| Step | Hyp | Ref
| Expression |
| 1 | | lgseisen.2 |
. . . . 5
⊢ (𝜑 → 𝑄 ∈ (ℙ ∖
{2})) |
| 2 | 1 | eldifad 3963 |
. . . 4
⊢ (𝜑 → 𝑄 ∈ ℙ) |
| 3 | | prmz 16712 |
. . . 4
⊢ (𝑄 ∈ ℙ → 𝑄 ∈
ℤ) |
| 4 | 2, 3 | syl 17 |
. . 3
⊢ (𝜑 → 𝑄 ∈ ℤ) |
| 5 | | lgseisen.1 |
. . 3
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖
{2})) |
| 6 | | lgsval3 27359 |
. . 3
⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (𝑄
/L 𝑃) =
((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1)) |
| 7 | 4, 5, 6 | syl2anc 584 |
. 2
⊢ (𝜑 → (𝑄 /L 𝑃) = ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1)) |
| 8 | 1 | gausslemma2dlem0a 27400 |
. . . . . . . 8
⊢ (𝜑 → 𝑄 ∈ ℕ) |
| 9 | | oddprm 16848 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((𝑃 − 1) / 2)
∈ ℕ) |
| 10 | 5, 9 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 − 1) / 2) ∈
ℕ) |
| 11 | 10 | nnnn0d 12587 |
. . . . . . . 8
⊢ (𝜑 → ((𝑃 − 1) / 2) ∈
ℕ0) |
| 12 | 8, 11 | nnexpcld 14284 |
. . . . . . 7
⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈
ℕ) |
| 13 | 12 | nnred 12281 |
. . . . . 6
⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈
ℝ) |
| 14 | | neg1rr 12381 |
. . . . . . . 8
⊢ -1 ∈
ℝ |
| 15 | 14 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → -1 ∈
ℝ) |
| 16 | | neg1ne0 12382 |
. . . . . . . 8
⊢ -1 ≠
0 |
| 17 | 16 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → -1 ≠
0) |
| 18 | | fzfid 14014 |
. . . . . . . 8
⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈
Fin) |
| 19 | 8 | nnred 12281 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑄 ∈ ℝ) |
| 20 | 5 | gausslemma2dlem0a 27400 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑃 ∈ ℕ) |
| 21 | 19, 20 | nndivred 12320 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑄 / 𝑃) ∈ ℝ) |
| 22 | 21 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 / 𝑃) ∈ ℝ) |
| 23 | | 2re 12340 |
. . . . . . . . . . 11
⊢ 2 ∈
ℝ |
| 24 | | elfznn 13593 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ) |
| 25 | 24 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ) |
| 26 | 25 | nnred 12281 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℝ) |
| 27 | | remulcl 11240 |
. . . . . . . . . . 11
⊢ ((2
∈ ℝ ∧ 𝑥
∈ ℝ) → (2 · 𝑥) ∈ ℝ) |
| 28 | 23, 26, 27 | sylancr 587 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈
ℝ) |
| 29 | 22, 28 | remulcld 11291 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℝ) |
| 30 | 29 | flcld 13838 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈
ℤ) |
| 31 | 18, 30 | fsumzcl 15771 |
. . . . . . 7
⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ) |
| 32 | 15, 17, 31 | reexpclzd 14288 |
. . . . . 6
⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℝ) |
| 33 | | 1re 11261 |
. . . . . . 7
⊢ 1 ∈
ℝ |
| 34 | 33 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 1 ∈
ℝ) |
| 35 | 20 | nnrpd 13075 |
. . . . . 6
⊢ (𝜑 → 𝑃 ∈
ℝ+) |
| 36 | | lgseisen.3 |
. . . . . . 7
⊢ (𝜑 → 𝑃 ≠ 𝑄) |
| 37 | | eqid 2737 |
. . . . . . 7
⊢ ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃) |
| 38 | | eqid 2737 |
. . . . . . 7
⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
((((-1↑((𝑄 · (2
· 𝑥)) mod 𝑃)) · ((𝑄 · (2 · 𝑥)) mod 𝑃)) mod 𝑃) / 2)) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
((((-1↑((𝑄 · (2
· 𝑥)) mod 𝑃)) · ((𝑄 · (2 · 𝑥)) mod 𝑃)) mod 𝑃) / 2)) |
| 39 | | eqid 2737 |
. . . . . . 7
⊢ ((𝑄 · (2 · 𝑦)) mod 𝑃) = ((𝑄 · (2 · 𝑦)) mod 𝑃) |
| 40 | | eqid 2737 |
. . . . . . 7
⊢
(ℤ/nℤ‘𝑃) = (ℤ/nℤ‘𝑃) |
| 41 | | eqid 2737 |
