Step | Hyp | Ref
| Expression |
1 | | 0re 10835 |
. . . . 5
⊢ 0 ∈
ℝ |
2 | | leloe 10919 |
. . . . 5
⊢ ((0
∈ ℝ ∧ 𝐴
∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴))) |
3 | 1, 2 | mpan 690 |
. . . 4
⊢ (𝐴 ∈ ℝ → (0 ≤
𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴))) |
4 | | elrp 12588 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ+
↔ (𝐴 ∈ ℝ
∧ 0 < 𝐴)) |
5 | | 01sqrex 14813 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ+
∧ 𝐴 ≤ 1) →
∃𝑥 ∈
ℝ+ (𝑥 ≤
1 ∧ (𝑥↑2) = 𝐴)) |
6 | | rprege0 12601 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℝ
∧ 0 ≤ 𝑥)) |
7 | 6 | anim1i 618 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ (𝑥↑2) = 𝐴) → ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑥↑2) = 𝐴)) |
8 | | anass 472 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℝ ∧ 0 ≤
𝑥) ∧ (𝑥↑2) = 𝐴) ↔ (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))) |
9 | 7, 8 | sylib 221 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ (𝑥↑2) = 𝐴) → (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))) |
10 | 9 | adantrl 716 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ (𝑥 ≤ 1 ∧
(𝑥↑2) = 𝐴)) → (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))) |
11 | 10 | reximi2 3167 |
. . . . . . . 8
⊢
(∃𝑥 ∈
ℝ+ (𝑥 ≤
1 ∧ (𝑥↑2) = 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)) |
12 | 5, 11 | syl 17 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ+
∧ 𝐴 ≤ 1) →
∃𝑥 ∈ ℝ (0
≤ 𝑥 ∧ (𝑥↑2) = 𝐴)) |
13 | 4, 12 | sylanbr 585 |
. . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 0 <
𝐴) ∧ 𝐴 ≤ 1) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)) |
14 | 13 | exp31 423 |
. . . . 5
⊢ (𝐴 ∈ ℝ → (0 <
𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))) |
15 | | sq0 13761 |
. . . . . . . . 9
⊢
(0↑2) = 0 |
16 | | id 22 |
. . . . . . . . 9
⊢ (0 =
𝐴 → 0 = 𝐴) |
17 | 15, 16 | syl5eq 2790 |
. . . . . . . 8
⊢ (0 =
𝐴 → (0↑2) = 𝐴) |
18 | | 0le0 11931 |
. . . . . . . 8
⊢ 0 ≤
0 |
19 | 17, 18 | jctil 523 |
. . . . . . 7
⊢ (0 =
𝐴 → (0 ≤ 0 ∧
(0↑2) = 𝐴)) |
20 | | breq2 5057 |
. . . . . . . . 9
⊢ (𝑥 = 0 → (0 ≤ 𝑥 ↔ 0 ≤
0)) |
21 | | oveq1 7220 |
. . . . . . . . . 10
⊢ (𝑥 = 0 → (𝑥↑2) = (0↑2)) |
22 | 21 | eqeq1d 2739 |
. . . . . . . . 9
⊢ (𝑥 = 0 → ((𝑥↑2) = 𝐴 ↔ (0↑2) = 𝐴)) |
23 | 20, 22 | anbi12d 634 |
. . . . . . . 8
⊢ (𝑥 = 0 → ((0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴) ↔ (0 ≤ 0 ∧ (0↑2) = 𝐴))) |
24 | 23 | rspcev 3537 |
. . . . . . 7
⊢ ((0
∈ ℝ ∧ (0 ≤ 0 ∧ (0↑2) = 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)) |
25 | 1, 19, 24 | sylancr 590 |
. . . . . 6
⊢ (0 =
𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)) |
26 | 25 | a1i13 27 |
. . . . 5
⊢ (𝐴 ∈ ℝ → (0 =
𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))) |
27 | 14, 26 | jaod 859 |
. . . 4
⊢ (𝐴 ∈ ℝ → ((0 <
𝐴 ∨ 0 = 𝐴) → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))) |
28 | 3, 27 | sylbid 243 |
. . 3
⊢ (𝐴 ∈ ℝ → (0 ≤
𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))) |
29 | 28 | imp 410 |
. 2
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))) |
30 | | 0lt1 11354 |
. . . . . . . . . 10
⊢ 0 <
1 |
31 | | 1re 10833 |
. . . . . . . . . . 11
⊢ 1 ∈
ℝ |
32 | | ltletr 10924 |
. . . . . . . . . . 11
⊢ ((0
∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 1 ∧ 1
≤ 𝐴) → 0 < 𝐴)) |
33 | 1, 31, 32 | mp3an12 1453 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ℝ → ((0 <
1 ∧ 1 ≤ 𝐴) → 0
< 𝐴)) |
34 | 30, 33 | mpani 696 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℝ → (1 ≤
𝐴 → 0 < 𝐴)) |
35 | 34 | imp 410 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) → 0 < 𝐴) |
36 | 4 | biimpri 231 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 0 <
𝐴) → 𝐴 ∈
ℝ+) |
37 | 35, 36 | syldan 594 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) → 𝐴 ∈
ℝ+) |
38 | 37 | rpreccld 12638 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) → (1 / 𝐴) ∈
ℝ+) |
39 | | simpr 488 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) → 1 ≤ 𝐴) |
40 | | lerec 11715 |
. . . . . . . . . 10
⊢ (((1
∈ ℝ ∧ 0 < 1) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1))) |
41 | 31, 30, 40 | mpanl12 702 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 0 <
𝐴) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1))) |
42 | 35, 41 | syldan 594 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1))) |
43 | 39, 42 | mpbid 235 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) → (1 / 𝐴) ≤ (1 / 1)) |
44 | | 1div1e1 11522 |
. . . . . . 7
