Proof of Theorem lcmineqlem23
Step | Hyp | Ref
| Expression |
1 | | lcmineqlem23.1 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℕ) |
2 | | 2nn 11732 |
. . . . . . . . . . . . 13
⊢ 2 ∈
ℕ |
3 | 2 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 2 ∈
ℕ) |
4 | 1, 3 | jca 516 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 2 ∈
ℕ)) |
5 | | nndivdvds 15649 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ ∧ 2 ∈
ℕ) → (2 ∥ 𝑁 ↔ (𝑁 / 2) ∈ ℕ)) |
6 | 4, 5 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (2 ∥ 𝑁 ↔ (𝑁 / 2) ∈ ℕ)) |
7 | 6 | biimpa 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (𝑁 / 2) ∈ ℕ) |
8 | 7 | nnzd 12110 |
. . . . . . . 8
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (𝑁 / 2) ∈ ℤ) |
9 | | 1zzd 12037 |
. . . . . . . 8
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → 1 ∈ ℤ) |
10 | 8, 9 | zsubcld 12116 |
. . . . . . 7
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → ((𝑁 / 2) − 1) ∈
ℤ) |
11 | | 0red 10667 |
. . . . . . . 8
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → 0 ∈ ℝ) |
12 | | 4re 11743 |
. . . . . . . . 9
⊢ 4 ∈
ℝ |
13 | 12 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → 4 ∈ ℝ) |
14 | 7 | nnred 11674 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (𝑁 / 2) ∈ ℝ) |
15 | | 1red 10665 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → 1 ∈ ℝ) |
16 | 14, 15 | resubcld 11091 |
. . . . . . . 8
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → ((𝑁 / 2) − 1) ∈
ℝ) |
17 | | 4pos 11766 |
. . . . . . . . 9
⊢ 0 <
4 |
18 | 17 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → 0 < 4) |
19 | | 5m1e4 11789 |
. . . . . . . . 9
⊢ (5
− 1) = 4 |
20 | | 5re 11746 |
. . . . . . . . . . 11
⊢ 5 ∈
ℝ |
21 | 20 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → 5 ∈ ℝ) |
22 | 2 | nncni 11669 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℂ |
23 | | 5cn 11747 |
. . . . . . . . . . . . . 14
⊢ 5 ∈
ℂ |
24 | 22, 23 | mulcomi 10672 |
. . . . . . . . . . . . 13
⊢ (2
· 5) = (5 · 2) |
25 | | 5t2e10 12222 |
. . . . . . . . . . . . 13
⊢ (5
· 2) = ;10 |
26 | 24, 25 | eqtri 2782 |
. . . . . . . . . . . 12
⊢ (2
· 5) = ;10 |
27 | | 10re 12141 |
. . . . . . . . . . . . . 14
⊢ ;10 ∈ ℝ |
28 | 27 | recni 10678 |
. . . . . . . . . . . . 13
⊢ ;10 ∈ ℂ |
29 | 2 | nnne0i 11699 |
. . . . . . . . . . . . 13
⊢ 2 ≠
0 |
30 | 28, 22, 23, 29 | divmuli 11417 |
. . . . . . . . . . . 12
⊢ ((;10 / 2) = 5 ↔ (2 · 5) =
;10) |
31 | 26, 30 | mpbir 234 |
. . . . . . . . . . 11
⊢ (;10 / 2) = 5 |
32 | 27 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → ;10 ∈ ℝ) |
33 | 1 | nnred 11674 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℝ) |
34 | 33 | adantr 485 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → 𝑁 ∈ ℝ) |
35 | | 2rp 12420 |
. . . . . . . . . . . . 13
⊢ 2 ∈
ℝ+ |
36 | 35 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → 2 ∈
ℝ+) |
37 | | 9p1e10 12124 |
. . . . . . . . . . . . 13
⊢ (9 + 1) =
;10 |
38 | | lcmineqlem23.2 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 9 ≤ 𝑁) |
39 | | 9re 11758 |
. . . . . . . . . . . . . . . . . . 19
⊢ 9 ∈
ℝ |
40 | 39 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 9 ∈
ℝ) |
41 | 40, 33 | leloed 10806 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (9 ≤ 𝑁 ↔ (9 < 𝑁 ∨ 9 = 𝑁))) |
42 | 38, 41 | mpbid 235 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (9 < 𝑁 ∨ 9 = 𝑁)) |
43 | 42 | adantr 485 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (9 < 𝑁 ∨ 9 = 𝑁)) |
44 | | 4cn 11744 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ 4 ∈
ℂ |
45 | 22, 44 | mulcomi 10672 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (2
· 4) = (4 · 2) |
46 | | 4t2e8 11827 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (4
· 2) = 8 |
47 | 45, 46 | eqtri 2782 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (2
· 4) = 8 |
48 | | 8re 11755 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ 8 ∈
ℝ |
49 | 48 | recni 10678 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 8 ∈
ℂ |
50 | 49, 22, 44, 29 | divmuli 11417 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((8 / 2)
= 4 ↔ (2 · 4) = 8) |
51 | 47, 50 | mpbir 234 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (8 / 2) =
4 |
52 | | 4nn 11742 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 4 ∈
ℕ |
53 | 51, 52 | eqeltri 2847 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (8 / 2)
∈ ℕ |
54 | | 8nn 11754 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 8 ∈
ℕ |
55 | | nndivdvds 15649 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((8
∈ ℕ ∧ 2 ∈ ℕ) → (2 ∥ 8 ↔ (8 / 2)
∈ ℕ)) |
56 | 54, 2, 55 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (2
∥ 8 ↔ (8 / 2) ∈ ℕ) |
57 | 53, 56 | mpbir 234 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 2 ∥
8 |
58 | | 9m1e8 11793 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (9
− 1) = 8 |
59 | 57, 58 | breqtrri 5052 |
. . . . . . . . . . . . . . . . . . 19
⊢ 2 ∥
(9 − 1) |
60 | | 9nn 11757 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 9 ∈
ℕ |
61 | 60 | nnzi 12030 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 9 ∈
ℤ |
62 | | oddm1even 15729 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (9 ∈
ℤ → (¬ 2 ∥ 9 ↔ 2 ∥ (9 −
1))) |
63 | 61, 62 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬ 2
∥ 9 ↔ 2 ∥ (9 − 1)) |
64 | 59, 63 | mpbir 234 |
. . . . . . . . . . . . . . . . . 18
⊢ ¬ 2
