Proof of Theorem 2lgslem3a
| Step | Hyp | Ref
| Expression |
| 1 | | 2lgslem2.n |
. . 3
⊢ 𝑁 = (((𝑃 − 1) / 2) −
(⌊‘(𝑃 /
4))) |
| 2 | | oveq1 7438 |
. . . . 5
⊢ (𝑃 = ((8 · 𝐾) + 1) → (𝑃 − 1) = (((8 · 𝐾) + 1) −
1)) |
| 3 | 2 | oveq1d 7446 |
. . . 4
⊢ (𝑃 = ((8 · 𝐾) + 1) → ((𝑃 − 1) / 2) = ((((8
· 𝐾) + 1) − 1)
/ 2)) |
| 4 | | fvoveq1 7454 |
. . . 4
⊢ (𝑃 = ((8 · 𝐾) + 1) →
(⌊‘(𝑃 / 4)) =
(⌊‘(((8 · 𝐾) + 1) / 4))) |
| 5 | 3, 4 | oveq12d 7449 |
. . 3
⊢ (𝑃 = ((8 · 𝐾) + 1) → (((𝑃 − 1) / 2) −
(⌊‘(𝑃 / 4))) =
(((((8 · 𝐾) + 1)
− 1) / 2) − (⌊‘(((8 · 𝐾) + 1) / 4)))) |
| 6 | 1, 5 | eqtrid 2789 |
. 2
⊢ (𝑃 = ((8 · 𝐾) + 1) → 𝑁 = (((((8 · 𝐾) + 1) − 1) / 2) −
(⌊‘(((8 · 𝐾) + 1) / 4)))) |
| 7 | | 8nn0 12549 |
. . . . . . . . . 10
⊢ 8 ∈
ℕ0 |
| 8 | 7 | a1i 11 |
. . . . . . . . 9
⊢ (𝐾 ∈ ℕ0
→ 8 ∈ ℕ0) |
| 9 | | id 22 |
. . . . . . . . 9
⊢ (𝐾 ∈ ℕ0
→ 𝐾 ∈
ℕ0) |
| 10 | 8, 9 | nn0mulcld 12592 |
. . . . . . . 8
⊢ (𝐾 ∈ ℕ0
→ (8 · 𝐾)
∈ ℕ0) |
| 11 | 10 | nn0cnd 12589 |
. . . . . . 7
⊢ (𝐾 ∈ ℕ0
→ (8 · 𝐾)
∈ ℂ) |
| 12 | | pncan1 11687 |
. . . . . . 7
⊢ ((8
· 𝐾) ∈ ℂ
→ (((8 · 𝐾) +
1) − 1) = (8 · 𝐾)) |
| 13 | 11, 12 | syl 17 |
. . . . . 6
⊢ (𝐾 ∈ ℕ0
→ (((8 · 𝐾) +
1) − 1) = (8 · 𝐾)) |
| 14 | 13 | oveq1d 7446 |
. . . . 5
⊢ (𝐾 ∈ ℕ0
→ ((((8 · 𝐾) +
1) − 1) / 2) = ((8 · 𝐾) / 2)) |
| 15 | | 4cn 12351 |
. . . . . . . . . . 11
⊢ 4 ∈
ℂ |
| 16 | | 2cn 12341 |
. . . . . . . . . . 11
⊢ 2 ∈
ℂ |
| 17 | | 4t2e8 12434 |
. . . . . . . . . . 11
⊢ (4
· 2) = 8 |
| 18 | 15, 16, 17 | mulcomli 11270 |
. . . . . . . . . 10
⊢ (2
· 4) = 8 |
| 19 | 18 | eqcomi 2746 |
. . . . . . . . 9
⊢ 8 = (2
· 4) |
| 20 | 19 | a1i 11 |
. . . . . . . 8
⊢ (𝐾 ∈ ℕ0
→ 8 = (2 · 4)) |
| 21 | 20 | oveq1d 7446 |
. . . . . . 7
⊢ (𝐾 ∈ ℕ0
→ (8 · 𝐾) = ((2
· 4) · 𝐾)) |
| 22 | 16 | a1i 11 |
. . . . . . . 8
⊢ (𝐾 ∈ ℕ0
→ 2 ∈ ℂ) |
| 23 | 15 | a1i 11 |
. . . . . . . 8
⊢ (𝐾 ∈ ℕ0
→ 4 ∈ ℂ) |
| 24 | | nn0cn 12536 |
. . . . . . . 8
⊢ (𝐾 ∈ ℕ0
→ 𝐾 ∈
ℂ) |
| 25 | 22, 23, 24 | mulassd 11284 |
. . . . . . 7
⊢ (𝐾 ∈ ℕ0
→ ((2 · 4) · 𝐾) = (2 · (4 · 𝐾))) |
| 26 | 21, 25 | eqtrd 2777 |
. . . . . 6
⊢ (𝐾 ∈ ℕ0
→ (8 · 𝐾) = (2
· (4 · 𝐾))) |
| 27 | 26 | oveq1d 7446 |
. . . . 5
⊢ (𝐾 ∈ ℕ0
→ ((8 · 𝐾) / 2)
= ((2 · (4 · 𝐾)) / 2)) |
