Proof of Theorem zeo
Step | Hyp | Ref
| Expression |
1 | | elz 8023 |
. 2
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ
∧ (𝑁 = 0 ∨
𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) |
2 | | oveq1 5462 |
. . . . . 6
⊢ (𝑁 = 0 → (𝑁 / 2) = (0 / 2)) |
3 | | 2cn 7766 |
. . . . . . . 8
⊢ 2 ∈ ℂ |
4 | | 2ap0 7789 |
. . . . . . . 8
⊢ 2 #
0 |
5 | 3, 4 | div0api 7504 |
. . . . . . 7
⊢ (0 / 2) =
0 |
6 | | 0z 8032 |
. . . . . . 7
⊢ 0 ∈ ℤ |
7 | 5, 6 | eqeltri 2107 |
. . . . . 6
⊢ (0 / 2)
∈ ℤ |
8 | 2, 7 | syl6eqel 2125 |
. . . . 5
⊢ (𝑁 = 0 → (𝑁 / 2) ∈
ℤ) |
9 | 8 | orcd 651 |
. . . 4
⊢ (𝑁 = 0 → ((𝑁 / 2) ∈
ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) |
10 | 9 | adantl 262 |
. . 3
⊢ ((𝑁 ∈ ℝ ∧ 𝑁 = 0) → ((𝑁 / 2) ∈
ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) |
11 | | nneoor 8116 |
. . . . 5
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈
ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ)) |
12 | | nnz 8040 |
. . . . . 6
⊢ ((𝑁 / 2) ∈ ℕ → (𝑁 / 2) ∈
ℤ) |
13 | | nnz 8040 |
. . . . . 6
⊢ (((𝑁 + 1) / 2) ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℤ) |
14 | 12, 13 | orim12i 675 |
. . . . 5
⊢ (((𝑁 / 2) ∈ ℕ ∨
((𝑁 + 1) / 2) ∈ ℕ) → ((𝑁 / 2) ∈
ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) |
15 | 11, 14 | syl 14 |
. . . 4
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈
ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) |
16 | 15 | adantl 262 |
. . 3
⊢ ((𝑁 ∈ ℝ ∧ 𝑁 ∈ ℕ) → ((𝑁 / 2) ∈
ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) |
17 | | nneoor 8116 |
. . . . 5
⊢ (-𝑁 ∈ ℕ → ((-𝑁 / 2) ∈
ℕ ∨ ((-𝑁 + 1) / 2) ∈ ℕ)) |
18 | 17 | adantl 262 |
. . . 4
⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → ((-𝑁 / 2) ∈
ℕ ∨ ((-𝑁 + 1) / 2) ∈ ℕ)) |
19 | | recn 6812 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℝ → 𝑁 ∈
ℂ) |
20 | | divnegap 7465 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℂ ∧ 2
∈ ℂ ∧
2 # 0) → -(𝑁 / 2) =
(-𝑁 / 2)) |
21 | 3, 4, 20 | mp3an23 1223 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℂ → -(𝑁 / 2) = (-𝑁 / 2)) |
22 | 19, 21 | syl 14 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℝ → -(𝑁 / 2) = (-𝑁 / 2)) |
23 | 22 | eleq1d 2103 |
. . . . . . . 8
⊢ (𝑁 ∈ ℝ → (-(𝑁 / 2) ∈
ℕ ↔ (-𝑁 / 2)
∈ ℕ)) |
24 | | nnnegz 8024 |
. . . . . . . 8
⊢ (-(𝑁 / 2) ∈ ℕ → --(𝑁 / 2) ∈
ℤ) |
25 | 23, 24 | syl6bir 153 |
. . . . . . 7
⊢ (𝑁 ∈ ℝ → ((-𝑁 / 2) ∈
ℕ → --(𝑁 / 2)
∈ ℤ)) |
26 | 19 | halfcld 7946 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℝ → (𝑁 / 2) ∈
ℂ) |
27 | 26 | negnegd 7109 |
. . . . . . . 8
⊢ (𝑁 ∈ ℝ → --(𝑁 / 2) = (𝑁 / 2)) |
28 | 27 | eleq1d 2103 |
. . . . . . 7
⊢ (𝑁 ∈ ℝ → (--(𝑁 / 2) ∈
ℤ ↔ (𝑁 / 2)
∈ ℤ)) |
29 | 25, 28 | sylibd 138 |
. . . . . 6
⊢ (𝑁 ∈ ℝ → ((-𝑁 / 2) ∈
ℕ → (𝑁 / 2)
∈ ℤ)) |
30 | | nnz 8040 |
