Step | Hyp | Ref
| Expression |
1 | | zringbas 20026 |
. . . . 5
⊢ ℤ =
(Base‘ℤring) |
2 | | zring0 20030 |
. . . . 5
⊢ 0 =
(0g‘ℤring) |
3 | | zringabl 20024 |
. . . . . 6
⊢
ℤring ∈ Abel |
4 | | ablcmn 18399 |
. . . . . 6
⊢
(ℤring ∈ Abel → ℤring ∈
CMnd) |
5 | 3, 4 | mp1i 13 |
. . . . 5
⊢ (𝜑 → ℤring
∈ CMnd) |
6 | | lgseisen.1 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖
{2})) |
7 | 6 | eldifad 3727 |
. . . . . . . . 9
⊢ (𝜑 → 𝑃 ∈ ℙ) |
8 | | lgseisen.7 |
. . . . . . . . . 10
⊢ 𝑌 =
(ℤ/nℤ‘𝑃) |
9 | 8 | znfld 20111 |
. . . . . . . . 9
⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field) |
10 | 7, 9 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ∈ Field) |
11 | | isfld 18958 |
. . . . . . . . 9
⊢ (𝑌 ∈ Field ↔ (𝑌 ∈ DivRing ∧ 𝑌 ∈ CRing)) |
12 | 11 | simprbi 483 |
. . . . . . . 8
⊢ (𝑌 ∈ Field → 𝑌 ∈ CRing) |
13 | 10, 12 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑌 ∈ CRing) |
14 | | lgseisen.8 |
. . . . . . . 8
⊢ 𝐺 = (mulGrp‘𝑌) |
15 | 14 | crngmgp 18755 |
. . . . . . 7
⊢ (𝑌 ∈ CRing → 𝐺 ∈ CMnd) |
16 | 13, 15 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐺 ∈ CMnd) |
17 | | cmnmnd 18408 |
. . . . . 6
⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) |
18 | 16, 17 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐺 ∈ Mnd) |
19 | | fzfid 12966 |
. . . . 5
⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈
Fin) |
20 | | m1expcl 13077 |
. . . . . . . 8
⊢ (𝑘 ∈ ℤ →
(-1↑𝑘) ∈
ℤ) |
21 | 20 | adantl 473 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (-1↑𝑘) ∈
ℤ) |
22 | | eqidd 2761 |
. . . . . . 7
⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) = (𝑘 ∈ ℤ ↦ (-1↑𝑘))) |
23 | | crngring 18758 |
. . . . . . . . . . 11
⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) |
24 | 13, 23 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑌 ∈ Ring) |
25 | | lgseisen.9 |
. . . . . . . . . . 11
⊢ 𝐿 = (ℤRHom‘𝑌) |
26 | 25 | zrhrhm 20062 |
. . . . . . . . . 10
⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring
RingHom 𝑌)) |
27 | 24, 26 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐿 ∈ (ℤring RingHom
𝑌)) |
28 | | eqid 2760 |
. . . . . . . . . 10
⊢
(Base‘𝑌) =
(Base‘𝑌) |
29 | 1, 28 | rhmf 18928 |
. . . . . . . . 9
⊢ (𝐿 ∈ (ℤring
RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌)) |
30 | 27, 29 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌)) |
31 | 30 | feqmptd 6411 |
. . . . . . 7
⊢ (𝜑 → 𝐿 = (𝑥 ∈ ℤ ↦ (𝐿‘𝑥))) |
32 | | fveq2 6352 |
. . . . . . 7
⊢ (𝑥 = (-1↑𝑘) → (𝐿‘𝑥) = (𝐿‘(-1↑𝑘))) |
33 | 21, 22, 31, 32 | fmptco 6559 |
. . . . . 6
⊢ (𝜑 → (𝐿 ∘ (𝑘 ∈ ℤ ↦ (-1↑𝑘))) = (𝑘 ∈ ℤ ↦ (𝐿‘(-1↑𝑘)))) |
34 | | zringmpg 20042 |
. . . . . . . . 9
⊢
((mulGrp‘ℂfld) ↾s ℤ) =
(mulGrp‘ℤring) |
35 | 34, 14 | rhmmhm 18924 |
. . . . . . . 8
⊢ (𝐿 ∈ (ℤring
RingHom 𝑌) → 𝐿 ∈
(((mulGrp‘ℂfld) ↾s ℤ) MndHom
𝐺)) |
36 | 27, 35 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐿 ∈
(((mulGrp‘ℂfld) ↾s ℤ) MndHom
𝐺)) |
37 | | neg1cn 11316 |
. . . . . . . . . . 11
⊢ -1 ∈
ℂ |
38 | | neg1ne0 11318 |
. . . . . . . . . . 11
⊢ -1 ≠
0 |
39 | | eqid 2760 |
. . . . . . . . . . . 12
⊢
(mulGrp‘ℂfld) =
(mulGrp‘ℂfld) |
40 | | eqid 2760 |
. . . . . . . . . . . 12
⊢
((mulGrp‘ℂfld) ↾s (ℂ
∖ {0})) = ((mulGrp‘ℂfld) ↾s
(ℂ ∖ {0})) |
41 | 39, 40 | expghm 20046 |
. . . . . . . . . . 11
⊢ ((-1
∈ ℂ ∧ -1 ≠ 0) → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈
(ℤring GrpHom ((mulGrp‘ℂfld)
↾s (ℂ ∖ {0})))) |
42 | 37, 38, 41 | mp2an 710 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℤ ↦
(-1↑𝑘)) ∈
(ℤring GrpHom ((mulGrp‘ℂfld)
↾s (ℂ ∖ {0}))) |
43 | | ghmmhm 17871 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ ℤ ↦
(-1↑𝑘)) ∈
(ℤring GrpHom ((mulGrp‘ℂfld)