| 28 | | 4nn0 12545 |
. . . . . . . . 9
⊢ 4 ∈
ℕ0 |
| 29 | 28 | a1i 11 |
. . . . . . . 8
⊢ (𝐾 ∈ ℕ0
→ 4 ∈ ℕ0) |
| 30 | 29, 9 | nn0mulcld 12592 |
. . . . . . 7
⊢ (𝐾 ∈ ℕ0
→ (4 · 𝐾)
∈ ℕ0) |
| 31 | 30 | nn0cnd 12589 |
. . . . . 6
⊢ (𝐾 ∈ ℕ0
→ (4 · 𝐾)
∈ ℂ) |
| 32 | | 2ne0 12370 |
. . . . . . 7
⊢ 2 ≠
0 |
| 33 | 32 | a1i 11 |
. . . . . 6
⊢ (𝐾 ∈ ℕ0
→ 2 ≠ 0) |
| 34 | 31, 22, 33 | divcan3d 12048 |
. . . . 5
⊢ (𝐾 ∈ ℕ0
→ ((2 · (4 · 𝐾)) / 2) = (4 · 𝐾)) |
| 35 | 14, 27, 34 | 3eqtrd 2781 |
. . . 4
⊢ (𝐾 ∈ ℕ0
→ ((((8 · 𝐾) +
1) − 1) / 2) = (4 · 𝐾)) |
| 36 | | 1cnd 11256 |
. . . . . . . 8
⊢ (𝐾 ∈ ℕ0
→ 1 ∈ ℂ) |
| 37 | | 4ne0 12374 |
. . . . . . . . . 10
⊢ 4 ≠
0 |
| 38 | 15, 37 | pm3.2i 470 |
. . . . . . . . 9
⊢ (4 ∈
ℂ ∧ 4 ≠ 0) |
| 39 | 38 | a1i 11 |
. . . . . . . 8
⊢ (𝐾 ∈ ℕ0
→ (4 ∈ ℂ ∧ 4 ≠ 0)) |
| 40 | | divdir 11947 |
. . . . . . . 8
⊢ (((8
· 𝐾) ∈ ℂ
∧ 1 ∈ ℂ ∧ (4 ∈ ℂ ∧ 4 ≠ 0)) → (((8
· 𝐾) + 1) / 4) =
(((8 · 𝐾) / 4) + (1
/ 4))) |
| 41 | 11, 36, 39, 40 | syl3anc 1373 |
. . . . . . 7
⊢ (𝐾 ∈ ℕ0
→ (((8 · 𝐾) +
1) / 4) = (((8 · 𝐾)
/ 4) + (1 / 4))) |
| 42 | | 8cn 12363 |
. . . . . . . . . . 11
⊢ 8 ∈
ℂ |
| 43 | 42 | a1i 11 |
. . . . . . . . . 10
⊢ (𝐾 ∈ ℕ0
→ 8 ∈ ℂ) |
| 44 | | div23 11941 |
. . . . . . . . . 10
⊢ ((8
∈ ℂ ∧ 𝐾
∈ ℂ ∧ (4 ∈ ℂ ∧ 4 ≠ 0)) → ((8 ·
𝐾) / 4) = ((8 / 4) ·
𝐾)) |
| 45 | 43, 24, 39, 44 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝐾 ∈ ℕ0
→ ((8 · 𝐾) / 4)
= ((8 / 4) · 𝐾)) |
| 46 | 17 | eqcomi 2746 |
. . . . . . . . . . . . 13
⊢ 8 = (4
· 2) |
| 47 | 46 | oveq1i 7441 |
. . . . . . . . . . . 12
⊢ (8 / 4) =
((4 · 2) / 4) |
| 48 | 16, 15, 37 | divcan3i 12013 |
. . . . . . . . . . . 12
⊢ ((4
· 2) / 4) = 2 |
| 49 | 47, 48 | eqtri 2765 |
. . . . . . . . . . 11
⊢ (8 / 4) =
2 |
| 50 | 49 | a1i 11 |
. . . . . . . . . 10
⊢ (𝐾 ∈ ℕ0
→ (8 / 4) = 2) |
| 51 | 50 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝐾 ∈ ℕ0
→ ((8 / 4) · 𝐾)
= (2 · 𝐾)) |
| 52 | 45, 51 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝐾 ∈ ℕ0
→ ((8 · 𝐾) / 4)
= (2 · 𝐾)) |
| 53 | 52 | oveq1d 7446 |
. . . . . . 7
⊢ (𝐾 ∈ ℕ0
→ (((8 · 𝐾) /
4) + (1 / 4)) = ((2 · 𝐾) + (1 / 4))) |
| 54 | 41, 53 | eqtrd 2777 |
. . . . . 6
⊢ (𝐾 ∈ ℕ0
→ (((8 · 𝐾) +
1) / 4) = ((2 · 𝐾) +