. . . . . . 7
⊢ (((-𝑁 + 1) / 2) ∈ ℕ → ((-𝑁 + 1) / 2) ∈ ℤ) |
31 | | peano2zm 8059 |
. . . . . . . . . 10
⊢ (((-𝑁 + 1) / 2) ∈ ℤ → (((-𝑁 + 1) / 2) − 1) ∈ ℤ) |
32 | | ax-1cn 6776 |
. . . . . . . . . . . . . . . . . . 19
⊢ 1 ∈ ℂ |
33 | 32, 3 | negsubdi2i 7093 |
. . . . . . . . . . . . . . . . . 18
⊢ -(1
− 2) = (2 − 1) |
34 | | 2m1e1 7812 |
. . . . . . . . . . . . . . . . . 18
⊢ (2
− 1) = 1 |
35 | 33, 34 | eqtr2i 2058 |
. . . . . . . . . . . . . . . . 17
⊢ 1 = -(1
− 2) |
36 | 32, 3 | subcli 7083 |
. . . . . . . . . . . . . . . . . 18
⊢ (1
− 2) ∈ ℂ |
37 | 32, 36 | negcon2i 7090 |
. . . . . . . . . . . . . . . . 17
⊢ (1 = -(1
− 2) ↔ (1 − 2) = -1) |
38 | 35, 37 | mpbi 133 |
. . . . . . . . . . . . . . . 16
⊢ (1
− 2) = -1 |
39 | 38 | oveq2i 5466 |
. . . . . . . . . . . . . . 15
⊢ (-𝑁 + (1 − 2)) = (-𝑁 + -1) |
40 | | negcl 7008 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈ ℂ → -𝑁 ∈
ℂ) |
41 | | addsubass 7018 |
. . . . . . . . . . . . . . . . 17
⊢ ((-𝑁 ∈ ℂ ∧ 1
∈ ℂ ∧
2 ∈ ℂ) → ((-𝑁 + 1) − 2) = (-𝑁 + (1 − 2))) |
42 | 32, 3, 41 | mp3an23 1223 |
. . . . . . . . . . . . . . . 16
⊢ (-𝑁 ∈ ℂ → ((-𝑁 + 1) − 2) = (-𝑁 + (1 − 2))) |
43 | 40, 42 | syl 14 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℂ → ((-𝑁 + 1) − 2) = (-𝑁 + (1 − 2))) |
44 | | negdi 7064 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℂ ∧ 1
∈ ℂ) → -(𝑁 + 1) = (-𝑁 + -1)) |
45 | 32, 44 | mpan2 401 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℂ → -(𝑁 + 1) = (-𝑁 + -1)) |
46 | 39, 43, 45 | 3eqtr4a 2095 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℂ → ((-𝑁 + 1) − 2) = -(𝑁 + 1)) |
47 | 46 | oveq1d 5470 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) − 2) / 2) = (-(𝑁 + 1) / 2)) |
48 | | peano2cn 6945 |
. . . . . . . . . . . . . . . 16
⊢ (-𝑁 ∈ ℂ → (-𝑁 + 1) ∈
ℂ) |
49 | 40, 48 | syl 14 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℂ → (-𝑁 + 1) ∈
ℂ) |
50 | 3, 4 | pm3.2i 257 |
. . . . . . . . . . . . . . . 16
⊢ (2 ∈ ℂ ∧ 2 #
0) |
51 | | divsubdirap 7466 |
. . . . . . . . . . . . . . . 16
⊢ (((-𝑁 + 1) ∈ ℂ ∧ 2
∈ ℂ ∧
(2 ∈ ℂ ∧ 2 # 0)) → (((-𝑁 + 1) − 2) / 2) = (((-𝑁 + 1) / 2) − (2 /
2))) |
52 | 3, 50, 51 | mp3an23 1223 |
. . . . . . . . . . . . . . 15
⊢ ((-𝑁 + 1) ∈ ℂ → (((-𝑁 + 1) − 2) / 2) = (((-𝑁 + 1) / 2) − (2 /
2))) |
53 | 49, 52 | syl 14 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) − 2) / 2) = (((-𝑁 + 1) / 2) − (2 /
2))) |
54 | | 2div2e1 7820 |
. . . . . . . . . . . . . . . 16
⊢ (2 / 2) =
1 |
55 | 54 | eqcomi 2041 |
. . . . . . . . . . . . . . 15
⊢ 1 = (2 /
2) |
56 | 55 | oveq2i 5466 |
. . . . . . . . . . . . . 14
⊢ (((-𝑁 + 1) / 2) − 1) =
(((-𝑁 + 1) / 2) − (2
/ 2)) |
57 | 53, 56 | syl6reqr 2088 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) / 2) − 1) = (((-𝑁 + 1) − 2) /
2)) |
