Step | Hyp | Ref
| Expression |
1 | | 2re 12030 |
. . . . 5
⊢ 2 ∈
ℝ |
2 | 1 | a1i 11 |
. . . 4
⊢ (⊤
→ 2 ∈ ℝ) |
3 | | 4re 12040 |
. . . . 5
⊢ 4 ∈
ℝ |
4 | 3 | a1i 11 |
. . . 4
⊢ (⊤
→ 4 ∈ ℝ) |
5 | | 0red 10962 |
. . . 4
⊢ (⊤
→ 0 ∈ ℝ) |
6 | | 2lt4 12131 |
. . . . 5
⊢ 2 <
4 |
7 | 6 | a1i 11 |
. . . 4
⊢ (⊤
→ 2 < 4) |
8 | | iccssre 13143 |
. . . . . . 7
⊢ ((2
∈ ℝ ∧ 4 ∈ ℝ) → (2[,]4) ⊆
ℝ) |
9 | 1, 3, 8 | mp2an 688 |
. . . . . 6
⊢ (2[,]4)
⊆ ℝ |
10 | | ax-resscn 10912 |
. . . . . 6
⊢ ℝ
⊆ ℂ |
11 | 9, 10 | sstri 3934 |
. . . . 5
⊢ (2[,]4)
⊆ ℂ |
12 | 11 | a1i 11 |
. . . 4
⊢ (⊤
→ (2[,]4) ⊆ ℂ) |
13 | | sincn 25584 |
. . . . 5
⊢ sin
∈ (ℂ–cn→ℂ) |
14 | 13 | a1i 11 |
. . . 4
⊢ (⊤
→ sin ∈ (ℂ–cn→ℂ)) |
15 | 9 | sseli 3921 |
. . . . . 6
⊢ (𝑦 ∈ (2[,]4) → 𝑦 ∈
ℝ) |
16 | 15 | resincld 15833 |
. . . . 5
⊢ (𝑦 ∈ (2[,]4) →
(sin‘𝑦) ∈
ℝ) |
17 | 16 | adantl 481 |
. . . 4
⊢
((⊤ ∧ 𝑦
∈ (2[,]4)) → (sin‘𝑦) ∈ ℝ) |
18 | | sin4lt0 15885 |
. . . . . 6
⊢
(sin‘4) < 0 |
19 | | sincos2sgn 15884 |
. . . . . . 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 24600 |
. . 3
⊢ (⊤
→ ∃𝑥 ∈
(2(,)4)(sin‘𝑥) =
0) |
24 | 23 | mptru 1548 |
. 2
⊢
∃𝑥 ∈
(2(,)4)(sin‘𝑥) =
0 |
25 | | df-pi 15763 |
. . . . . . 7
⊢ π =
inf((ℝ+ ∩ (◡sin
“ {0})), ℝ, < ) |
26 | | inss1 4167 |
. . . . . . . . 9
⊢
(ℝ+ ∩ (◡sin
“ {0})) ⊆ ℝ+ |
27 | | rpssre 12719 |
. . . . . . . . 9
⊢
ℝ+ ⊆ ℝ |
28 | 26, 27 | sstri 3934 |
. . . . . . . 8
⊢
(ℝ+ ∩ (◡sin
“ {0})) ⊆ ℝ |
29 | | 0re 10961 |
. . . . . . . . 9
⊢ 0 ∈
ℝ |
30 | 26 | sseli 3921 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ (ℝ+
∩ (◡sin “ {0})) → 𝑧 ∈
ℝ+) |
31 | 30 | rpge0d 12758 |
. . . . . . . . . 10
⊢ (𝑧 ∈ (ℝ+
∩ (◡sin “ {0})) → 0 ≤
𝑧) |
32 | 31 | rgen 3075 |
. . . . . . . . 9
⊢
∀𝑧 ∈
(ℝ+ ∩ (◡sin
“ {0}))0 ≤ 𝑧 |
33 | | breq1 5081 |
. . . . . . . . . . 11
⊢ (𝑦 = 0 → (𝑦 ≤ 𝑧 ↔ 0 ≤ 𝑧)) |
34 | 33 | ralbidv 3122 |
. . . . . . . . . 10
⊢ (𝑦 = 0 → (∀𝑧 ∈ (ℝ+
∩ (◡sin “ {0}))𝑦 ≤ 𝑧 ↔ ∀𝑧 ∈ (ℝ+ ∩ (◡sin “ {0}))0 ≤ 𝑧)) |
35 | 34 | rspcev 3560 |
. . . . . . . . 9
⊢ ((0
∈ ℝ ∧ ∀𝑧 ∈ (ℝ+ ∩ (◡sin “ {0}))0 ≤ 𝑧) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ (ℝ+ ∩ (◡sin “ {0}))𝑦 ≤ 𝑧) |
36 | 29, 32, 35 | mp2an 688 |
. . . . . . . 8
⊢
∃𝑦 ∈
ℝ ∀𝑧 ∈
(ℝ+ ∩ (◡sin
“ {0}))𝑦 ≤ 𝑧 |
37 | | elioore 13091 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (2(,)4) → 𝑥 ∈
ℝ) |
38 | 37 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
𝑥 ∈
ℝ) |
39 | | 0red 10962 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) → 0
∈ ℝ) |
40 | 1 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) → 2
∈ ℝ) |
41 | | 2pos 12059 |
. . . . . . . . . . . 12
⊢ 0 <
2 |
42 | 41 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) → 0
< 2) |
43 | | eliooord 13120 |
. . . . . . . . . . . . 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 11119 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) → 0
< 𝑥) |
47 | 38, 46 | elrpd 12751 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
𝑥 ∈
ℝ+) |
48 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(sin‘𝑥) =
0) |
49 | | pilem1 25591 |
. . . . . . . . 9
⊢ (𝑥 ∈ (ℝ+
∩ (◡sin “ {0})) ↔ (𝑥 ∈ ℝ+
∧ (sin‘𝑥) =
0)) |
50 | 47, 48, 49 | sylanbrc 582 |
. . . . . . . 8
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
𝑥 ∈
(ℝ+ ∩ (◡sin
“ {0}))) |
51 | | infrelb 11943 |
. . . . . . . 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 1463 |
