Proof of Theorem zeo
Step | Hyp | Ref
| Expression |
1 | | elz 12066 |
. . 3
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) |
2 | | oveq1 7179 |
. . . . . . 7
⊢ (𝑁 = 0 → (𝑁 / 2) = (0 / 2)) |
3 | | 2cn 11793 |
. . . . . . . . 9
⊢ 2 ∈
ℂ |
4 | | 2ne0 11822 |
. . . . . . . . 9
⊢ 2 ≠
0 |
5 | 3, 4 | div0i 11454 |
. . . . . . . 8
⊢ (0 / 2) =
0 |
6 | | 0z 12075 |
. . . . . . . 8
⊢ 0 ∈
ℤ |
7 | 5, 6 | eqeltri 2829 |
. . . . . . 7
⊢ (0 / 2)
∈ ℤ |
8 | 2, 7 | eqeltrdi 2841 |
. . . . . 6
⊢ (𝑁 = 0 → (𝑁 / 2) ∈ ℤ) |
9 | 8 | pm2.24d 154 |
. . . . 5
⊢ (𝑁 = 0 → (¬ (𝑁 / 2) ∈ ℤ →
((𝑁 + 1) / 2) ∈
ℤ)) |
10 | 9 | adantl 485 |
. . . 4
⊢ ((𝑁 ∈ ℝ ∧ 𝑁 = 0) → (¬ (𝑁 / 2) ∈ ℤ →
((𝑁 + 1) / 2) ∈
ℤ)) |
11 | | nnz 12087 |
. . . . . . 7
⊢ ((𝑁 / 2) ∈ ℕ →
(𝑁 / 2) ∈
ℤ) |
12 | 11 | con3i 157 |
. . . . . 6
⊢ (¬
(𝑁 / 2) ∈ ℤ
→ ¬ (𝑁 / 2) ∈
ℕ) |
13 | | nneo 12149 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ↔
¬ ((𝑁 + 1) / 2) ∈
ℕ)) |
14 | 13 | biimprd 251 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → (¬
((𝑁 + 1) / 2) ∈
ℕ → (𝑁 / 2)
∈ ℕ)) |
15 | 14 | con1d 147 |
. . . . . 6
⊢ (𝑁 ∈ ℕ → (¬
(𝑁 / 2) ∈ ℕ
→ ((𝑁 + 1) / 2) ∈
ℕ)) |
16 | | nnz 12087 |
. . . . . 6
⊢ (((𝑁 + 1) / 2) ∈ ℕ →
((𝑁 + 1) / 2) ∈
ℤ) |
17 | 12, 15, 16 | syl56 36 |
. . . . 5
⊢ (𝑁 ∈ ℕ → (¬
(𝑁 / 2) ∈ ℤ
→ ((𝑁 + 1) / 2) ∈
ℤ)) |
18 | 17 | adantl 485 |
. . . 4
⊢ ((𝑁 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (¬
(𝑁 / 2) ∈ ℤ
→ ((𝑁 + 1) / 2) ∈
ℤ)) |
19 | | recn 10707 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℝ → 𝑁 ∈
ℂ) |
20 | | divneg 11412 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℂ ∧ 2 ∈
ℂ ∧ 2 ≠ 0) → -(𝑁 / 2) = (-𝑁 / 2)) |
21 | 3, 4, 20 | mp3an23 1454 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℂ → -(𝑁 / 2) = (-𝑁 / 2)) |
22 | 19, 21 | syl 17 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℝ → -(𝑁 / 2) = (-𝑁 / 2)) |
23 | 22 | eleq1d 2817 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℝ → (-(𝑁 / 2) ∈ ℕ ↔
(-𝑁 / 2) ∈
ℕ)) |
24 | | nnnegz 12067 |
. . . . . . . . 9
⊢ (-(𝑁 / 2) ∈ ℕ →
--(𝑁 / 2) ∈
ℤ) |
25 | 23, 24 | syl6bir 257 |
. . . . . . . 8
⊢ (𝑁 ∈ ℝ → ((-𝑁 / 2) ∈ ℕ →
--(𝑁 / 2) ∈
ℤ)) |
26 | 19 | halfcld 11963 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℝ → (𝑁 / 2) ∈
ℂ) |
27 | 26 | negnegd 11068 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℝ → --(𝑁 / 2) = (𝑁 / 2)) |
28 | 27 | eleq1d 2817 |
. . . . . . . 8
⊢ (𝑁 ∈ ℝ → (--(𝑁 / 2) ∈ ℤ ↔
(𝑁 / 2) ∈
ℤ)) |
29 | 25, 28 | sylibd 242 |
. . . . . . 7
⊢ (𝑁 ∈ ℝ → ((-𝑁 / 2) ∈ ℕ →
(𝑁 / 2) ∈
ℤ)) |
30 | 29 | adantr 484 |
. . . . . 6
⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → ((-𝑁 / 2) ∈ ℕ →
(𝑁 / 2) ∈
ℤ)) |
31 | 30 | con3d 155 |
. . . . 5
⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (¬
(𝑁 / 2) ∈ ℤ
→ ¬ (-𝑁 / 2)
∈ ℕ)) |
32 | | nneo 12149 |
. . . . . . . 8
⊢ (-𝑁 ∈ ℕ → ((-𝑁 / 2) ∈ ℕ ↔
¬ ((-𝑁 + 1) / 2) ∈
ℕ)) |
33 | 32 | biimprd 251 |
. . . . . . 7
⊢ (-𝑁 ∈ ℕ → (¬
((-𝑁 + 1) / 2) ∈
ℕ → (-𝑁 / 2)
∈ ℕ)) |
34 | 33 | con1d 147 |
. . . . . 6
⊢ (-𝑁 ∈ ℕ → (¬
(-𝑁 / 2) ∈ ℕ
→ ((-𝑁 + 1) / 2)
∈ ℕ)) |
35 | | nnz 12087 |
. . . . . . 7
⊢ (((-𝑁 + 1) / 2) ∈ ℕ →
((-𝑁 + 1) / 2) ∈
ℤ) |
36 | | peano2zm 12108 |
. . . . . . . . . 10
⊢ (((-𝑁 + 1) / 2) ∈ ℤ →
(((-𝑁 + 1) / 2) − 1)
∈ ℤ) |
37 | | ax-1cn 10675 |
. . . . . . . . . . . . . . . . . . 19
⊢ 1 ∈
ℂ |
38 | 37, 3 | negsubdi2i 11052 |
. . . . . . . . . . . . . . . . . 18
⊢ -(1
− 2) = (2 − 1) |
39 | | 2m1e1 11844 |
. . . . . . . . . . . . . . . . . 18
⊢ (2
− 1) = 1 |
40 | 38, 39 | eqtr2i 2762 |
. . . . . . . . . . . . . . . . 17
⊢ 1 = -(1
− 2) |
41 | 37, 3 | subcli 11042 |
. . . . . . . . . . . . . . . . . 18
⊢ (1
− 2) ∈ ℂ |
42 | 37, 41 | negcon2i 11049 |
. . . . . . . . . . . . . . . . 17
⊢ (1 = -(1
− 2) ↔ (1 − 2) = -1) |
43 | 40, 42 | mpbi 233 |
. . . . . . . . . . . . . . . 16
⊢ (1
− 2) = -1 |
44 | 43 | oveq2i 7183 |
. . . . . . . . . . . . . . 15
⊢ (-𝑁 + (1 − 2)) = (-𝑁 + -1) |
45 | | negcl 10966 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈ ℂ → -𝑁 ∈
ℂ) |
46 | | addsubass 10976 |
. . . . . . . . . . . . . . . . 17
⊢ ((-𝑁 ∈ ℂ ∧ 1 ∈
ℂ ∧ 2 ∈ ℂ) → ((-𝑁 + 1) − 2) = (-𝑁 + (1 − 2))) |
47 | 37, 3, 46 | mp3an23 1454 |
. . . . . . . . . . . . . . . 16
⊢ (-𝑁 ∈ ℂ → ((-𝑁 + 1) − 2) = (-𝑁 + (1 −
2))) |
48 | 45, 47 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℂ → ((-𝑁 + 1) − 2) = (-𝑁 + (1 −
2))) |
49 | | negdi 11023 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℂ ∧ 1 ∈
ℂ) → -(𝑁 + 1) =
(-𝑁 + -1)) |
50 | 37, 49 | mpan2 691 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℂ → -(𝑁 + 1) = (-𝑁 + -1)) |
51 | 44, 48, 50 | 3eqtr4a 2799 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℂ → ((-𝑁 + 1) − 2) = -(𝑁 + 1)) |
52 | 51 | oveq1d 7187 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) − 2) / 2) = (-(𝑁 + 1) / 2)) |
53 | | 2div2e1 11859 |
. . . . . . . . . . . . . . . 16
⊢ (2 / 2) =
1 |
54 | 53 | eqcomi 2747 |
. . . . . . . . . . . . . . 15
⊢ 1 = (2 /
2) |
55 | 54 | oveq2i 7183 |
. . . . . . . . . . . . . 14
⊢ (((-𝑁 + 1) / 2) − 1) =
(((-𝑁 + 1) / 2) − (2
/ 2)) |
56 | | peano2cn 10892 |
. . . . . . . . . . . . . . . 16
⊢ (-𝑁 ∈ ℂ → (-𝑁 + 1) ∈
ℂ) |
57 | 45, 56 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℂ → (-𝑁 + 1) ∈
ℂ) |
58 | | 2cnne0 11928 |
. . . . . . . . . . . . . . . 16
⊢ (2 ∈
ℂ ∧ 2 ≠ 0) |
59 | | divsubdir 11414 |
. . . . . . . . . . . . . . . 16
⊢ (((-𝑁 + 1) ∈ ℂ ∧ 2
∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((-𝑁 + 1) − 2) / 2) =
(((-𝑁 + 1) / 2) − (2
/ 2))) |
60 | 3, 58, 59 | mp3an23 1454 |
. . . . . . . . . . . . . . 15