↾s (ℂ ∖ {0}))) → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈
(ℤring MndHom ((mulGrp‘ℂfld)
↾s (ℂ ∖ {0})))) |
44 | 42, 43 | ax-mp 5 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℤ ↦
(-1↑𝑘)) ∈
(ℤring MndHom ((mulGrp‘ℂfld)
↾s (ℂ ∖ {0}))) |
45 | | cnring 19970 |
. . . . . . . . . 10
⊢
ℂfld ∈ Ring |
46 | | cnfldbas 19952 |
. . . . . . . . . . . 12
⊢ ℂ =
(Base‘ℂfld) |
47 | | cnfld0 19972 |
. . . . . . . . . . . 12
⊢ 0 =
(0g‘ℂfld) |
48 | | cndrng 19977 |
. . . . . . . . . . . 12
⊢
ℂfld ∈ DivRing |
49 | 46, 47, 48 | drngui 18955 |
. . . . . . . . . . 11
⊢ (ℂ
∖ {0}) = (Unit‘ℂfld) |
50 | 49, 39 | unitsubm 18870 |
. . . . . . . . . 10
⊢
(ℂfld ∈ Ring → (ℂ ∖ {0}) ∈
(SubMnd‘(mulGrp‘ℂfld))) |
51 | 45, 50 | ax-mp 5 |
. . . . . . . . 9
⊢ (ℂ
∖ {0}) ∈
(SubMnd‘(mulGrp‘ℂfld)) |
52 | 40 | resmhm2 17561 |
. . . . . . . . 9
⊢ (((𝑘 ∈ ℤ ↦
(-1↑𝑘)) ∈
(ℤring MndHom ((mulGrp‘ℂfld)
↾s (ℂ ∖ {0}))) ∧ (ℂ ∖ {0}) ∈
(SubMnd‘(mulGrp‘ℂfld))) → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈
(ℤring MndHom
(mulGrp‘ℂfld))) |
53 | 44, 51, 52 | mp2an 710 |
. . . . . . . 8
⊢ (𝑘 ∈ ℤ ↦
(-1↑𝑘)) ∈
(ℤring MndHom
(mulGrp‘ℂfld)) |
54 | | zsubrg 20001 |
. . . . . . . . . 10
⊢ ℤ
∈ (SubRing‘ℂfld) |
55 | 39 | subrgsubm 18995 |
. . . . . . . . . 10
⊢ (ℤ
∈ (SubRing‘ℂfld) → ℤ ∈
(SubMnd‘(mulGrp‘ℂfld))) |
56 | 54, 55 | ax-mp 5 |
. . . . . . . . 9
⊢ ℤ
∈ (SubMnd‘(mulGrp‘ℂfld)) |
57 | | eqid 2760 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℤ ↦
(-1↑𝑘)) = (𝑘 ∈ ℤ ↦
(-1↑𝑘)) |
58 | 21, 57 | fmptd 6548 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (-1↑𝑘)):ℤ⟶ℤ) |
59 | | frn 6214 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ ℤ ↦
(-1↑𝑘)):ℤ⟶ℤ → ran (𝑘 ∈ ℤ ↦
(-1↑𝑘)) ⊆
ℤ) |
60 | 58, 59 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ran (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ⊆
ℤ) |
61 | | eqid 2760 |
. . . . . . . . . 10
⊢
((mulGrp‘ℂfld) ↾s ℤ) =
((mulGrp‘ℂfld) ↾s
ℤ) |
62 | 61 | resmhm2b 17562 |
. . . . . . . . 9
⊢ ((ℤ
∈ (SubMnd‘(mulGrp‘ℂfld)) ∧ ran (𝑘 ∈ ℤ ↦
(-1↑𝑘)) ⊆
ℤ) → ((𝑘 ∈
ℤ ↦ (-1↑𝑘)) ∈ (ℤring MndHom
(mulGrp‘ℂfld)) ↔ (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈
(ℤring MndHom ((mulGrp‘ℂfld)
↾s ℤ)))) |
63 | 56, 60, 62 | sylancr 698 |
. . . . . . . 8
⊢ (𝜑 → ((𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈
(ℤring MndHom (mulGrp‘ℂfld)) ↔
(𝑘 ∈ ℤ ↦
(-1↑𝑘)) ∈
(ℤring MndHom ((mulGrp‘ℂfld)
↾s ℤ)))) |
64 | 53, 63 | mpbii 223 |
. . . . . . 7
⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈
(ℤring MndHom ((mulGrp‘ℂfld)
↾s ℤ))) |
65 | | mhmco 17563 |
. . . . . . 7
⊢ ((𝐿 ∈
(((mulGrp‘ℂfld) ↾s ℤ) MndHom
𝐺) ∧ (𝑘 ∈ ℤ ↦
(-1↑𝑘)) ∈
(ℤring MndHom ((mulGrp‘ℂfld)
↾s ℤ))) → (𝐿 ∘ (𝑘 ∈ ℤ ↦ (-1↑𝑘))) ∈
(ℤring MndHom 𝐺)) |
66 | 36, 64, 65 | syl2anc 696 |
. . . . . 6
⊢ (𝜑 → (𝐿 ∘ (𝑘 ∈ ℤ ↦ (-1↑𝑘))) ∈
(ℤring MndHom 𝐺)) |
67 | 33, 66 | eqeltrrd 2840 |
. . . . 5
⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (𝐿‘(-1↑𝑘))) ∈ (ℤring MndHom
𝐺)) |
68 | | lgseisen.2 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑄 ∈ (ℙ ∖
{2})) |
69 | 68 | eldifad 3727 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑄 ∈ ℙ) |
70 | | prmnn 15590 |
. . . . . . . . . . 11
⊢ (𝑄 ∈ ℙ → 𝑄 ∈
ℕ) |
71 | 69, 70 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ∈ ℕ) |
72 | 71 | nnred 11227 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄 ∈ ℝ) |
73 | | prmnn 15590 |
. . . . . . . . . 10
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
74 | 7, 73 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑃 ∈ ℕ) |
75 | 72, 74 | nndivred 11261 |
. . . . . . . 8
⊢ (𝜑 → (𝑄 / 𝑃) ∈ ℝ) |
76 | 75 | adantr 472 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 / 𝑃) ∈ ℝ) |
77 | | 2nn 11377 |
. . . . . . . . 9
⊢ 2 ∈
ℕ |