. . . . . . 7
⊢
(mulGrp‘(ℤ/nℤ‘𝑃)) =
(mulGrp‘(ℤ/nℤ‘𝑃)) |
| 42 | | eqid 2737 |
. . . . . . 7
⊢
(ℤRHom‘(ℤ/nℤ‘𝑃)) =
(ℤRHom‘(ℤ/nℤ‘𝑃)) |
| 43 | 5, 1, 36, 37, 38, 39, 40, 41, 42 | lgseisenlem4 27422 |
. . . . . 6
⊢ (𝜑 → ((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃)) |
| 44 | | modadd1 13948 |
. . . . . 6
⊢ ((((𝑄↑((𝑃 − 1) / 2)) ∈ ℝ ∧
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ∈ ℝ) ∧
(1 ∈ ℝ ∧ 𝑃
∈ ℝ+) ∧ ((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃)) → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃)) |
| 45 | 13, 32, 34, 35, 43, 44 | syl221anc 1383 |
. . . . 5
⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃)) |
| 46 | | peano2re 11434 |
. . . . . . 7
⊢
((-1↑Σ𝑥
∈ (1...((𝑃 − 1)
/ 2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ∈ ℝ →
((-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) + 1) ∈
ℝ) |
| 47 | 32, 46 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) ∈ ℝ) |
| 48 | | df-neg 11495 |
. . . . . . . 8
⊢ -1 = (0
− 1) |
| 49 | | neg1cn 12380 |
. . . . . . . . . . . . 13
⊢ -1 ∈
ℂ |
| 50 | | absexpz 15344 |
. . . . . . . . . . . . 13
⊢ ((-1
∈ ℂ ∧ -1 ≠ 0 ∧ Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ) →
(abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
| 51 | 49, 16, 31, 50 | mp3an12i 1467 |
. . . . . . . . . . . 12
⊢ (𝜑 →
(abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
| 52 | | ax-1cn 11213 |
. . . . . . . . . . . . . . . 16
⊢ 1 ∈
ℂ |
| 53 | 52 | absnegi 15439 |
. . . . . . . . . . . . . . 15
⊢
(abs‘-1) = (abs‘1) |
| 54 | | abs1 15336 |
. . . . . . . . . . . . . . 15
⊢
(abs‘1) = 1 |
| 55 | 53, 54 | eqtri 2765 |
. . . . . . . . . . . . . 14
⊢
(abs‘-1) = 1 |
| 56 | 55 | oveq1i 7441 |
. . . . . . . . . . . . 13
⊢
((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) |
| 57 | | 1exp 14132 |
. . . . . . . . . . . . . 14
⊢
(Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥))) ∈ ℤ →
(1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) = 1) |
| 58 | 31, 57 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = 1) |
| 59 | 56, 58 | eqtrid 2789 |
. . . . . . . . . . . 12
⊢ (𝜑 →
((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = 1) |
| 60 | 51, 59 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ (𝜑 →
(abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = 1) |
| 61 | | 1le1 11891 |
. . . . . . . . . . 11
⊢ 1 ≤
1 |
| 62 | 60, 61 | eqbrtrdi 5182 |
. . . . . . . . . 10
⊢ (𝜑 →
(abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ≤ 1) |
| 63 | | absle 15354 |
. . . . . . . . . . 11
⊢
(((-1↑Σ𝑥
∈ (1...((𝑃 − 1)
/ 2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ∈ ℝ ∧
1 ∈ ℝ) → ((abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ≤ 1 ↔ (-1 ≤
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ∧
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ≤
1))) |
| 64 | 32, 33, 63 | sylancl 586 |
. . . . . . . . . 10
⊢ (𝜑 →
((abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ≤ 1 ↔ (-1 ≤
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ∧
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ≤
1))) |
| 65 | 62, 64 | mpbid 232 |
. . . . . . . . 9
⊢ (𝜑 → (-1 ≤
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ∧
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ≤
1)) |
| 66 | 65 | simpld 494 |
. . . . . . . 8
⊢ (𝜑 → -1 ≤
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥))))) |
| 67 | 48, 66 | eqbrtrrid 5179 |
. . . . . . 7
⊢ (𝜑 → (0 − 1) ≤