58 | | peano2cn 6945 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℂ → (𝑁 + 1) ∈
ℂ) |
59 | | divnegap 7465 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 + 1) ∈ ℂ ∧ 2
∈ ℂ ∧
2 # 0) → -((𝑁 + 1) /
2) = (-(𝑁 + 1) /
2)) |
60 | 3, 4, 59 | mp3an23 1223 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 + 1) ∈ ℂ → -((𝑁 + 1) / 2) = (-(𝑁 + 1) / 2)) |
61 | 58, 60 | syl 14 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℂ → -((𝑁 + 1) / 2) = (-(𝑁 + 1) / 2)) |
62 | 47, 57, 61 | 3eqtr4d 2079 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) / 2) − 1) = -((𝑁 + 1) / 2)) |
63 | 19, 62 | syl 14 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) − 1) = -((𝑁 + 1) / 2)) |
64 | 63 | eleq1d 2103 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℝ → ((((-𝑁 + 1) / 2) − 1) ∈ ℤ ↔ -((𝑁 + 1) / 2) ∈ ℤ)) |
65 | 31, 64 | syl5ib 143 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℤ → -((𝑁 + 1) / 2) ∈ ℤ)) |
66 | | znegcl 8052 |
. . . . . . . . 9
⊢ (-((𝑁 + 1) / 2) ∈ ℤ → --((𝑁 + 1) / 2) ∈ ℤ) |
67 | 65, 66 | syl6 29 |
. . . . . . . 8
⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℤ → --((𝑁 + 1) / 2) ∈ ℤ)) |
68 | | peano2re 6946 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈
ℝ) |
69 | 68 | recnd 6851 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈
ℂ) |
70 | 69 | halfcld 7946 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℝ → ((𝑁 + 1) / 2) ∈ ℂ) |
71 | 70 | negnegd 7109 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℝ → --((𝑁 + 1) / 2) = ((𝑁 + 1) / 2)) |
72 | 71 | eleq1d 2103 |
. . . . . . . 8
⊢ (𝑁 ∈ ℝ → (--((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 + 1) / 2) ∈ ℤ)) |
73 | 67, 72 | sylibd 138 |
. . . . . . 7
⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℤ → ((𝑁 + 1) / 2) ∈ ℤ)) |
74 | 30, 73 | syl5 28 |
. . . . . 6
⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℤ)) |
75 | 29, 74 | orim12d 699 |
. . . . 5
⊢ (𝑁 ∈ ℝ → (((-𝑁 / 2) ∈
ℕ ∨ ((-𝑁 + 1) / 2) ∈ ℕ) → ((𝑁 / 2) ∈
ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))) |
76 | 75 | adantr 261 |
. . . 4
⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (((-𝑁 / 2) ∈
ℕ ∨ ((-𝑁 + 1) / 2) ∈ ℕ) → ((𝑁 / 2) ∈
ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))) |
77 | 18, 76 | mpd 13 |
. . 3
⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → ((𝑁 / 2) ∈
ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) |
78 | 10, 16, 77 | 3jaodan 1200 |
. 2
⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0
∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) → ((𝑁 / 2) ∈
ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) |
79 | 1, 78 | sylbi 114 |
1
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈
ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) |