78 | | elfznn 12563 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ) |
79 | 78 | adantl 473 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ) |
80 | | nnmulcl 11235 |
. . . . . . . . 9
⊢ ((2
∈ ℕ ∧ 𝑥
∈ ℕ) → (2 · 𝑥) ∈ ℕ) |
81 | 77, 79, 80 | sylancr 698 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈
ℕ) |
82 | 81 | nnred 11227 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈
ℝ) |
83 | 76, 82 | remulcld 10262 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℝ) |
84 | 83 | flcld 12793 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈
ℤ) |
85 | | eqid 2760 |
. . . . . 6
⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) |
86 | | fvexd 6364 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ V) |
87 | | c0ex 10226 |
. . . . . . 7
⊢ 0 ∈
V |
88 | 87 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 0 ∈
V) |
89 | 85, 19, 86, 88 | fsuppmptdm 8451 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) finSupp 0) |
90 | | oveq2 6821 |
. . . . . 6
⊢ (𝑘 = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) → (-1↑𝑘) = (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
91 | 90 | fveq2d 6356 |
. . . . 5
⊢ (𝑘 = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) → (𝐿‘(-1↑𝑘)) = (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) |
92 | | oveq2 6821 |
. . . . . 6
⊢ (𝑘 = (ℤring
Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) → (-1↑𝑘) =
(-1↑(ℤring Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) |
93 | 92 | fveq2d 6356 |
. . . . 5
⊢ (𝑘 = (ℤring
Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) → (𝐿‘(-1↑𝑘)) = (𝐿‘(-1↑(ℤring
Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))) |
94 | 1, 2, 5, 18, 19, 67, 84, 89, 91, 93 | gsummhm2 18539 |
. . . 4
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) = (𝐿‘(-1↑(ℤring
Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))) |
95 | 14, 28 | mgpbas 18695 |
. . . . . . 7
⊢
(Base‘𝑌) =
(Base‘𝐺) |
96 | | eqid 2760 |
. . . . . . . 8
⊢
(.r‘𝑌) = (.r‘𝑌) |
97 | 14, 96 | mgpplusg 18693 |
. . . . . . 7
⊢
(.r‘𝑌) = (+g‘𝐺) |
98 | 30 | adantr 472 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿:ℤ⟶(Base‘𝑌)) |
99 | | m1expcl 13077 |
. . . . . . . . 9
⊢
((⌊‘((𝑄
/ 𝑃) · (2 ·
𝑥))) ∈ ℤ →
(-1↑(⌊‘((𝑄
/ 𝑃) · (2 ·
𝑥)))) ∈
ℤ) |
100 | 84, 99 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(-1↑(⌊‘((𝑄
/ 𝑃) · (2 ·
𝑥)))) ∈
ℤ) |
101 | 98, 100 | ffvelrnd 6523 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ∈ (Base‘𝑌)) |
102 | | neg1z 11605 |
. . . . . . . . . 10
⊢ -1 ∈
ℤ |
103 | | lgseisen.4 |
. . . . . . . . . . 11
⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃) |
104 | 69 | adantr 472 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℙ) |
105 | | prmz 15591 |
. . . . . . . . . . . . . 14
⊢ (𝑄 ∈ ℙ → 𝑄 ∈
ℤ) |
106 | 104, 105 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℤ) |
107 | 81 | nnzd 11673 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈
ℤ) |
108 | 106, 107 | zmulcld 11680 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℤ) |
109 | 7 | adantr 472 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℙ) |
110 | 109, 73 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ) |
111 | 108, 110 | zmodcld 12885 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) ∈
ℕ0) |
112 | 103, 111 | syl5eqel 2843 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈
ℕ0) |
113 | | zexpcl 13069 |
. . . . . . . . . 10
⊢ ((-1
∈ ℤ ∧ 𝑅
∈ ℕ0) → (-1↑𝑅) ∈ ℤ) |
114 | 102, 112,
113 | sylancr 698 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈
ℤ) |
115 | 114, 106 | zmulcld 11680 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑄) ∈ ℤ) |
116 | 98, 115 | ffvelrnd 6523 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝑌)) |
