, 82 | bitr4d 282 |
. . . . . . 7
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(((π + 𝑥) / 2) ≤ π
↔ 𝑥 ≤
π)) |
| 84 | 75, 83 | mpbid 232 |
. . . . . 6
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
𝑥 ≤
π) |
| 85 | 69, 38 | letri3d 11403 |
. . . . . 6
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(π = 𝑥 ↔ (π ≤
𝑥 ∧ 𝑥 ≤ π))) |
| 86 | 53, 84, 85 | mpbir2and 713 |
. . . . 5
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
π = 𝑥) |
| 87 | | simpl 482 |
. . . . 5
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
𝑥 ∈
(2(,)4)) |
| 88 | 86, 87 | eqeltrd 2841 |
. . . 4
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
π ∈ (2(,)4)) |
| 89 | 86 | fveq2d 6910 |
. . . . 5
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(sin‘π) = (sin‘𝑥)) |
| 90 | 89, 48 | eqtrd 2777 |
. . . 4
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(sin‘π) = 0) |
| 91 | 88, 90 | jca 511 |
. . 3
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(π ∈ (2(,)4) ∧ (sin‘π) = 0)) |
| 92 | 91 | rexlimiva 3147 |
. 2
⊢
(∃𝑥 ∈
(2(,)4)(sin‘𝑥) = 0
→ (π ∈ (2(,)4) ∧ (sin‘π) = 0)) |
| 93 | 24, 92 | ax-mp 5 |
1
⊢ (π
∈ (2(,)4) ∧ (sin‘π) = 0) |