∥ 9 |
65 | | breq2 5029 |
. . . . . . . . . . . . . . . . . 18
⊢ (9 =
𝑁 → (2 ∥ 9
↔ 2 ∥ 𝑁)) |
66 | 64, 65 | mtbii 330 |
. . . . . . . . . . . . . . . . 17
⊢ (9 =
𝑁 → ¬ 2 ∥
𝑁) |
67 | 66 | con2i 141 |
. . . . . . . . . . . . . . . 16
⊢ (2
∥ 𝑁 → ¬ 9 =
𝑁) |
68 | 67 | adantl 486 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → ¬ 9 = 𝑁) |
69 | 43, 68 | olcnd 875 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → 9 < 𝑁) |
70 | 1 | nnzd 12110 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈ ℤ) |
71 | | zltp1le 12056 |
. . . . . . . . . . . . . . . . 17
⊢ ((9
∈ ℤ ∧ 𝑁
∈ ℤ) → (9 < 𝑁 ↔ (9 + 1) ≤ 𝑁)) |
72 | 61, 71 | mpan 690 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈ ℤ → (9 <
𝑁 ↔ (9 + 1) ≤ 𝑁)) |
73 | 70, 72 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (9 < 𝑁 ↔ (9 + 1) ≤ 𝑁)) |
74 | 73 | adantr 485 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (9 < 𝑁 ↔ (9 + 1) ≤ 𝑁)) |
75 | 69, 74 | mpbid 235 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (9 + 1) ≤ 𝑁) |
76 | 37, 75 | eqbrtrrid 5061 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → ;10 ≤ 𝑁) |
77 | 32, 34, 36, 76 | lediv1dd 12515 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (;10 / 2) ≤ (𝑁 / 2)) |
78 | 31, 77 | eqbrtrrid 5061 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → 5 ≤ (𝑁 / 2)) |
79 | 21, 14, 15, 78 | lesub1dd 11279 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (5 − 1) ≤ ((𝑁 / 2) −
1)) |
80 | 19, 79 | eqbrtrrid 5061 |
. . . . . . . 8
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → 4 ≤ ((𝑁 / 2) − 1)) |
81 | 11, 13, 16, 18, 80 | ltletrd 10823 |
. . . . . . 7
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → 0 < ((𝑁 / 2) − 1)) |
82 | 10, 81 | jca 516 |
. . . . . 6
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (((𝑁 / 2) − 1) ∈ ℤ ∧ 0 <
((𝑁 / 2) −
1))) |
83 | | elnnz 12015 |
. . . . . 6
⊢ (((𝑁 / 2) − 1) ∈ ℕ
↔ (((𝑁 / 2) − 1)
∈ ℤ ∧ 0 < ((𝑁 / 2) − 1))) |
84 | 82, 83 | sylibr 237 |
. . . . 5
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → ((𝑁 / 2) − 1) ∈
ℕ) |
85 | 84, 80 | lcmineqlem22 39602 |
. . . 4
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → ((2↑((2 · ((𝑁 / 2) − 1)) + 1)) ≤
(lcm‘(1...((2 · ((𝑁 / 2) − 1)) + 1))) ∧ (2↑((2
· ((𝑁 / 2) −
1)) + 2)) ≤ (lcm‘(1...((2 · ((𝑁 / 2) − 1)) + 2))))) |
86 | 85 | simprd 500 |
. . 3
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (2↑((2 · ((𝑁 / 2) − 1)) + 2)) ≤
(lcm‘(1...((2 · ((𝑁 / 2) − 1)) + 2)))) |
87 | 3 | nncnd 11675 |
. . . . . . . . . 10
⊢ (𝜑 → 2 ∈
ℂ) |
88 | 1 | nncnd 11675 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℂ) |
89 | 88 | halfcld 11904 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁 / 2) ∈ ℂ) |