117 | | eqid 2760 |
. . . . . . 7
⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) |
118 | | eqid 2760 |
. . . . . . 7
⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) |
119 | 95, 97, 16, 19, 101, 116, 117, 118 | gsummptfidmadd2 18526 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) ∘𝑓
(.r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))) = ((𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))(.r‘𝑌)(𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))))) |
120 | | eqidd 2761 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) |
121 | | eqidd 2761 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) |
122 | 19, 101, 116, 120, 121 | offval2 7079 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) ∘𝑓
(.r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))(.r‘𝑌)(𝐿‘((-1↑𝑅) · 𝑄))))) |
123 | 27 | adantr 472 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿 ∈ (ℤring RingHom
𝑌)) |
124 | | zringmulr 20029 |
. . . . . . . . . . . 12
⊢ ·
= (.r‘ℤring) |
125 | 1, 124, 96 | rhmmul 18929 |
. . . . . . . . . . 11
⊢ ((𝐿 ∈ (ℤring
RingHom 𝑌) ∧
(-1↑(⌊‘((𝑄
/ 𝑃) · (2 ·
𝑥)))) ∈ ℤ ∧
((-1↑𝑅) · 𝑄) ∈ ℤ) → (𝐿‘((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄))) = ((𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))(.r‘𝑌)(𝐿‘((-1↑𝑅) · 𝑄)))) |
126 | 123, 100,
115, 125 | syl3anc 1477 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄))) = ((𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))(.r‘𝑌)(𝐿‘((-1↑𝑅) · 𝑄)))) |
127 | 108 | zred 11674 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℝ) |
128 | 110 | nnrpd 12063 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈
ℝ+) |
129 | | modval 12864 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑄 · (2 · 𝑥)) ∈ ℝ ∧ 𝑃 ∈ ℝ+)
→ ((𝑄 · (2
· 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃))))) |
130 | 127, 128,
129 | syl2anc 696 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃))))) |
131 | 103, 130 | syl5eq 2806 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃))))) |
132 | 106 | zcnd 11675 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℂ) |
133 | 81 | nncnd 11228 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈
ℂ) |
134 | 110 | nncnd 11228 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℂ) |
135 | 110 | nnne0d 11257 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ≠ 0) |
136 | 132, 133,
134, 135 | div23d 11030 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) / 𝑃) = ((𝑄 / 𝑃) · (2 · 𝑥))) |
137 | 136 | fveq2d 6356 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(⌊‘((𝑄 ·
(2 · 𝑥)) / 𝑃)) = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) |
138 | 137 | oveq2d 6829 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃))) = (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
139 | 138 | oveq2d 6829 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) |
140 | 131, 139 | eqtrd 2794 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) |
141 | 140 | oveq2d 6829 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅) = ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) |
142 | | prmz 15591 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℤ) |
143 | 109, 142 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℤ) |
144 | 143, 84 | zmulcld 11680 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ) |
145 | 144 | zcnd 11675 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ) |
146 | 108 | zcnd 11675 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℂ) |
147 | 145, 146 | pncan3d 10587 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑄 · (2 · 𝑥))) |
148 | | 2cnd 11285 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈
ℂ) |