. . . . . . 7
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
inf((ℝ+ ∩ (◡sin
“ {0})), ℝ, < ) ≤ 𝑥) |
53 | 25, 52 | eqbrtrid 5113 |
. . . . . 6
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
π ≤ 𝑥) |
54 | | simpll 763 |
. . . . . . . . . . 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 25591 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (ℝ+
∩ (◡sin “ {0})) ↔ (𝑦 ∈ ℝ+
∧ (sin‘𝑦) =
0)) |
57 | 55, 56 | sylib 217 |
. . . . . . . . . . . 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 765 |
. . . . . . . . . . 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 25592 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) ∧
𝑦 ∈
(ℝ+ ∩ (◡sin
“ {0}))) → ((π + 𝑥) / 2) ≤ 𝑦) |
62 | 61 | ralrimiva 3109 |
. . . . . . . . 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 4274 |
. . . . . . . . . 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 11940 |
. . . . . . . . . . . . . . 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 1450 |
. . . . . . . . . . . . . 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 2844 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
π ∈ ℝ) |
70 | 69, 38 | readdcld 10988 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(π + 𝑥) ∈
ℝ) |
71 | 70 | rehalfcld 12203 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
((π + 𝑥) / 2) ∈
ℝ) |
72 | | infregelb 11942 |
. . . . . . . . . 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 1371 |
. . . . . . . . 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 256 |
. . . . . . . 8
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
((π + 𝑥) / 2) ≤
inf((ℝ+ ∩ (◡sin
“ {0})), ℝ, < )) |
75 | 74, 25 | breqtrrdi 5120 |
. . . . . . 7
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
((π + 𝑥) / 2) ≤
π) |
76 | 69 | recnd 10987 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
π ∈ ℂ) |
77 | 38 | recnd 10987 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
𝑥 ∈
ℂ) |
78 | 76, 77 | addcomd 11160 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(π + 𝑥) = (𝑥 + π)) |
79 | 78 | oveq1d 7283 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
((π + 𝑥) / 2) = ((𝑥 + π) / 2)) |
80 | 79 | breq1d 5088 |
. . . . . . . 8
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(((π + 𝑥) / 2) ≤ π
↔ ((𝑥 + π) / 2)
≤ π)) |
81 | | avgle2 12197 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧ π
∈ ℝ) → (𝑥
≤ π ↔ ((𝑥 +
π) / 2) ≤ π)) |
82 | 38, 69, 81 | syl2anc 583 |
. . . . . . . 8
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(𝑥 ≤ π ↔ ((𝑥 + π) / 2) ≤
π)) |
83 | 80, 82 | bitr4d 281 |
. . . . . . 7
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(((π + 𝑥) / 2) ≤ π
↔ 𝑥 ≤
π)) |
84 | 75, 83 | mpbid 231 |
. . . . . 6
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
𝑥 ≤
π) |
85 | 69, 38 | letri3d 11100 |
. . . . . 6
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(π = 𝑥 ↔ (π ≤
𝑥 ∧ 𝑥 ≤ π))) |
86 | 53, 84, 85 | mpbir2and 709 |
. . . . 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 2840 |
. . . 4
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
π ∈ (2(,)4)) |
89 | 86 | fveq2d 6772 |
. . . . 5
⊢ ((𝑥 ∈ (2(,)4) ∧
(sin‘𝑥) = 0) →
(sin‘π) = (sin‘𝑥)) |
90 | 89, 48 | eqtrd 2779 |
. . . 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 3211 |
. 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) |