90 | 87, 89 | muls1d 11123 |
. . . . . . . . 9
⊢ (𝜑 → (2 · ((𝑁 / 2) − 1)) = ((2 ·
(𝑁 / 2)) −
2)) |
91 | 90 | oveq1d 7158 |
. . . . . . . 8
⊢ (𝜑 → ((2 · ((𝑁 / 2) − 1)) + 2) = (((2
· (𝑁 / 2)) −
2) + 2)) |
92 | 87, 89 | mulcld 10684 |
. . . . . . . . 9
⊢ (𝜑 → (2 · (𝑁 / 2)) ∈
ℂ) |
93 | 92, 87 | npcand 11024 |
. . . . . . . 8
⊢ (𝜑 → (((2 · (𝑁 / 2)) − 2) + 2) = (2
· (𝑁 /
2))) |
94 | 91, 93 | eqtrd 2794 |
. . . . . . 7
⊢ (𝜑 → ((2 · ((𝑁 / 2) − 1)) + 2) = (2
· (𝑁 /
2))) |
95 | 3 | nnne0d 11709 |
. . . . . . . 8
⊢ (𝜑 → 2 ≠ 0) |
96 | 88, 87, 95 | divcan2d 11441 |
. . . . . . 7
⊢ (𝜑 → (2 · (𝑁 / 2)) = 𝑁) |
97 | 94, 96 | eqtrd 2794 |
. . . . . 6
⊢ (𝜑 → ((2 · ((𝑁 / 2) − 1)) + 2) = 𝑁) |
98 | 97 | oveq2d 7159 |
. . . . 5
⊢ (𝜑 → (2↑((2 ·
((𝑁 / 2) − 1)) + 2))
= (2↑𝑁)) |
99 | 97 | oveq2d 7159 |
. . . . . 6
⊢ (𝜑 → (1...((2 · ((𝑁 / 2) − 1)) + 2)) =
(1...𝑁)) |
100 | 99 | fveq2d 6655 |
. . . . 5
⊢ (𝜑 → (lcm‘(1...((2
· ((𝑁 / 2) −
1)) + 2))) = (lcm‘(1...𝑁))) |
101 | 98, 100 | breq12d 5038 |
. . . 4
⊢ (𝜑 → ((2↑((2 ·
((𝑁 / 2) − 1)) + 2))
≤ (lcm‘(1...((2 · ((𝑁 / 2) − 1)) + 2))) ↔
(2↑𝑁) ≤
(lcm‘(1...𝑁)))) |
102 | 101 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → ((2↑((2 · ((𝑁 / 2) − 1)) + 2)) ≤
(lcm‘(1...((2 · ((𝑁 / 2) − 1)) + 2))) ↔
(2↑𝑁) ≤
(lcm‘(1...𝑁)))) |
103 | 86, 102 | mpbid 235 |
. 2
⊢ ((𝜑 ∧ 2 ∥ 𝑁) → (2↑𝑁) ≤ (lcm‘(1...𝑁))) |
104 | | oddm1even 15729 |
. . . . . . . 8
⊢ (𝑁 ∈ ℤ → (¬ 2
∥ 𝑁 ↔ 2 ∥
(𝑁 −
1))) |
105 | 70, 104 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1))) |
106 | 105 | biimpa 481 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → 2 ∥ (𝑁 − 1)) |
107 | 2 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → 2 ∈
ℕ) |
108 | | 1zzd 12037 |
. . . . . . . . . . 11
⊢ (𝜑 → 1 ∈
ℤ) |
109 | 70, 108 | zsubcld 12116 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
110 | | 0red 10667 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 ∈
ℝ) |
111 | 48 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → 8 ∈
ℝ) |
112 | | 1red 10665 |
. . . . . . . . . . . 12
⊢ (𝜑 → 1 ∈
ℝ) |
113 | 33, 112 | resubcld 11091 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
114 | | 8pos 11771 |
. . . . . . . . . . . 12
⊢ 0 <
8 |
115 | 114 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 < 8) |
116 | 40, 33, 112, 38 | lesub1dd 11279 |
. . . . . . . . . . . 12
⊢ (𝜑 → (9 − 1) ≤ (𝑁 − 1)) |
117 | 58, 116 | eqbrtrrid 5061 |
. . . . . . . . . . 11
⊢ (𝜑 → 8 ≤ (𝑁 − 1)) |
118 | 110, 111,
113, 115, 117 | ltletrd 10823 |
. . . . . . . . . 10
⊢ (𝜑 → 0 < (𝑁 − 1)) |
119 | 109, 118 | jca 516 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑁 − 1) ∈ ℤ ∧ 0 <
(𝑁 −
1))) |
120 | | elnnz 12015 |
. . . . . . . . 9
⊢ ((𝑁 − 1) ∈ ℕ
↔ ((𝑁 − 1)
∈ ℤ ∧ 0 < (𝑁 − 1))) |
121 | 119, 120 | sylibr 237 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 − 1) ∈ ℕ) |
122 | 121 | adantr 485 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → (𝑁 − 1) ∈ ℕ) |
123 | 107, 122 | nndivdvdsd 39552 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → (2 ∥ (𝑁 − 1) ↔ ((𝑁 − 1) / 2) ∈
ℕ)) |
124 | 106, 123 | mpbid 235 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → ((𝑁 − 1) / 2) ∈
ℕ) |
125 | 44, 22 | mulcomi 10672 |
. . . . . . . . 9
⊢ (4
· 2) = (2 · 4) |
126 | 125, 46 | eqtr3i 2784 |
. . . . . . . 8
⊢ (2
· 4) = 8 |
127 | 126, 50 | mpbir 234 |
. . . . . . 7
⊢ (8 / 2) =
4 |
128 | 3 | nnrpd 12455 |
. . . . . . . 8
⊢ (𝜑 → 2 ∈
ℝ+) |
129 | 111, 113,
128, 117 | lediv1dd 12515 |
. . . . . . 7
⊢ (𝜑 → (8 / 2) ≤ ((𝑁 − 1) /
2)) |
130 | 127, 129 | eqbrtrrid 5061 |
. . . . . 6
⊢ (𝜑 → 4 ≤ ((𝑁 − 1) / 2)) |
131 | 130 | adantr 485 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → 4 ≤ ((𝑁 − 1) /
2)) |
132 | 124, 131 | lcmineqlem22 39602 |
. . . 4
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → ((2↑((2 ·
((𝑁 − 1) / 2)) + 1))
≤ (lcm‘(1...((2 · ((𝑁 − 1) / 2)) + 1))) ∧ (2↑((2
· ((𝑁 − 1) /
2)) + 2)) ≤ (lcm‘(1...((2 · ((𝑁 − 1) / 2)) + 2))))) |
133 | 132 | simpld 499 |
. . 3
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → (2↑((2 ·
((𝑁 − 1) / 2)) + 1))
≤ (lcm‘(1...((2 · ((𝑁 − 1) / 2)) + 1)))) |
134 | | 1cnd 10659 |
. . . . . . . . . 10
⊢ (𝜑 → 1 ∈
ℂ) |
135 | 88, 134 | subcld 11020 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁 − 1) ∈ ℂ) |
136 | 135, 87, 95 | divcan2d 11441 |
. . . . . . . 8
⊢ (𝜑 → (2 · ((𝑁 − 1) / 2)) = (𝑁 − 1)) |
137 | 136 | oveq1d 7158 |
. . . . . . 7
⊢ (𝜑 → ((2 · ((𝑁 − 1) / 2)) + 1) = ((𝑁 − 1) +
1)) |
138 | 88, 134 | npcand 11024 |
. . . . . . 7
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
139 | 137, 138 | eqtrd 2794 |
. . . . . 6
⊢ (𝜑 → ((2 · ((𝑁 − 1) / 2)) + 1) = 𝑁) |
140 | 139 | oveq2d 7159 |
. . . . 5
⊢ (𝜑 → (2↑((2 ·
((𝑁 − 1) / 2)) + 1))
= (2↑𝑁)) |
141 | 139 | oveq2d 7159 |
. . . . . 6
⊢ (𝜑 → (1...((2 · ((𝑁 − 1) / 2)) + 1)) =
(1...𝑁)) |
142 | 141 | fveq2d 6655 |
. . . . 5
⊢ (𝜑 → (lcm‘(1...((2
· ((𝑁 − 1) /
2)) + 1))) = (lcm‘(1...𝑁))) |
143 | 140, 142 | breq12d 5038 |
. . . 4
⊢ (𝜑 → ((2↑((2 ·
((𝑁 − 1) / 2)) + 1))
≤ (lcm‘(1...((2 · ((𝑁 − 1) / 2)) + 1))) ↔
(2↑𝑁) ≤
(lcm‘(1...𝑁)))) |
144 | 143 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → ((2↑((2 ·
((𝑁 − 1) / 2)) + 1))
≤ (lcm‘(1...((2 · ((𝑁 − 1) / 2)) + 1))) ↔
(2↑𝑁) ≤
(lcm‘(1...𝑁)))) |
145 | 133, 144 | mpbid 235 |
. 2
⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑁) → (2↑𝑁) ≤
(lcm‘(1...𝑁))) |
146 | 103, 145 | pm2.61dan 813 |
1
⊢ (𝜑 → (2↑𝑁) ≤ (lcm‘(1...𝑁))) |