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥))))) |
| 68 | | 0red 11264 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℝ) |
| 69 | 68, 34, 32 | lesubaddd 11860 |
. . . . . . 7
⊢ (𝜑 → ((0 − 1) ≤
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ↔ 0 ≤
((-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) +
1))) |
| 70 | 67, 69 | mpbid 232 |
. . . . . 6
⊢ (𝜑 → 0 ≤
((-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) +
1)) |
| 71 | 20 | nnred 12281 |
. . . . . . . . 9
⊢ (𝜑 → 𝑃 ∈ ℝ) |
| 72 | | peano2rem 11576 |
. . . . . . . . 9
⊢ (𝑃 ∈ ℝ → (𝑃 − 1) ∈
ℝ) |
| 73 | 71, 72 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑃 − 1) ∈ ℝ) |
| 74 | 65 | simprd 495 |
. . . . . . . 8
⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ≤ 1) |
| 75 | | df-2 12329 |
. . . . . . . . . 10
⊢ 2 = (1 +
1) |
| 76 | 23 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → 2 ∈
ℝ) |
| 77 | 5 | eldifad 3963 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑃 ∈ ℙ) |
| 78 | | prmuz2 16733 |
. . . . . . . . . . . 12
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
(ℤ≥‘2)) |
| 79 | | eluzle 12891 |
. . . . . . . . . . . 12
⊢ (𝑃 ∈
(ℤ≥‘2) → 2 ≤ 𝑃) |
| 80 | 77, 78, 79 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → 2 ≤ 𝑃) |
| 81 | | eldifsni 4790 |
. . . . . . . . . . . 12
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ≠
2) |
| 82 | 5, 81 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑃 ≠ 2) |
| 83 | 76, 71, 80, 82 | leneltd 11415 |
. . . . . . . . . 10
⊢ (𝜑 → 2 < 𝑃) |
| 84 | 75, 83 | eqbrtrrid 5179 |
. . . . . . . . 9
⊢ (𝜑 → (1 + 1) < 𝑃) |
| 85 | 34, 34, 71 | ltaddsubd 11863 |
. . . . . . . . 9
⊢ (𝜑 → ((1 + 1) < 𝑃 ↔ 1 < (𝑃 − 1))) |
| 86 | 84, 85 | mpbid 232 |
. . . . . . . 8
⊢ (𝜑 → 1 < (𝑃 − 1)) |
| 87 | 32, 34, 73, 74, 86 | lelttrd 11419 |
. . . . . . 7
⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) < (𝑃 − 1)) |
| 88 | 32, 34, 71 | ltaddsubd 11863 |
. . . . . . 7
⊢ (𝜑 → (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) < 𝑃 ↔ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) < (𝑃 − 1))) |
| 89 | 87, 88 | mpbird 257 |
. . . . . 6
⊢ (𝜑 → ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) < 𝑃) |
| 90 | | modid 13936 |
. . . . . 6
⊢
(((((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) ∈ ℝ ∧ 𝑃 ∈ ℝ+)
∧ (0 ≤ ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) ∧ ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) < 𝑃)) → (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1)) |
| 91 | 47, 35, 70, 89, 90 | syl22anc 839 |
. . . . 5
⊢ (𝜑 → (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1)) |
| 92 | 45, 91 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1)) |
| 93 | 92 | oveq1d 7446 |
. . 3
⊢ (𝜑 → ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) − 1)) |
| 94 | 32 | recnd 11289 |
. . . 4
⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ) |
| 95 | | pncan 11514 |
. . . 4
⊢
(((-1↑Σ𝑥
∈ (1...((𝑃 − 1)
/ 2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ∈ ℂ ∧
1 ∈ ℂ) → (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) − 1) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
| 96 | 94, 52, 95 | sylancl 586 |
. . 3
⊢ (𝜑 → (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) − 1) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
| 97 | 93, 96 | eqtrd 2777 |
. 2
⊢ (𝜑 → ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
| 98 | 7, 97 | eqtrd 2777 |
1
⊢ (𝜑 → (𝑄 /L 𝑃) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |