Proof of Theorem lgsdir2lem2
Step | Hyp | Ref
| Expression |
1 | | lgsdir2lem2.3 |
. . 3
⊢ 𝑁 = (𝑀 + 1) |
2 | | lgsdir2lem2.2 |
. . . . 5
⊢ 𝑀 = (𝐾 + 1) |
3 | | lgsdir2lem2.1 |
. . . . . . 7
⊢ (𝐾 ∈ ℤ ∧ 2 ∥
(𝐾 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2
∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝐾) → (𝐴 mod 8) ∈ 𝑆))) |
4 | 3 | simp1i 1140 |
. . . . . 6
⊢ 𝐾 ∈ ℤ |
5 | | peano2z 12104 |
. . . . . 6
⊢ (𝐾 ∈ ℤ → (𝐾 + 1) ∈
ℤ) |
6 | 4, 5 | ax-mp 5 |
. . . . 5
⊢ (𝐾 + 1) ∈
ℤ |
7 | 2, 6 | eqeltri 2829 |
. . . 4
⊢ 𝑀 ∈ ℤ |
8 | | peano2z 12104 |
. . . 4
⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈
ℤ) |
9 | 7, 8 | ax-mp 5 |
. . 3
⊢ (𝑀 + 1) ∈
ℤ |
10 | 1, 9 | eqeltri 2829 |
. 2
⊢ 𝑁 ∈ ℤ |
11 | 3 | simp2i 1141 |
. . . 4
⊢ 2 ∥
(𝐾 + 1) |
12 | | 2z 12095 |
. . . . 5
⊢ 2 ∈
ℤ |
13 | | dvdsadd 15747 |
. . . . 5
⊢ ((2
∈ ℤ ∧ (𝐾 +
1) ∈ ℤ) → (2 ∥ (𝐾 + 1) ↔ 2 ∥ (2 + (𝐾 + 1)))) |
14 | 12, 6, 13 | mp2an 692 |
. . . 4
⊢ (2
∥ (𝐾 + 1) ↔ 2
∥ (2 + (𝐾 +
1))) |
15 | 11, 14 | mpbi 233 |
. . 3
⊢ 2 ∥
(2 + (𝐾 +
1)) |
16 | | zcn 12067 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ ℤ → 𝐾 ∈
ℂ) |
17 | 4, 16 | ax-mp 5 |
. . . . . . . . . 10
⊢ 𝐾 ∈ ℂ |
18 | | ax-1cn 10673 |
. . . . . . . . . 10
⊢ 1 ∈
ℂ |
19 | 17, 18 | addcomi 10909 |
. . . . . . . . 9
⊢ (𝐾 + 1) = (1 + 𝐾) |
20 | 2, 19 | eqtri 2761 |
. . . . . . . 8
⊢ 𝑀 = (1 + 𝐾) |
21 | 20 | oveq1i 7180 |
. . . . . . 7
⊢ (𝑀 + 1) = ((1 + 𝐾) + 1) |
22 | 1, 21 | eqtri 2761 |
. . . . . 6
⊢ 𝑁 = ((1 + 𝐾) + 1) |
23 | | df-2 11779 |
. . . . . . . 8
⊢ 2 = (1 +
1) |
24 | 23 | oveq1i 7180 |
. . . . . . 7
⊢ (2 +
𝐾) = ((1 + 1) + 𝐾) |
25 | 18, 17, 18 | add32i 10941 |
. . . . . . 7
⊢ ((1 +
𝐾) + 1) = ((1 + 1) + 𝐾) |
26 | 24, 25 | eqtr4i 2764 |
. . . . . 6
⊢ (2 +
𝐾) = ((1 + 𝐾) + 1) |
27 | 22, 26 | eqtr4i 2764 |
. . . . 5
⊢ 𝑁 = (2 + 𝐾) |
28 | 27 | oveq1i 7180 |
. . . 4
⊢ (𝑁 + 1) = ((2 + 𝐾) + 1) |
29 | | 2cn 11791 |
. . . . 5
⊢ 2 ∈
ℂ |
30 | 29, 17, 18 | addassi 10729 |
. . . 4
⊢ ((2 +
𝐾) + 1) = (2 + (𝐾 + 1)) |
31 | 28, 30 | eqtri 2761 |
. . 3
⊢ (𝑁 + 1) = (2 + (𝐾 + 1)) |
32 | 15, 31 | breqtrri 5057 |
. 2
⊢ 2 ∥
(𝑁 + 1) |
33 | | elfzuz2 13003 |
. . . . 5
⊢ ((𝐴 mod 8) ∈ (0...𝑁) → 𝑁 ∈
(ℤ≥‘0)) |
