Step | Hyp | Ref
| Expression |
1 | | zringbas 20308 |
. . . . 5
⊢ ℤ =
(Base‘ℤring) |
2 | | zring0 20312 |
. . . . 5
⊢ 0 =
(0g‘ℤring) |
3 | | zringabl 20306 |
. . . . . 6
⊢
ℤring ∈ Abel |
4 | | ablcmn 19044 |
. . . . . 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 3865 |
. . . . . . . . 9
⊢ (𝜑 → 𝑃 ∈ ℙ) |
8 | | lgseisen.7 |
. . . . . . . . . 10
⊢ 𝑌 =
(ℤ/nℤ‘𝑃) |
9 | 8 | znfld 20392 |
. . . . . . . . 9
⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field) |
10 | 7, 9 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ∈ Field) |
11 | | isfld 19643 |
. . . . . . . . 9
⊢ (𝑌 ∈ Field ↔ (𝑌 ∈ DivRing ∧ 𝑌 ∈ CRing)) |
12 | 11 | simprbi 500 |
. . . . . . . 8
⊢ (𝑌 ∈ Field → 𝑌 ∈ CRing) |
13 | 10, 12 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑌 ∈ CRing) |
14 | | lgseisen.8 |
. . . . . . . 8
⊢ 𝐺 = (mulGrp‘𝑌) |
15 | 14 | crngmgp 19437 |
. . . . . . 7
⊢ (𝑌 ∈ CRing → 𝐺 ∈ CMnd) |
16 | 13, 15 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐺 ∈ CMnd) |
17 | | cmnmnd 19053 |
. . . . . 6
⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) |
18 | 16, 17 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐺 ∈ Mnd) |
19 | | fzfid 13445 |
. . . . 5
⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈
Fin) |
20 | | crngring 19441 |
. . . . . . . . . 10
⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) |
21 | 13, 20 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑌 ∈ Ring) |
22 | | lgseisen.9 |
. . . . . . . . . 10
⊢ 𝐿 = (ℤRHom‘𝑌) |
23 | 22 | zrhrhm 20345 |
. . . . . . . . 9
⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring
RingHom 𝑌)) |
24 | 21, 23 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐿 ∈ (ℤring RingHom
𝑌)) |
25 | | eqid 2739 |
. . . . . . . . 9
⊢
(Base‘𝑌) =
(Base‘𝑌) |
26 | 1, 25 | rhmf 19613 |
. . . . . . . 8
⊢ (𝐿 ∈ (ℤring
RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌)) |
27 | 24, 26 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌)) |
28 | | m1expcl 13557 |
. . . . . . . 8
⊢ (𝑘 ∈ ℤ →
(-1↑𝑘) ∈
ℤ) |
29 | 28 | adantl 485 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (-1↑𝑘) ∈
ℤ) |
30 | 27, 29 | cofmpt 6917 |
. . . . . 6
⊢ (𝜑 → (𝐿 ∘ (𝑘 ∈ ℤ ↦ (-1↑𝑘))) = (𝑘 ∈ ℤ ↦ (𝐿‘(-1↑𝑘)))) |
31 | | zringmpg 20325 |
. . . . . . . . 9
⊢
((mulGrp‘ℂfld) ↾s ℤ) =
(mulGrp‘ℤring) |
32 | 31, 14 | rhmmhm 19609 |
. . . . . . . 8
⊢ (𝐿 ∈ (ℤring
RingHom 𝑌) → 𝐿 ∈
(((mulGrp‘ℂfld) ↾s ℤ) MndHom
𝐺)) |
33 | 24, 32 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐿 ∈
(((mulGrp‘ℂfld) ↾s ℤ) MndHom
𝐺)) |
34 | | neg1cn 11843 |
. . . . . . . . . . 11
⊢ -1 ∈
ℂ |
35 | | neg1ne0 11845 |
. . . . . . . . . . 11
⊢ -1 ≠
0 |
36 | | eqid 2739 |
. . . . . . . . . . . 12
⊢
(mulGrp‘ℂfld) =
(mulGrp‘ℂfld) |
37 | | eqid 2739 |
. . . . . . . . . . . 12
⊢
((mulGrp‘ℂfld) ↾s (ℂ
∖ {0})) = ((mulGrp‘ℂfld) ↾s
(ℂ ∖ {0})) |
38 | 36, 37 | expghm 20329 |
. . . . . . . . . . 11
⊢ ((-1
∈ ℂ ∧ -1 ≠ 0) → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈
(ℤring GrpHom ((mulGrp‘ℂfld)
↾s (ℂ ∖ {0})))) |
39 | 34, 35, 38 | mp2an 692 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℤ ↦
(-1↑𝑘)) ∈
(ℤring GrpHom ((mulGrp‘ℂfld)
↾s (ℂ ∖ {0}))) |
40 | | ghmmhm 18499 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ ℤ ↦
(-1↑𝑘)) ∈
(ℤring GrpHom ((mulGrp‘ℂfld)
↾s (ℂ ∖ {0}))) → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈
(ℤring MndHom ((mulGrp‘ℂfld)
↾s (ℂ ∖ {0})))) |
41 | 39, 40 | ax-mp 5 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℤ ↦
(-1↑𝑘)) ∈
(ℤring MndHom ((mulGrp‘ℂfld)
↾s (ℂ ∖ {0}))) |
42 | | cnring 20252 |
. . . . . . . . . 10
⊢
ℂfld ∈ Ring |
43 | | cnfldbas 20234 |
. . . . . . . . . . . 12
⊢ ℂ =
(Base‘ℂfld) |
44 | | cnfld0 20254 |
. . . . . . . . . . . 12
⊢ 0 =
(0g‘ℂfld) |
45 | | cndrng 20259 |
. . . . . . . . . . . 12
⊢
ℂfld ∈ DivRing |
46 | 43, 44, 45 | drngui 19640 |
. . . . . . . . . . 11
⊢ (ℂ
∖ {0}) = (Unit‘ℂfld) |
47 | 46, 36 | unitsubm 19555 |
. . . . . . . . . 10
⊢
(ℂfld ∈ Ring → (ℂ ∖ {0}) ∈
(SubMnd‘(mulGrp‘ℂfld))) |
48 | 42, 47 | ax-mp 5 |
. . . . . . . . 9
⊢ (ℂ
∖ {0}) ∈
(SubMnd‘(mulGrp‘ℂfld)) |
49 | 37 | resmhm2 18115 |
. . . . . . . . 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))) |
50 | 41, 48, 49 | mp2an 692 |
. . . . . . . 8
⊢ (𝑘 ∈ ℤ ↦
(-1↑𝑘)) ∈
(ℤring MndHom
(mulGrp‘ℂfld)) |
51 | | zsubrg 20283 |
. . . . . . . . . 10
⊢ ℤ
∈ (SubRing‘ℂfld) |
52 | 36 | subrgsubm 19680 |
. . . . . . . . . 10
⊢ (ℤ
∈ (SubRing‘ℂfld) → ℤ ∈
(SubMnd‘(mulGrp‘ℂfld))) |
53 | 51, 52 | ax-mp 5 |
. . . . . . . . 9
⊢ ℤ
∈ (SubMnd‘(mulGrp‘ℂfld)) |
54 | 29 | fmpttd 6902 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (-1↑𝑘)):ℤ⟶ℤ) |
55 | 54 | frnd 6523 |
. . . . . . . . 9
⊢ (𝜑 → ran (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ⊆
ℤ) |
56 | | eqid 2739 |
. . . . . . . . . 10
⊢
((mulGrp‘ℂfld) ↾s ℤ) =
((mulGrp‘ℂfld) ↾s
ℤ) |
57 | 56 | resmhm2b 18116 |
. . . . . . . . 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)))) |
58 | 53, 55, 57 | sylancr 590 |
. . . . . . . 8
⊢ (𝜑 → ((𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈
(ℤring MndHom (mulGrp‘ℂfld)) ↔
(𝑘 ∈ ℤ ↦
(-1↑𝑘)) ∈
(ℤring MndHom ((mulGrp‘ℂfld)
↾s ℤ)))) |
59 | 50, 58 | mpbii 236 |
. . . . . . 7
⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈
(ℤring MndHom ((mulGrp‘ℂfld)
↾s ℤ))) |
60 | | mhmco 18117 |
. . . . . . 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)) |
61 | 33, 59, 60 | syl2anc 587 |
. . . . . 6
⊢ (𝜑 → (𝐿 ∘ (𝑘 ∈ ℤ ↦ (-1↑𝑘))) ∈
(ℤring MndHom 𝐺)) |
62 | 30, 61 | eqeltrrd 2835 |
. . . . 5
⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (𝐿‘(-1↑𝑘))) ∈ (ℤring MndHom
𝐺)) |
63 | | lgseisen.2 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑄 ∈ (ℙ ∖
{2})) |
64 | 63 | gausslemma2dlem0a 26105 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ∈ ℕ) |
65 | 64 | nnred 11744 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄 ∈ ℝ) |
66 | 6 | gausslemma2dlem0a 26105 |
. . . . . . . . 9
⊢ (𝜑 → 𝑃 ∈ ℕ) |
67 | 65, 66 | nndivred 11783 |
. . . . . . . 8
⊢ (𝜑 → (𝑄 / 𝑃) ∈ ℝ) |
68 | 67 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 / 𝑃) ∈ ℝ) |
69 | | 2nn 11802 |
. . . . . . . . 9
⊢ 2 ∈
ℕ |
70 | | elfznn 13040 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ) |
71 | 70 | adantl 485 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ) |
72 | | nnmulcl 11753 |
. . . . . . . . 9
⊢ ((2
∈ ℕ ∧ 𝑥
∈ ℕ) → (2 · 𝑥) ∈ ℕ) |
73 | 69, 71, 72 | sylancr 590 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈
ℕ) |
74 | 73 | nnred 11744 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈
ℝ) |
75 | 68, 74 | remulcld 10762 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℝ) |
76 | 75 | flcld 13272 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈
ℤ) |
77 | | eqid 2739 |
. . . . . 6
⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) |
78 | | fvexd 6702 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ V) |
79 | | c0ex 10726 |
. . . . . . 7
⊢ 0 ∈
V |
80 | 79 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 0 ∈
V) |
81 | 77, 19, 78, 80 | fsuppmptdm 8930 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) finSupp 0) |
82 | | oveq2 7191 |
. . . . . 6
⊢ (𝑘 = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) → (-1↑𝑘) = (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
83 | 82 | fveq2d 6691 |
. . . . 5
⊢ (𝑘 = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) → (𝐿‘(-1↑𝑘)) = (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) |
84 | | oveq2 7191 |
. . . . . 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))))))) |
85 | 84 | fveq2d 6691 |
. . . . 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)))))))) |
86 | 1, 2, 5, 18, 19, 62, 76, 81, 83, 85 | gsummhm2 19191 |
. . . 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)))))))) |
87 | 14, 25 | mgpbas 19377 |
. . . . . . 7
⊢
(Base‘𝑌) =
(Base‘𝐺) |
88 | | eqid 2739 |
. . . . . . . 8
⊢
(.r‘𝑌) = (.r‘𝑌) |
89 | 14, 88 | mgpplusg 19375 |
. . . . . . 7
⊢
(.r‘𝑌) = (+g‘𝐺) |
90 | 27 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿:ℤ⟶(Base‘𝑌)) |
91 | | m1expcl 13557 |
. . . . . . . . 9
⊢
((⌊‘((𝑄
/ 𝑃) · (2 ·
𝑥))) ∈ ℤ →
(-1↑(⌊‘((𝑄
/ 𝑃) · (2 ·
𝑥)))) ∈
ℤ) |
92 | 76, 91 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(-1↑(⌊‘((𝑄
/ 𝑃) · (2 ·
𝑥)))) ∈
ℤ) |
93 | 90, 92 | ffvelrnd 6875 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ∈ (Base‘𝑌)) |
94 | | neg1z 12112 |
. . . . . . . . . 10
⊢ -1 ∈
ℤ |
95 | | lgseisen.4 |
. . . . . . . . . . 11
⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃) |
96 | 63 | eldifad 3865 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑄 ∈ ℙ) |
97 | 96 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℙ) |
98 | | prmz 16129 |
. . . . . . . . . . . . . 14
⊢ (𝑄 ∈ ℙ → 𝑄 ∈
ℤ) |
99 | 97, 98 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℤ) |
100 | 73 | nnzd 12180 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈
ℤ) |
101 | 99, 100 | zmulcld 12187 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℤ) |
102 | 7 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℙ) |
103 | | prmnn 16128 |
. . . . . . . . . . . . 13
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
104 | 102, 103 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ) |
105 | 101, 104 | zmodcld 13364 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) ∈
ℕ0) |
106 | 95, 105 | eqeltrid 2838 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈
ℕ0) |
107 | | zexpcl 13549 |
. . . . . . . . . 10
⊢ ((-1
∈ ℤ ∧ 𝑅
∈ ℕ0) → (-1↑𝑅) ∈ ℤ) |
108 | 94, 106, 107 | sylancr 590 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈
ℤ) |
109 | 108, 99 | zmulcld 12187 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑄) ∈ ℤ) |
110 | 90, 109 | ffvelrnd 6875 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝑌)) |
111 | | eqid 2739 |
. . . . . . 7
⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) |
112 | | eqid 2739 |
. . . . . . 7
⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) |
113 | 87, 89, 16, 19, 93, 110, 111, 112 | gsummptfidmadd2 19178 |
. . . . . 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)))))) |
114 | | eqidd 2740 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) |
115 | | eqidd 2740 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) |
116 | 19, 93, 110, 114, 115 | offval2 7457 |
. . . . . . . 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))))) |
117 | 24 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿 ∈ (ℤring RingHom
𝑌)) |
118 | | zringmulr 20311 |
. . . . . . . . . . . 12
⊢ ·
= (.r‘ℤring) |
119 | 1, 118, 88 | rhmmul 19614 |
. . . . . . . . . . 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)))) |
120 | 117, 92, 109, 119 | syl3anc 1372 |
. . . . . . . . . 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)))) |
121 | 101 | zred 12181 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℝ) |
122 | 104 | nnrpd 12525 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈
ℝ+) |
123 | | modval 13343 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑄 · (2 · 𝑥)) ∈ ℝ ∧ 𝑃 ∈ ℝ+)
→ ((𝑄 · (2
· 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃))))) |
124 | 121, 122,
123 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃))))) |
125 | 95, 124 | syl5eq 2786 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃))))) |
126 | 99 | zcnd 12182 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℂ) |
127 | 73 | nncnd 11745 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈
ℂ) |
128 | 104 | nncnd 11745 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℂ) |
129 | 104 | nnne0d 11779 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ≠ 0) |
130 | 126, 127,
128, 129 | div23d 11544 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) / 𝑃) = ((𝑄 / 𝑃) · (2 · 𝑥))) |
131 | 130 | fveq2d 6691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(⌊‘((𝑄 ·
(2 · 𝑥)) / 𝑃)) = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) |
132 | 131 | oveq2d 7199 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃))) = (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
133 | 132 | oveq2d 7199 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) |
134 | 125, 133 | eqtrd 2774 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) |
135 | 134 | oveq2d 7199 |
. . . . . . . . . . . . . . . 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))))))) |
136 | | prmz 16129 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℤ) |
137 | 102, 136 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℤ) |
138 | 137, 76 | zmulcld 12187 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ) |
139 | 138 | zcnd 12182 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ) |
140 | 101 | zcnd 12182 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℂ) |
141 | 139, 140 | pncan3d 11091 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑄 · (2 · 𝑥))) |
142 | | 2cnd 11807 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈
ℂ) |
143 | 71 | nncnd 11745 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℂ) |
144 | 126, 142,
143 | mul12d 10940 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) = (2 · (𝑄 · 𝑥))) |
145 | 135, 141,
144 | 3eqtrd 2778 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅) = (2 · (𝑄 · 𝑥))) |
146 | 145 | oveq2d 7199 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = (-1↑(2 · (𝑄 · 𝑥)))) |
147 | 34 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -1 ∈
ℂ) |
148 | 35 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -1 ≠
0) |
149 | 106 | nn0zd 12179 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℤ) |
150 | | expaddz 13578 |
. . . . . . . . . . . . . . . 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))) |
151 | 147, 148,
138, 149, 150 | syl22anc 838 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = ((-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) · (-1↑𝑅))) |
152 | | expmulz 13580 |
. . . . . . . . . . . . . . . . . 18
⊢ (((-1
∈ ℂ ∧ -1 ≠ 0) ∧ (𝑃 ∈ ℤ ∧ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)) → (-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ((-1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
153 | 147, 148,
137, 76, 152 | syl22anc 838 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ((-1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
154 | | 1cnd 10727 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 1 ∈
ℂ) |
155 | | eldifsni 4688 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ≠
2) |
156 | 6, 155 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝑃 ≠ 2) |
157 | 156 | necomd 2990 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 2 ≠ 𝑃) |
158 | 157 | neneqd 2940 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ¬ 2 = 𝑃) |
159 | 158 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 2 = 𝑃) |
160 | | 2z 12108 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 2 ∈
ℤ |
161 | | uzid 12352 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (2 ∈
ℤ → 2 ∈ (ℤ≥‘2)) |
162 | 160, 161 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ∈
(ℤ≥‘2) |
163 | | dvdsprm 16157 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((2
∈ (ℤ≥‘2) ∧ 𝑃 ∈ ℙ) → (2 ∥ 𝑃 ↔ 2 = 𝑃)) |
164 | 162, 102,
163 | sylancr 590 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 ∥ 𝑃 ↔ 2 = 𝑃)) |
165 | 159, 164 | mtbird 328 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 2 ∥
𝑃) |
166 | | oexpneg 15803 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((1
∈ ℂ ∧ 𝑃
∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (-1↑𝑃) = -(1↑𝑃)) |
167 | 154, 104,
165, 166 | syl3anc 1372 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑃) = -(1↑𝑃)) |
168 | | 1exp 13563 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑃 ∈ ℤ →
(1↑𝑃) =
1) |
169 | 137, 168 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1↑𝑃) = 1) |
170 | 169 | negeqd 10971 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -(1↑𝑃) = -1) |
171 | 167, 170 | eqtrd 2774 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑃) = -1) |
172 | 171 | oveq1d 7198 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
173 | 153, 172 | eqtrd 2774 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
174 | 173 | oveq1d 7198 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) · (-1↑𝑅)) = ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅))) |
175 | 151, 174 | eqtrd 2774 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅))) |
176 | | nnmulcl 11753 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑄 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (𝑄 · 𝑥) ∈ ℕ) |
177 | 64, 70, 176 | syl2an 599 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℕ) |
178 | 177 | nnnn0d 12049 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈
ℕ0) |
179 | | 2nn0 12006 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ∈
ℕ0 |
180 | 179 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈
ℕ0) |
181 | 147, 178,
180 | expmuld 13618 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(2
· (𝑄 · 𝑥))) = ((-1↑2)↑(𝑄 · 𝑥))) |
182 | | neg1sqe1 13664 |
. . . . . . . . . . . . . . . . 17
⊢
(-1↑2) = 1 |
183 | 182 | oveq1i 7193 |
. . . . . . . . . . . . . . . 16
⊢
((-1↑2)↑(𝑄
· 𝑥)) =
(1↑(𝑄 · 𝑥)) |
184 | 177 | nnzd 12180 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℤ) |
185 | | 1exp 13563 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑄 · 𝑥) ∈ ℤ → (1↑(𝑄 · 𝑥)) = 1) |
186 | 184, 185 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1↑(𝑄 · 𝑥)) = 1) |
187 | 183, 186 | syl5eq 2786 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
((-1↑2)↑(𝑄
· 𝑥)) =
1) |
188 | 181, 187 | eqtrd 2774 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(2
· (𝑄 · 𝑥))) = 1) |
189 | 146, 175,
188 | 3eqtr3d 2782 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)) = 1) |
190 | 189 | oveq1d 7198 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)) · 𝑄) = (1 · 𝑄)) |
191 | 92 | zcnd 12182 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(-1↑(⌊‘((𝑄
/ 𝑃) · (2 ·
𝑥)))) ∈
ℂ) |
192 | 108 | zcnd 12182 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈
ℂ) |
193 | 191, 192,
126 | mulassd 10755 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)) · 𝑄) = ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄))) |
194 | 126 | mulid2d 10750 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1 · 𝑄) = 𝑄) |
195 | 190, 193,
194 | 3eqtr3d 2782 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄)) = 𝑄) |
196 | 195 | fveq2d 6691 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄))) = (𝐿‘𝑄)) |
197 | 120, 196 | eqtr3d 2776 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))(.r‘𝑌)(𝐿‘((-1↑𝑅) · 𝑄))) = (𝐿‘𝑄)) |
198 | 197 | mpteq2dva 5135 |
. . . . . . . 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))) |
199 | 116, 198 | eqtrd 2774 |
. . . . . . 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))) |
200 | 199 | oveq2d 7199 |
. . . . . 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)))) |
201 | | lgseisen.3 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ≠ 𝑄) |
202 | | lgseisen.5 |
. . . . . . . 8
⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)) |
203 | | lgseisen.6 |
. . . . . . . 8
⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃) |
204 | 6, 63, 201, 95, 202, 203, 8, 14, 22 | lgseisenlem3 26126 |
. . . . . . 7
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) = (1r‘𝑌)) |
205 | 204 | oveq2d 7199 |
. . . . . 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))) |
206 | 113, 200,
205 | 3eqtr3rd 2783 |
. . . . 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)))) |
207 | | eqid 2739 |
. . . . . . 7
⊢
(0g‘𝐺) = (0g‘𝐺) |
208 | 93 | fmpttd 6902 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌)) |
209 | | fvexd 6702 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ∈ V) |
210 | | fvexd 6702 |
. . . . . . . 8
⊢ (𝜑 → (0g‘𝐺) ∈ V) |
211 | 111, 19, 209, 210 | fsuppmptdm 8930 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) finSupp (0g‘𝐺)) |
212 | 87, 207, 16, 19, 208, 211 | gsumcl 19167 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) ∈ (Base‘𝑌)) |
213 | | eqid 2739 |
. . . . . . 7
⊢
(1r‘𝑌) = (1r‘𝑌) |
214 | 25, 88, 213 | ringridm 19457 |
. . . . . 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)))))))) |
215 | 21, 212, 214 | syl2anc 587 |
. . . . 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)))))))) |
216 | 96, 98 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑄 ∈ ℤ) |
217 | 27, 216 | ffvelrnd 6875 |
. . . . . . 7
⊢ (𝜑 → (𝐿‘𝑄) ∈ (Base‘𝑌)) |
218 | | eqid 2739 |
. . . . . . . 8
⊢
(.g‘𝐺) = (.g‘𝐺) |
219 | 87, 218 | gsumconst 19186 |
. . . . . . 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))) |
220 | 18, 19, 217, 219 | syl3anc 1372 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = ((♯‘(1...((𝑃 − 1) /
2)))(.g‘𝐺)(𝐿‘𝑄))) |
221 | | oddprm 16260 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((𝑃 − 1) / 2)
∈ ℕ) |
222 | 6, 221 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 − 1) / 2) ∈
ℕ) |
223 | 222 | nnnn0d 12049 |
. . . . . . . 8
⊢ (𝜑 → ((𝑃 − 1) / 2) ∈
ℕ0) |
224 | | hashfz1 13811 |
. . . . . . . 8
⊢ (((𝑃 − 1) / 2) ∈
ℕ0 → (♯‘(1...((𝑃 − 1) / 2))) = ((𝑃 − 1) / 2)) |
225 | 223, 224 | syl 17 |
. . . . . . 7
⊢ (𝜑 →
(♯‘(1...((𝑃
− 1) / 2))) = ((𝑃
− 1) / 2)) |
226 | 225 | oveq1d 7198 |
. . . . . 6
⊢ (𝜑 →
((♯‘(1...((𝑃
− 1) / 2)))(.g‘𝐺)(𝐿‘𝑄)) = (((𝑃 − 1) / 2)(.g‘𝐺)(𝐿‘𝑄))) |
227 | 31, 1 | mgpbas 19377 |
. . . . . . . . 9
⊢ ℤ =
(Base‘((mulGrp‘ℂfld) ↾s
ℤ)) |
228 | | eqid 2739 |
. . . . . . . . 9
⊢
(.g‘((mulGrp‘ℂfld)
↾s ℤ)) =
(.g‘((mulGrp‘ℂfld) ↾s
ℤ)) |
229 | 227, 228,
218 | mhmmulg 18399 |
. . . . . . . 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))) |
230 | 33, 223, 216, 229 | syl3anc 1372 |
. . . . . . 7
⊢ (𝜑 → (𝐿‘(((𝑃 − 1) /
2)(.g‘((mulGrp‘ℂfld)
↾s ℤ))𝑄)) = (((𝑃 − 1) / 2)(.g‘𝐺)(𝐿‘𝑄))) |
231 | 53 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℤ ∈
(SubMnd‘(mulGrp‘ℂfld))) |
232 | | eqid 2739 |
. . . . . . . . . . 11
⊢
(.g‘(mulGrp‘ℂfld)) =
(.g‘(mulGrp‘ℂfld)) |
233 | 232, 56, 228 | submmulg 18402 |
. . . . . . . . . 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)) |
234 | 231, 223,
216, 233 | syl3anc 1372 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑃 − 1) /
2)(.g‘(mulGrp‘ℂfld))𝑄) = (((𝑃 − 1) /
2)(.g‘((mulGrp‘ℂfld)
↾s ℤ))𝑄)) |
235 | 216 | zcnd 12182 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ∈ ℂ) |
236 | | cnfldexp 20263 |
. . . . . . . . . 10
⊢ ((𝑄 ∈ ℂ ∧ ((𝑃 − 1) / 2) ∈
ℕ0) → (((𝑃 − 1) /
2)(.g‘(mulGrp‘ℂfld))𝑄) = (𝑄↑((𝑃 − 1) / 2))) |
237 | 235, 223,
236 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑃 − 1) /
2)(.g‘(mulGrp‘ℂfld))𝑄) = (𝑄↑((𝑃 − 1) / 2))) |
238 | 234, 237 | eqtr3d 2776 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 − 1) /
2)(.g‘((mulGrp‘ℂfld)
↾s ℤ))𝑄) = (𝑄↑((𝑃 − 1) / 2))) |
239 | 238 | fveq2d 6691 |
. . . . . . 7
⊢ (𝜑 → (𝐿‘(((𝑃 − 1) /
2)(.g‘((mulGrp‘ℂfld)
↾s ℤ))𝑄)) = (𝐿‘(𝑄↑((𝑃 − 1) / 2)))) |
240 | 230, 239 | eqtr3d 2776 |
. . . . . 6
⊢ (𝜑 → (((𝑃 − 1) / 2)(.g‘𝐺)(𝐿‘𝑄)) = (𝐿‘(𝑄↑((𝑃 − 1) / 2)))) |
241 | 220, 226,
240 | 3eqtrd 2778 |
. . . . 5
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = (𝐿‘(𝑄↑((𝑃 − 1) / 2)))) |
242 | 206, 215,
241 | 3eqtr3d 2782 |
. . . 4
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) = (𝐿‘(𝑄↑((𝑃 − 1) / 2)))) |
243 | | subrgsubg 19673 |
. . . . . . . . . 10
⊢ (ℤ
∈ (SubRing‘ℂfld) → ℤ ∈
(SubGrp‘ℂfld)) |
244 | 51, 243 | ax-mp 5 |
. . . . . . . . 9
⊢ ℤ
∈ (SubGrp‘ℂfld) |
245 | | subgsubm 18432 |
. . . . . . . . 9
⊢ (ℤ
∈ (SubGrp‘ℂfld) → ℤ ∈
(SubMnd‘ℂfld)) |
246 | 244, 245 | mp1i 13 |
. . . . . . . 8
⊢ (𝜑 → ℤ ∈
(SubMnd‘ℂfld)) |
247 | 76 | fmpttd 6902 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))):(1...((𝑃 − 1) /
2))⟶ℤ) |
248 | | df-zring 20303 |
. . . . . . . 8
⊢
ℤring = (ℂfld ↾s
ℤ) |
249 | 19, 246, 247, 248 | gsumsubm 18128 |
. . . . . . 7
⊢ (𝜑 → (ℂfld
Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = (ℤring
Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) |
250 | 76 | zcnd 12182 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) →
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈
ℂ) |
251 | 19, 250 | gsumfsum 20297 |
. . . . . . 7
⊢ (𝜑 → (ℂfld
Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) |
252 | 249, 251 | eqtr3d 2776 |
. . . . . 6
⊢ (𝜑 → (ℤring
Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) |
253 | 252 | oveq2d 7199 |
. . . . 5
⊢ (𝜑 →
(-1↑(ℤring Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦
(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) |
254 | 253 | fveq2d 6691 |
. . . 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)))))) |
255 | 86, 242, 254 | 3eqtr3d 2782 |
. . 3
⊢ (𝜑 → (𝐿‘(𝑄↑((𝑃 − 1) / 2))) = (𝐿‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) |
256 | 66 | nnnn0d 12049 |
. . . 4
⊢ (𝜑 → 𝑃 ∈
ℕ0) |
257 | | zexpcl 13549 |
. . . . 5
⊢ ((𝑄 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈
ℕ0) → (𝑄↑((𝑃 − 1) / 2)) ∈
ℤ) |
258 | 216, 223,
257 | syl2anc 587 |
. . . 4
⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈
ℤ) |
259 | 19, 76 | fsumzcl 15198 |
. . . . 5
⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ) |
260 | | m1expcl 13557 |
. . . . 5
⊢
(Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥))) ∈ ℤ →
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))) ∈
ℤ) |
261 | 259, 260 | syl 17 |
. . . 4
⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ) |
262 | 8, 22 | zndvds 20381 |
. . . 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))))))) |
263 | 256, 258,
261, 262 | syl3anc 1372 |
. . 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))))))) |
264 | 255, 263 | mpbid 235 |
. 2
⊢ (𝜑 → 𝑃 ∥ ((𝑄↑((𝑃 − 1) / 2)) −
(-1↑Σ𝑥 ∈
(1...((𝑃 − 1) /
2))(⌊‘((𝑄 /
𝑃) · (2 ·
𝑥)))))) |
265 | | moddvds 15723 |
. . 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))))))) |
266 | 66, 258, 261, 265 | syl3anc 1372 |
. 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))))))) |
267 | 264, 266 | mpbird 260 |
1
⊢ (𝜑 → ((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃)) |