⊢ (1 / 1) =
1 |
45 | 43, 44 | breqtrdi 5094 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) → (1 / 𝐴) ≤ 1) |
46 | | 01sqrex 14813 |
. . . . . 6
⊢ (((1 /
𝐴) ∈
ℝ+ ∧ (1 / 𝐴) ≤ 1) → ∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) |
47 | 38, 45, 46 | syl2anc 587 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) → ∃𝑦 ∈ ℝ+
(𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) |
48 | | rpre 12594 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℝ+
→ 𝑦 ∈
ℝ) |
49 | 48 | 3ad2ant2 1136 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 𝑦 ∈ ℝ) |
50 | | rpgt0 12598 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℝ+
→ 0 < 𝑦) |
51 | 50 | 3ad2ant2 1136 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 0 < 𝑦) |
52 | | gt0ne0 11297 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℝ ∧ 0 <
𝑦) → 𝑦 ≠ 0) |
53 | | rereccl 11550 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℝ ∧ 𝑦 ≠ 0) → (1 / 𝑦) ∈
ℝ) |
54 | 52, 53 | syldan 594 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℝ ∧ 0 <
𝑦) → (1 / 𝑦) ∈
ℝ) |
55 | 49, 51, 54 | syl2anc 587 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / 𝑦) ∈
ℝ) |
56 | | recgt0 11678 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℝ ∧ 0 <
𝑦) → 0 < (1 / 𝑦)) |
57 | | ltle 10921 |
. . . . . . . . . 10
⊢ ((0
∈ ℝ ∧ (1 / 𝑦) ∈ ℝ) → (0 < (1 / 𝑦) → 0 ≤ (1 / 𝑦))) |
58 | 1, 57 | mpan 690 |
. . . . . . . . 9
⊢ ((1 /
𝑦) ∈ ℝ → (0
< (1 / 𝑦) → 0 ≤
(1 / 𝑦))) |
59 | 54, 56, 58 | sylc 65 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℝ ∧ 0 <
𝑦) → 0 ≤ (1 / 𝑦)) |
60 | 49, 51, 59 | syl2anc 587 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 0 ≤ (1 / 𝑦)) |
61 | | recn 10819 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℝ → 𝑦 ∈
ℂ) |
62 | 61 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ℝ ∧ 0 <
𝑦) → 𝑦 ∈
ℂ) |
63 | 62, 52 | sqrecd 13720 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℝ ∧ 0 <
𝑦) → ((1 / 𝑦)↑2) = (1 / (𝑦↑2))) |
64 | 49, 51, 63 | syl2anc 587 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ((1 / 𝑦)↑2) = (1 / (𝑦↑2))) |
65 | | simp3r 1204 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (𝑦↑2) = (1 / 𝐴)) |
66 | 65 | oveq2d 7229 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / (𝑦↑2)) = (1 / (1 / 𝐴))) |
67 | | recn 10819 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ℝ → 𝐴 ∈
ℂ) |
68 | | gt0ne0 11297 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 0 <
𝐴) → 𝐴 ≠ 0) |
69 | 35, 68 | syldan 594 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) → 𝐴 ≠ 0) |
70 | | recrec 11529 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / (1 / 𝐴)) = 𝐴) |
71 | 67, 69, 70 | syl2an2r 685 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) → (1 / (1 / 𝐴)) = 𝐴) |
72 | 71 | 3ad2ant1 1135 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / (1 / 𝐴)) = 𝐴) |
73 | 64, 66, 72 | 3eqtrd 2781 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ((1 / 𝑦)↑2) = 𝐴) |
74 | | breq2 5057 |
. . . . . . . . 9
⊢ (𝑥 = (1 / 𝑦) → (0 ≤ 𝑥 ↔ 0 ≤ (1 / 𝑦))) |
75 | | oveq1 7220 |
. . . . . . . . . 10
⊢ (𝑥 = (1 / 𝑦) → (𝑥↑2) = ((1 / 𝑦)↑2)) |
76 | 75 | eqeq1d 2739 |
. . . . . . . . 9
⊢ (𝑥 = (1 / 𝑦) → ((𝑥↑2) = 𝐴 ↔ ((1 / 𝑦)↑2) = 𝐴)) |
77 | 74, 76 | anbi12d 634 |
. . . . . . . 8
⊢ (𝑥 = (1 / 𝑦) → ((0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴) ↔ (0 ≤ (1 / 𝑦) ∧ ((1 / 𝑦)↑2) = 𝐴))) |
78 | 77 | rspcev 3537 |
. . . . . . 7
⊢ (((1 /
𝑦) ∈ ℝ ∧ (0
≤ (1 / 𝑦) ∧ ((1 /
𝑦)↑2) = 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)) |
79 | 55, 60, 73, 78 | syl12anc 837 |
. . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)) |
80 | 79 | rexlimdv3a 3205 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) → (∃𝑦 ∈ ℝ+
(𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))) |
81 | 47, 80 | mpd 15 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)) |
82 | 81 | ex 416 |
. . 3
⊢ (𝐴 ∈ ℝ → (1 ≤
𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))) |
83 | 82 | adantr 484 |
. 2
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) → (1 ≤ 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))) |
84 | | simpl 486 |
. . 3
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) → 𝐴 ∈ ℝ) |
85 | | letric 10932 |
. . 3
⊢ ((𝐴 ∈ ℝ ∧ 1 ∈
ℝ) → (𝐴 ≤ 1
∨ 1 ≤ 𝐴)) |
86 | 84, 31, 85 | sylancl 589 |
. 2
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) → (𝐴 ≤ 1 ∨ 1 ≤ 𝐴)) |
87 | 29, 83, 86 | mpjaod 860 |
1
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)) |