34 | | fzm1 13078 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘0) → ((𝐴 mod 8) ∈ (0...𝑁) ↔ ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) ∨ (𝐴 mod 8) = 𝑁))) |
35 | 33, 34 | syl 17 |
. . . 4
⊢ ((𝐴 mod 8) ∈ (0...𝑁) → ((𝐴 mod 8) ∈ (0...𝑁) ↔ ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) ∨ (𝐴 mod 8) = 𝑁))) |
36 | 35 | ibi 270 |
. . 3
⊢ ((𝐴 mod 8) ∈ (0...𝑁) → ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) ∨ (𝐴 mod 8) = 𝑁)) |
37 | | elfzuz2 13003 |
. . . . . . . 8
⊢ ((𝐴 mod 8) ∈ (0...𝑀) → 𝑀 ∈
(ℤ≥‘0)) |
38 | | fzm1 13078 |
. . . . . . . 8
⊢ (𝑀 ∈
(ℤ≥‘0) → ((𝐴 mod 8) ∈ (0...𝑀) ↔ ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀))) |
39 | 37, 38 | syl 17 |
. . . . . . 7
⊢ ((𝐴 mod 8) ∈ (0...𝑀) → ((𝐴 mod 8) ∈ (0...𝑀) ↔ ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀))) |
40 | 39 | ibi 270 |
. . . . . 6
⊢ ((𝐴 mod 8) ∈ (0...𝑀) → ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀)) |
41 | | zcn 12067 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
ℂ) |
42 | 7, 41 | ax-mp 5 |
. . . . . . . 8
⊢ 𝑀 ∈ ℂ |
43 | 42, 18, 1 | mvrraddi 10981 |
. . . . . . 7
⊢ (𝑁 − 1) = 𝑀 |
44 | 43 | oveq2i 7181 |
. . . . . 6
⊢
(0...(𝑁 − 1))
= (0...𝑀) |
45 | 40, 44 | eleq2s 2851 |
. . . . 5
⊢ ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) → ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀)) |
46 | 17, 18, 2 | mvrraddi 10981 |
. . . . . . . . 9
⊢ (𝑀 − 1) = 𝐾 |
47 | 46 | oveq2i 7181 |
. . . . . . . 8
⊢
(0...(𝑀 − 1))
= (0...𝐾) |
48 | 47 | eleq2i 2824 |
. . . . . . 7
⊢ ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ↔ (𝐴 mod 8) ∈ (0...𝐾)) |
49 | 3 | simp3i 1142 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2
∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝐾) → (𝐴 mod 8) ∈ 𝑆)) |
50 | 48, 49 | syl5bi 245 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2
∥ 𝐴) → ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) → (𝐴 mod 8) ∈ 𝑆)) |
51 | | 2nn 11789 |
. . . . . . . . . . 11
⊢ 2 ∈
ℕ |
52 | | 8nn 11811 |
. . . . . . . . . . 11
⊢ 8 ∈
ℕ |
53 | | 4z 12097 |
. . . . . . . . . . . . . 14
⊢ 4 ∈
ℤ |
54 | | dvdsmul2 15724 |
. . . . . . . . . . . . . 14
⊢ ((4
∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (4 ·
2)) |
55 | 53, 12, 54 | mp2an 692 |
. . . . . . . . . . . . 13
⊢ 2 ∥
(4 · 2) |
56 | | 4t2e8 11884 |