⊢ ((-𝑁 + 1) ∈ ℂ →
(((-𝑁 + 1) − 2) / 2)
= (((-𝑁 + 1) / 2) −
(2 / 2))) |
61 | 57, 60 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) − 2) / 2) =
(((-𝑁 + 1) / 2) − (2
/ 2))) |
62 | 55, 61 | eqtr4id 2792 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) / 2) − 1) =
(((-𝑁 + 1) − 2) /
2)) |
63 | | peano2cn 10892 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℂ → (𝑁 + 1) ∈
ℂ) |
64 | | divneg 11412 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 + 1) ∈ ℂ ∧ 2
∈ ℂ ∧ 2 ≠ 0) → -((𝑁 + 1) / 2) = (-(𝑁 + 1) / 2)) |
65 | 3, 4, 64 | mp3an23 1454 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 + 1) ∈ ℂ →
-((𝑁 + 1) / 2) = (-(𝑁 + 1) / 2)) |
66 | 63, 65 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℂ → -((𝑁 + 1) / 2) = (-(𝑁 + 1) / 2)) |
67 | 52, 62, 66 | 3eqtr4d 2783 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) / 2) − 1) = -((𝑁 + 1) / 2)) |
68 | 19, 67 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) − 1) = -((𝑁 + 1) / 2)) |
69 | 68 | eleq1d 2817 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℝ →
((((-𝑁 + 1) / 2) − 1)
∈ ℤ ↔ -((𝑁
+ 1) / 2) ∈ ℤ)) |
70 | 36, 69 | syl5ib 247 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℤ →
-((𝑁 + 1) / 2) ∈
ℤ)) |
71 | | znegcl 12100 |
. . . . . . . . 9
⊢ (-((𝑁 + 1) / 2) ∈ ℤ →
--((𝑁 + 1) / 2) ∈
ℤ) |
72 | 70, 71 | syl6 35 |
. . . . . . . 8
⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℤ →
--((𝑁 + 1) / 2) ∈
ℤ)) |
73 | | peano2re 10893 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈
ℝ) |
74 | 73 | recnd 10749 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈
ℂ) |
75 | 74 | halfcld 11963 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℝ → ((𝑁 + 1) / 2) ∈
ℂ) |
76 | 75 | negnegd 11068 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℝ → --((𝑁 + 1) / 2) = ((𝑁 + 1) / 2)) |
77 | 76 | eleq1d 2817 |
. . . . . . . 8
⊢ (𝑁 ∈ ℝ →
(--((𝑁 + 1) / 2) ∈
ℤ ↔ ((𝑁 + 1) /
2) ∈ ℤ)) |
78 | 72, 77 | sylibd 242 |
. . . . . . 7
⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℤ →
((𝑁 + 1) / 2) ∈
ℤ)) |
79 | 35, 78 | syl5 34 |
. . . . . 6
⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℕ →
((𝑁 + 1) / 2) ∈
ℤ)) |
80 | 34, 79 | sylan9r 512 |
. . . . 5
⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (¬
(-𝑁 / 2) ∈ ℕ
→ ((𝑁 + 1) / 2) ∈
ℤ)) |
81 | 31, 80 | syld 47 |
. . . 4
⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (¬
(𝑁 / 2) ∈ ℤ
→ ((𝑁 + 1) / 2) ∈
ℤ)) |
82 | 10, 18, 81 | 3jaodan 1431 |
. . 3
⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) → (¬ (𝑁 / 2) ∈ ℤ →
((𝑁 + 1) / 2) ∈
ℤ)) |
83 | 1, 82 | sylbi 220 |
. 2
⊢ (𝑁 ∈ ℤ → (¬
(𝑁 / 2) ∈ ℤ
→ ((𝑁 + 1) / 2) ∈
ℤ)) |
84 | 83 | orrd 862 |
1
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨
((𝑁 + 1) / 2) ∈
ℤ)) |