149 | 79 | nncnd 11228 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℂ) |
150 | 132, 148,
149 | mul12d 10437 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) = (2 · (𝑄 · 𝑥))) |
151 | 141, 147,
150 | 3eqtrd 2798 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅) = (2 · (𝑄 · 𝑥))) |
152 | 151 | oveq2d 6829 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = (-1↑(2 · (𝑄 · 𝑥)))) |
153 | 37 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -1 ∈
ℂ) |
154 | 38 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -1 ≠
0) |
155 | 112 | nn0zd 11672 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℤ) |
156 | | expaddz 13098 |
. . . . . . . . . . . . . . . 16
⊢ (((-1
∈ ℂ ∧ -1 ≠ 0) ∧ ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ ∧ 𝑅 ∈ ℤ)) → (-1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = ((-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) · (-1↑𝑅))) |
157 | 153, 154,
144, 155, 156 | syl22anc 1478 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = ((-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) · (-1↑𝑅))) |
158 | | expmulz 13100 |
. . . . . . . . . . . . . . . . . 18
⊢ (((-1
∈ ℂ ∧ -1 ≠ 0) ∧ (𝑃 ∈ ℤ ∧ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)) → (-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ((-1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
159 | 153, 154,
143, 84, 158 | syl22anc 1478 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ((-1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
160 | | 1cnd 10248 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 1 ∈
ℂ) |
161 | | eldifsni 4466 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ≠
2) |
162 | 6, 161 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝑃 ≠ 2) |
163 | 162 | necomd 2987 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 2 ≠ 𝑃) |
164 | 163 | neneqd 2937 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ¬ 2 = 𝑃) |
165 | 164 | adantr 472 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 2 = 𝑃) |
166 | | 2z 11601 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 2 ∈
ℤ |
167 | | uzid 11894 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (2 ∈
ℤ → 2 ∈ (ℤ≥‘2)) |
168 | 166, 167 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ∈
(ℤ≥‘2) |
169 | | dvdsprm 15617 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((2
∈ (ℤ≥‘2) ∧ 𝑃 ∈ ℙ) → (2 ∥ 𝑃 ↔ 2 = 𝑃)) |
170 | 168, 109,
169 | sylancr 698 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 ∥ 𝑃 ↔ 2 = 𝑃)) |
171 | 165, 170 | mtbird 314 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 2 ∥
𝑃) |
172 | | oexpneg 15271 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((1
∈ ℂ ∧ 𝑃
∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (-1↑𝑃) = -(1↑𝑃)) |
173 | 160, 110,
171, 172 | syl3anc 1477 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑃) = -(1↑𝑃)) |
174 | | 1exp 13083 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑃 ∈ ℤ →
(1↑𝑃) =
1) |
175 | 143, 174 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1↑𝑃) = 1) |
176 | 175 | negeqd 10467 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -(1↑𝑃) = -1) |
177 | 173, 176 | eqtrd 2794 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑃) = -1) |
178 | 177 | oveq1d 6828 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
179 | 159, 178 | eqtrd 2794 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
180 | 179 | oveq1d 6828 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) · (-1↑𝑅)) = ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅))) |
181 | 157, 180 | eqtrd 2794 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅))) |
182 | | nnmulcl 11235 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑄 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (𝑄 · 𝑥) ∈ ℕ) |
183 | 71, 78, 182 | syl2an 495 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℕ) |
184 | 183 | nnnn0d 11543 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈
ℕ0) |
185 | | 2nn0 11501 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ∈
ℕ0 |
186 | 185 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈
ℕ0) |
187 | 153, 184,
186 | expmuld 13205 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(2
· (𝑄 · 𝑥))) = ((-1↑2)↑(𝑄 · 𝑥))) |
188 | | neg1sqe1 13153 |
. . . . . . . . . . . . . . . . 17
⊢
(-1↑2) = 1 |
189 | 188 | oveq1i 6823 |
. . . . . . . . . . . . . . . 16
⊢
((-1↑2)↑(𝑄
· 𝑥)) =
(1↑(𝑄 · 𝑥)) |
190 | 183 | nnzd 11673 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℤ) |
191 | | 1exp 13083 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑄 · 𝑥) ∈ ℤ → (1↑(𝑄 · 𝑥)) = 1) |
192 | 190, 191 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1↑(𝑄 · 𝑥)) = 1) |
193 | 189, 192 | syl5eq 2806 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
((-1↑2)↑(𝑄
· 𝑥)) =
1) |
194 | 187, 193 | eqtrd 2794 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(2
· (𝑄 · 𝑥))) = 1) |
195 | 152, 181,
194 | 3eqtr3d 2802 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)) = 1) |
196 | 195 | oveq1d 6828 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)) · 𝑄) = (1 · 𝑄)) |
197 | 100 | zcnd 11675 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(-1↑(⌊‘((𝑄
/ 𝑃) · (2 ·
𝑥)))) ∈
ℂ) |
198 | 114 | zcnd 11675 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈
ℂ) |
199 | 197, 198,
132 | mulassd 10255 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)) · 𝑄) = ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄))) |
200 | 132 | mulid2d 10250 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1 · 𝑄) = 𝑄) |
201 | 196, 199,
200 | 3eqtr3d 2802 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄)) = 𝑄) |
202 | 201 | fveq2d 6356 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄))) = (𝐿‘𝑄)) |
203 | 126, 202 | eqtr3d 2796 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))(.r‘𝑌)(𝐿‘((-1↑𝑅) · 𝑄))) = (𝐿‘𝑄)) |
204 | 203 | mpteq2dva 4896 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))(.r‘𝑌)(𝐿‘((-1↑𝑅) · 𝑄)))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) |
205 | 122, 204 | eqtrd 2794 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) ∘𝑓
(.r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) |
206 | 205 | oveq2d 6829 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) ∘𝑓
(.r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))) = (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄)))) |
207 | | lgseisen.3 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ≠ 𝑄) |
208 | | lgseisen.5 |
. . . . . . . 8
⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)) |
209 | | lgseisen.6 |
. . . . . . . 8
⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃) |
210 | 6, 68, 207, 103, 208, 209, 8, 14, 25 | lgseisenlem3 25301 |
. . . . . . 7
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) = (1r‘𝑌)) |
211 | 210 | oveq2d 6829 |
. . . . . 6
⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))(.r‘𝑌)(𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))) = ((𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))(.r‘𝑌)(1r‘𝑌))) |
212 | 119, 206,
211 | 3eqtr3rd 2803 |
. . . . 5
⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))(.r‘𝑌)(1r‘𝑌)) = (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄)))) |
213 | | eqid 2760 |
. . . . . . 7
⊢
(0g‘𝐺) = (0g‘𝐺) |
214 | 101, 117 | fmptd 6548 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌)) |
215 | | fvexd 6364 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ∈ V) |
216 | | fvexd 6364 |
. . . . . . . 8
⊢ (𝜑 → (0g‘𝐺) ∈ V) |
217 | 117, 19, 215, 216 | fsuppmptdm 8451 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) finSupp (0g‘𝐺)) |
218 | 95, 213, 16, 19, 214, 217 | gsumcl 18516 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) ∈ (Base‘𝑌)) |
219 | | eqid 2760 |
. . . . . . 7
⊢
(1r‘𝑌) = (1r‘𝑌) |
220 | 28, 96, 219 | ringridm 18772 |
. . . . . 6
⊢ ((𝑌 ∈ Ring ∧ (𝐺 Σg
(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) ∈ (Base‘𝑌)) → ((𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))(.r‘𝑌)(1r‘𝑌)) = (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))) |
221 | 24, 218, 220 | syl2anc 696 |
. . . . 5
⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))(.r‘𝑌)(1r‘𝑌)) = (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))) |
222 | 69, 105 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑄 ∈ ℤ) |
223 | 30, 222 | ffvelrnd 6523 |
. . . . . . 7
⊢ (𝜑 → (𝐿‘𝑄) ∈ (Base‘𝑌)) |
224 | | eqid 2760 |
. . . . . . . 8
⊢
(.g‘𝐺) = (.g‘𝐺) |
225 | 95, 224 | gsumconst 18534 |
. . . . . . 7
⊢ ((𝐺 ∈ Mnd ∧ (1...((𝑃 − 1) / 2)) ∈ Fin
∧ (𝐿‘𝑄) ∈ (Base‘𝑌)) → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = ((♯‘(1...((𝑃 − 1) /
2)))(.g‘𝐺)(𝐿‘𝑄))) |
226 | 18, 19, 223, 225 | syl3anc 1477 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = ((♯‘(1...((𝑃 − 1) /
2)))(.g‘𝐺)(𝐿‘𝑄))) |
227 | | oddprm 15717 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((𝑃 − 1) / 2)
∈ ℕ) |
228 | 6, 227 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 − 1) / 2) ∈
ℕ) |
229 | 228 | nnnn0d 11543 |
. . . . . . . 8
⊢ (𝜑 → ((𝑃 − 1) / 2) ∈
ℕ0) |
230 | | hashfz1 13328 |
. . . . . . . 8
⊢ (((𝑃 − 1) / 2) ∈
ℕ0 → (♯‘(1...((𝑃 − 1) / 2))) = ((𝑃 − 1) / 2)) |
231 | 229, 230 | syl 17 |
. . . . . . 7
⊢ (𝜑 →
(♯‘(1...((𝑃
− 1) / 2))) = ((𝑃
− 1) / 2)) |
232 | 231 | oveq1d 6828 |
. . . . . 6
⊢ (𝜑 →
((♯‘(1...((𝑃
− 1) / 2)))(.g‘𝐺)(𝐿‘𝑄)) = (((𝑃 − 1) / 2)(.g‘𝐺)(𝐿‘𝑄))) |
233 | 34, 1 | mgpbas 18695 |
. . . . . . . . 9
⊢ ℤ =
(Base‘((mulGrp‘ℂfld) ↾s
ℤ)) |
234 | | eqid 2760 |
. . . . . . . . 9
⊢
(.g‘((mulGrp‘ℂfld)
↾s ℤ)) =
(.g‘((mulGrp‘ℂfld) ↾s
ℤ)) |
235 | 233, 234,
224 | mhmmulg 17784 |
. . . . . . . 8
⊢ ((𝐿 ∈
(((mulGrp‘ℂfld) ↾s ℤ) MndHom
𝐺) ∧ ((𝑃 − 1) / 2) ∈
ℕ0 ∧ 𝑄
∈ ℤ) → (𝐿‘(((𝑃 − 1) /
2)(.g‘((mulGrp‘ℂfld)
↾s ℤ))𝑄)) = (((𝑃 − 1) / 2)(.g‘𝐺)(𝐿‘𝑄))) |
236 | 36, 229, 222, 235 | syl3anc 1477 |
. . . . . . 7
⊢ (𝜑 → (𝐿‘(((𝑃 − 1) /
2)(.g‘((mulGrp‘ℂfld)
↾s ℤ))𝑄)) = (((𝑃 − 1) / 2)(.g‘𝐺)(𝐿‘𝑄))) |
237 | 56 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℤ ∈
(SubMnd‘(mulGrp‘ℂfld))) |
238 | | eqid 2760 |
. . . . . . . . . . 11
⊢
(.g‘(mulGrp‘ℂfld)) =
(.g‘(mulGrp‘ℂfld)) |
239 | 238, 61, 234 | submmulg 17787 |
. . . . . . . . . 10
⊢ ((ℤ
∈ (SubMnd‘(mulGrp‘ℂfld)) ∧ ((𝑃 − 1) / 2) ∈
ℕ0 ∧ 𝑄
∈ ℤ) → (((𝑃
− 1) / 2)(.g‘(mulGrp‘ℂfld))𝑄) = (((𝑃 − 1) /
2)(.g‘((mulGrp‘ℂfld)
↾s ℤ))𝑄)) |
240 | 237, 229,
222, 239 | syl3anc 1477 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑃 − 1) /
2)(.g‘(mulGrp‘ℂfld))𝑄) = (((𝑃 − 1) /
2)(.g‘((mulGrp‘ℂfld)
↾s ℤ))𝑄)) |
241 | 222 | zcnd 11675 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ∈ ℂ) |
242 | | cnfldexp 19981 |
. . . . . . . . . 10
⊢ ((𝑄 ∈ ℂ ∧ ((𝑃 − 1) / 2) ∈
ℕ0) → (((𝑃 − 1) /
2)(.g‘(mulGrp‘ℂfld))𝑄) = (𝑄↑((𝑃 − 1) / 2))) |
243 | 241, 229,
242 | syl2anc 696 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑃 − 1) /
2)(.g‘(mulGrp‘ℂfld))𝑄) = (𝑄↑((𝑃 − 1) / 2))) |
244 | 240, 243 | eqtr3d 2796 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 − 1) /
2)(.g‘((mulGrp‘ℂfld)
↾s ℤ))𝑄) = (𝑄↑((𝑃 − 1) / 2))) |
245 | 244 | fveq2d 6356 |
. . . . . . 7
⊢ (𝜑 → (𝐿‘(((𝑃 − 1) /
2)(.g‘((mulGrp‘ℂfld)
↾s ℤ))𝑄)) = (𝐿‘(𝑄↑((𝑃 − 1) / 2)))) |
246 | 236, 245 | eqtr3d 2796 |
. . . . . 6
⊢ (𝜑 → (((𝑃 − 1) / 2)(.g‘𝐺)(𝐿‘𝑄)) = (𝐿‘(𝑄↑((𝑃 − 1) / 2)))) |
247 | 226, 232,
246 | 3eqtrd 2798 |
. . . . 5
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = (𝐿‘(𝑄↑((𝑃 − 1) / 2)))) |
248 | 212, 221,
247 | 3eqtr3d 2802 |
. . . 4
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) = (𝐿‘(𝑄↑((𝑃 − 1) / 2)))) |
249 | | subrgsubg 18988 |
. . . . . . . . . 10
⊢ (ℤ
∈ (SubRing‘ℂfld) → ℤ ∈
(SubGrp‘ℂfld)) |
250 | 54, 249 | ax-mp 5 |
. . . . . . . . 9
⊢ ℤ
∈ (SubGrp‘ℂfld) |
251 | | subgsubm 17817 |
. . . . . . . . 9
⊢ (ℤ
∈ (SubGrp‘ℂfld) → ℤ ∈
(SubMnd‘ℂfld)) |
252 | 250, 251 | mp1i 13 |
. . . . . . . 8
⊢ (𝜑 → ℤ ∈
(SubMnd‘ℂfld)) |
253 | 84, 85 | fmptd 6548 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))):(1...((𝑃 − 1) /
2))⟶ℤ) |
254 | | df-zring 20021 |
. . . . . . . 8
⊢
ℤring = (ℂfld ↾s
ℤ) |
255 | 19, 252, 253, 254 | gsumsubm 17574 |
. . . . . . 7
⊢ (𝜑 → (ℂfld
Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = (ℤring
Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) |
256 | 84 | zcnd 11675 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈
ℂ) |
257 | 19, 256 | gsumfsum 20015 |
. . . . . . 7
⊢ (𝜑 → (ℂfld
Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) |
258 | 255, 257 | eqtr3d 2796 |
. . . . . 6
⊢ (𝜑 → (ℤring
Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) |
259 | 258 | oveq2d 6829 |
. . . . 5
⊢ (𝜑 →
(-1↑(ℤring Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
260 | 259 | fveq2d 6356 |
. . . 4
⊢ (𝜑 → (𝐿‘(-1↑(ℤring
Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) = (𝐿‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) |
261 | 94, 248, 260 | 3eqtr3d 2802 |
. . 3
⊢ (𝜑 → (𝐿‘(𝑄↑((𝑃 − 1) / 2))) = (𝐿‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) |
262 | 74 | nnnn0d 11543 |
. . . 4
⊢ (𝜑 → 𝑃 ∈
ℕ0) |
263 | | zexpcl 13069 |
. . . . 5
⊢ ((𝑄 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈
ℕ0) → (𝑄↑((𝑃 − 1) / 2)) ∈
ℤ) |
264 | 222, 229,
263 | syl2anc 696 |
. . . 4
⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈
ℤ) |
265 | 19, 84 | fsumzcl 14665 |
. . . . 5
⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ) |
266 | | m1expcl 13077 |
. . . . 5
⊢
(Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥))) ∈ ℤ →
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ∈
ℤ) |
267 | 265, 266 | syl 17 |
. . . 4
⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ) |
268 | 8, 25 | zndvds 20100 |
. . . 4
⊢ ((𝑃 ∈ ℕ0
∧ (𝑄↑((𝑃 − 1) / 2)) ∈ ℤ
∧ (-1↑Σ𝑥
∈ (1...((𝑃 − 1)
/ 2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ∈ ℤ)
→ ((𝐿‘(𝑄↑((𝑃 − 1) / 2))) = (𝐿‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ↔ 𝑃 ∥ ((𝑄↑((𝑃 − 1) / 2)) −
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥))))))) |
269 | 262, 264,
267, 268 | syl3anc 1477 |
. . 3
⊢ (𝜑 → ((𝐿‘(𝑄↑((𝑃 − 1) / 2))) = (𝐿‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ↔ 𝑃 ∥ ((𝑄↑((𝑃 − 1) / 2)) −
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥))))))) |
270 | 261, 269 | mpbid 222 |
. 2
⊢ (𝜑 → 𝑃 ∥ ((𝑄↑((𝑃 − 1) / 2)) −
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))))) |
271 | | moddvds 15193 |
. . 3
⊢ ((𝑃 ∈ ℕ ∧ (𝑄↑((𝑃 − 1) / 2)) ∈ ℤ ∧
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ∈ ℤ)
→ (((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃) ↔ 𝑃 ∥ ((𝑄↑((𝑃 − 1) / 2)) −
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥))))))) |
272 | 74, 264, 267, 271 | syl3anc 1477 |
. 2
⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃) ↔ 𝑃 ∥ ((𝑄↑((𝑃 − 1) / 2)) −
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥))))))) |
273 | 270, 272 | mpbird 247 |
1
⊢ (𝜑 → ((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃)) |