(1 / 4))) |
| 55 | 54 | fveq2d 6910 |
. . . . 5
⊢ (𝐾 ∈ ℕ0
→ (⌊‘(((8 · 𝐾) + 1) / 4)) = (⌊‘((2 ·
𝐾) + (1 /
4)))) |
| 56 | | 1lt4 12442 |
. . . . . 6
⊢ 1 <
4 |
| 57 | | 2nn0 12543 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ0 |
| 58 | 57 | a1i 11 |
. . . . . . . . 9
⊢ (𝐾 ∈ ℕ0
→ 2 ∈ ℕ0) |
| 59 | 58, 9 | nn0mulcld 12592 |
. . . . . . . 8
⊢ (𝐾 ∈ ℕ0
→ (2 · 𝐾)
∈ ℕ0) |
| 60 | 59 | nn0zd 12639 |
. . . . . . 7
⊢ (𝐾 ∈ ℕ0
→ (2 · 𝐾)
∈ ℤ) |
| 61 | | 1nn0 12542 |
. . . . . . . 8
⊢ 1 ∈
ℕ0 |
| 62 | 61 | a1i 11 |
. . . . . . 7
⊢ (𝐾 ∈ ℕ0
→ 1 ∈ ℕ0) |
| 63 | | 4nn 12349 |
. . . . . . . 8
⊢ 4 ∈
ℕ |
| 64 | 63 | a1i 11 |
. . . . . . 7
⊢ (𝐾 ∈ ℕ0
→ 4 ∈ ℕ) |
| 65 | | adddivflid 13858 |
. . . . . . 7
⊢ (((2
· 𝐾) ∈ ℤ
∧ 1 ∈ ℕ0 ∧ 4 ∈ ℕ) → (1 < 4
↔ (⌊‘((2 · 𝐾) + (1 / 4))) = (2 · 𝐾))) |
| 66 | 60, 62, 64, 65 | syl3anc 1373 |
. . . . . 6
⊢ (𝐾 ∈ ℕ0
→ (1 < 4 ↔ (⌊‘((2 · 𝐾) + (1 / 4))) = (2 · 𝐾))) |
| 67 | 56, 66 | mpbii 233 |
. . . . 5
⊢ (𝐾 ∈ ℕ0
→ (⌊‘((2 · 𝐾) + (1 / 4))) = (2 · 𝐾)) |
| 68 | 55, 67 | eqtrd 2777 |
. . . 4
⊢ (𝐾 ∈ ℕ0
→ (⌊‘(((8 · 𝐾) + 1) / 4)) = (2 · 𝐾)) |
| 69 | 35, 68 | oveq12d 7449 |
. . 3
⊢ (𝐾 ∈ ℕ0
→ (((((8 · 𝐾) +
1) − 1) / 2) − (⌊‘(((8 · 𝐾) + 1) / 4))) = ((4 · 𝐾) − (2 · 𝐾))) |
| 70 | | 2t2e4 12430 |
. . . . . . . 8
⊢ (2
· 2) = 4 |
| 71 | 70 | eqcomi 2746 |
. . . . . . 7
⊢ 4 = (2
· 2) |
| 72 | 71 | a1i 11 |
. . . . . 6
⊢ (𝐾 ∈ ℕ0
→ 4 = (2 · 2)) |
| 73 | 72 | oveq1d 7446 |
. . . . 5
⊢ (𝐾 ∈ ℕ0
→ (4 · 𝐾) = ((2
· 2) · 𝐾)) |
| 74 | 22, 22, 24 | mulassd 11284 |
. . . . 5
⊢ (𝐾 ∈ ℕ0
→ ((2 · 2) · 𝐾) = (2 · (2 · 𝐾))) |
| 75 | 73, 74 | eqtrd 2777 |
. . . 4
⊢ (𝐾 ∈ ℕ0
→ (4 · 𝐾) = (2
· (2 · 𝐾))) |
| 76 | 75 | oveq1d 7446 |
. . 3
⊢ (𝐾 ∈ ℕ0
→ ((4 · 𝐾)
− (2 · 𝐾)) =
((2 · (2 · 𝐾)) − (2 · 𝐾))) |
| 77 | 59 | nn0cnd 12589 |
. . . 4
⊢ (𝐾 ∈ ℕ0
→ (2 · 𝐾)
∈ ℂ) |
| 78 | | 2txmxeqx 12406 |
. . . 4
⊢ ((2
· 𝐾) ∈ ℂ
→ ((2 · (2 · 𝐾)) − (2 · 𝐾)) = (2 · 𝐾)) |
| 79 | 77, 78 | syl 17 |
. . 3
⊢ (𝐾 ∈ ℕ0
→ ((2 · (2 · 𝐾)) − (2 · 𝐾)) = (2 · 𝐾)) |
| 80 | 69, 76, 79 | 3eqtrd 2781 |
. 2
⊢ (𝐾 ∈ ℕ0
→ (((((8 · 𝐾) +
1) − 1) / 2) − (⌊‘(((8 · 𝐾) + 1) / 4))) = (2 · 𝐾)) |
| 81 | 6, 80 | sylan9eqr 2799 |
1
⊢ ((𝐾 ∈ ℕ0
∧ 𝑃 = ((8 ·
𝐾) + 1)) → 𝑁 = (2 · 𝐾)) |