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 12658 |
. . . . . 6
⊢ (𝐾 ∈ ℤ → (𝐾 + 1) ∈
ℤ) |
| 6 | 4, 5 | ax-mp 5 |
. . . . 5
⊢ (𝐾 + 1) ∈
ℤ |
| 7 | 2, 6 | eqeltri 2837 |
. . . 4
⊢ 𝑀 ∈ ℤ |
| 8 | | peano2z 12658 |
. . . 4
⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈
ℤ) |
| 9 | 7, 8 | ax-mp 5 |
. . 3
⊢ (𝑀 + 1) ∈
ℤ |
| 10 | 1, 9 | eqeltri 2837 |
. 2
⊢ 𝑁 ∈ ℤ |
| 11 | 3 | simp2i 1141 |
. . . 4
⊢ 2 ∥
(𝐾 + 1) |
| 12 | | 2z 12649 |
. . . . 5
⊢ 2 ∈
ℤ |
| 13 | | dvdsadd 16339 |
. . . . 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 230 |
. . 3
⊢ 2 ∥
(2 + (𝐾 +
1)) |
| 16 | | zcn 12618 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ ℤ → 𝐾 ∈
ℂ) |
| 17 | 4, 16 | ax-mp 5 |
. . . . . . . . . 10
⊢ 𝐾 ∈ ℂ |
| 18 | | ax-1cn 11213 |
. . . . . . . . . 10
⊢ 1 ∈
ℂ |
| 19 | 17, 18 | addcomi 11452 |
. . . . . . . . 9
⊢ (𝐾 + 1) = (1 + 𝐾) |
| 20 | 2, 19 | eqtri 2765 |
. . . . . . . 8
⊢ 𝑀 = (1 + 𝐾) |
| 21 | 20 | oveq1i 7441 |
. . . . . . 7
⊢ (𝑀 + 1) = ((1 + 𝐾) + 1) |
| 22 | 1, 21 | eqtri 2765 |
. . . . . 6
⊢ 𝑁 = ((1 + 𝐾) + 1) |
| 23 | | df-2 12329 |
. . . . . . . 8
⊢ 2 = (1 +
1) |
| 24 | 23 | oveq1i 7441 |
. . . . . . 7
⊢ (2 +
𝐾) = ((1 + 1) + 𝐾) |
| 25 | 18, 17, 18 | add32i 11485 |
. . . . . . 7
⊢ ((1 +
𝐾) + 1) = ((1 + 1) + 𝐾) |
| 26 | 24, 25 | eqtr4i 2768 |
. . . . . 6
⊢ (2 +
𝐾) = ((1 + 𝐾) + 1) |
| 27 | 22, 26 | eqtr4i 2768 |
. . . . 5
⊢ 𝑁 = (2 + 𝐾) |
| 28 | 27 | oveq1i 7441 |
. . . 4
⊢ (𝑁 + 1) = ((2 + 𝐾) + 1) |
| 29 | | 2cn 12341 |
. . . . 5
⊢ 2 ∈
ℂ |
| 30 | 29, 17, 18 | addassi 11271 |
. . . 4
⊢ ((2 +
𝐾) + 1) = (2 + (𝐾 + 1)) |
| 31 | 28, 30 | eqtri 2765 |
. . 3
⊢ (𝑁 + 1) = (2 + (𝐾 + 1)) |
| 32 | 15, 31 | breqtrri 5170 |
. 2
⊢ 2 ∥
(𝑁 + 1) |
| 33 | | elfzuz2 13569 |
. . . . 5
⊢ ((𝐴 mod 8) ∈ (0...𝑁) → 𝑁 ∈
(ℤ≥‘0)) |
| 34 | | fzm1 13647 |
. . . . 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 267 |
. . 3
⊢ ((𝐴 mod 8) ∈ (0...𝑁) → ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) ∨ (𝐴 mod 8) = 𝑁)) |
| 37 | | elfzuz2 13569 |
. . . . . . . 8
⊢ ((𝐴 mod 8) ∈ (0...𝑀) → 𝑀 ∈
(ℤ≥‘0)) |
| 38 | | fzm1 13647 |
. . . . . . . 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 267 |
. . . . . 6
⊢ ((𝐴 mod 8) ∈ (0...𝑀) → ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀)) |
| 41 | | zcn 12618 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
ℂ) |
| 42 | 7, 41 | ax-mp 5 |
. . . . . . . 8
⊢ 𝑀 ∈ ℂ |
| 43 | 42, 18, 1 | mvrraddi 11525 |
. . . . . . 7
⊢ (𝑁 − 1) = 𝑀 |
| 44 | 43 | oveq2i 7442 |
. . . . . 6
⊢
(0...(𝑁 − 1))
= (0...𝑀) |
| 45 | 40, 44 | eleq2s 2859 |
. . . . 5
⊢ ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) → ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀)) |
| 46 | 17, 18, 2 | mvrraddi 11525 |
. . . . . . . . 9
⊢ (𝑀 − 1) = 𝐾 |
| 47 | 46 | oveq2i 7442 |
. . . . . . . 8
⊢
(0...(𝑀 − 1))
= (0...𝐾) |
| 48 | 47 | eleq2i 2833 |
. . . . . . 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 | biimtrid 242 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2
∥ 𝐴) → ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) → (𝐴 mod 8) ∈ 𝑆)) |
| 51 | | 2nn 12339 |
. . . . . . . . . . 11
⊢ 2 ∈
ℕ |
| 52 | | 8nn 12361 |
. . . . . . . . . . 11
⊢ 8 ∈
ℕ |
| 53 | | 4z 12651 |
. . . . . . . . . . . . . 14
⊢ 4 ∈
ℤ |
| 54 | | dvdsmul2 16316 |
. . . . . . . . . . . . . 14
⊢ ((4
∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (4 ·
2)) |
| 55 | 53, 12, 54 | mp2an 692 |
. . . . . . . . . . . . 13
⊢ 2 ∥
(4 · 2) |
| 56 | | 4t2e8 12434 |
. . . . . . . . . . . . 13
⊢ (4
· 2) = 8 |
| 57 | 55, 56 | breqtri 5168 |
. . . . . . . . . . . 12
⊢ 2 ∥
8 |
| 58 | | dvdsmod 16366 |
. . . . . . . . . . . 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 1453 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ℤ → (2
∥ (𝐴 mod 8) ↔ 2
∥ 𝐴)) |
| 61 | 60 | notbid 318 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℤ → (¬ 2
∥ (𝐴 mod 8) ↔
¬ 2 ∥ 𝐴)) |
| 62 | 61 | biimpar 477 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2
∥ 𝐴) → ¬ 2
∥ (𝐴 mod
8)) |
| 63 | 11, 2 | breqtrri 5170 |
. . . . . . . . 9
⊢ 2 ∥
𝑀 |
| 64 | | id 22 |
. . . . . . . . 9
⊢ ((𝐴 mod 8) = 𝑀 → (𝐴 mod 8) = 𝑀) |
| 65 | 63, 64 | breqtrrid 5181 |
. . . . . . . 8
⊢ ((𝐴 mod 8) = 𝑀 → 2 ∥ (𝐴 mod 8)) |
| 66 | 62, 65 | nsyl 140 |
. . . . . . 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 860 |
. . . . 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 2829 |
. . . . . 6
⊢ ((𝐴 mod 8) = 𝑁 → ((𝐴 mod 8) ∈ 𝑆 ↔ 𝑁 ∈ 𝑆)) |
| 72 | 70, 71 | mpbiri 258 |
. . . . 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 860 |
. . 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))) |