. . . . . . . . . . . . 13
⊢ (4
· 2) = 8 |
57 | 55, 56 | breqtri 5055 |
. . . . . . . . . . . 12
⊢ 2 ∥
8 |
58 | | dvdsmod 15774 |
. . . . . . . . . . . 12
⊢ (((2
∈ ℕ ∧ 8 ∈ ℕ ∧ 𝐴 ∈ ℤ) ∧ 2 ∥ 8) →
(2 ∥ (𝐴 mod 8) ↔
2 ∥ 𝐴)) |
59 | 57, 58 | mpan2 691 |
. . . . . . . . . . 11
⊢ ((2
∈ ℕ ∧ 8 ∈ ℕ ∧ 𝐴 ∈ ℤ) → (2 ∥ (𝐴 mod 8) ↔ 2 ∥ 𝐴)) |
60 | 51, 52, 59 | mp3an12 1452 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ℤ → (2
∥ (𝐴 mod 8) ↔ 2
∥ 𝐴)) |
61 | 60 | notbid 321 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℤ → (¬ 2
∥ (𝐴 mod 8) ↔
¬ 2 ∥ 𝐴)) |
62 | 61 | biimpar 481 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2
∥ 𝐴) → ¬ 2
∥ (𝐴 mod
8)) |
63 | 11, 2 | breqtrri 5057 |
. . . . . . . . 9
⊢ 2 ∥
𝑀 |
64 | | id 22 |
. . . . . . . . 9
⊢ ((𝐴 mod 8) = 𝑀 → (𝐴 mod 8) = 𝑀) |
65 | 63, 64 | breqtrrid 5068 |
. . . . . . . 8
⊢ ((𝐴 mod 8) = 𝑀 → 2 ∥ (𝐴 mod 8)) |
66 | 62, 65 | nsyl 142 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2
∥ 𝐴) → ¬
(𝐴 mod 8) = 𝑀) |
67 | 66 | pm2.21d 121 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2
∥ 𝐴) → ((𝐴 mod 8) = 𝑀 → (𝐴 mod 8) ∈ 𝑆)) |
68 | 50, 67 | jaod 858 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2
∥ 𝐴) → (((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀) → (𝐴 mod 8) ∈ 𝑆)) |
69 | 45, 68 | syl5 34 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2
∥ 𝐴) → ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) → (𝐴 mod 8) ∈ 𝑆)) |
70 | | lgsdir2lem2.4 |
. . . . . 6
⊢ 𝑁 ∈ 𝑆 |
71 | | eleq1 2820 |
. . . . . 6
⊢ ((𝐴 mod 8) = 𝑁 → ((𝐴 mod 8) ∈ 𝑆 ↔ 𝑁 ∈ 𝑆)) |
72 | 70, 71 | mpbiri 261 |
. . . . 5
⊢ ((𝐴 mod 8) = 𝑁 → (𝐴 mod 8) ∈ 𝑆) |
73 | 72 | a1i 11 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2
∥ 𝐴) → ((𝐴 mod 8) = 𝑁 → (𝐴 mod 8) ∈ 𝑆)) |
74 | 69, 73 | jaod 858 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2
∥ 𝐴) → (((𝐴 mod 8) ∈ (0...(𝑁 − 1)) ∨ (𝐴 mod 8) = 𝑁) → (𝐴 mod 8) ∈ 𝑆)) |
75 | 36, 74 | syl5 34 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2
∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝑁) → (𝐴 mod 8) ∈ 𝑆)) |
76 | 10, 32, 75 | 3pm3.2i 1340 |
1
⊢ (𝑁 ∈ ℤ ∧ 2 ∥
(𝑁 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2
∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝑁) → (𝐴 mod 8) ∈ 𝑆))) |