| Step | Hyp | Ref
| Expression |
| 1 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝑘 = 𝑥 → (2 · 𝑘) = (2 · 𝑥)) |
| 2 | 1 | fveq2d 6910 |
. . . . . . . 8
⊢ (𝑘 = 𝑥 → (𝐿‘(2 · 𝑘)) = (𝐿‘(2 · 𝑥))) |
| 3 | 2 | cbvmptv 5255 |
. . . . . . 7
⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))) |
| 4 | 3 | oveq2i 7442 |
. . . . . 6
⊢ (𝐺 Σg
(𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘)))) = (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) |
| 5 | | lgseisen.8 |
. . . . . . . 8
⊢ 𝐺 = (mulGrp‘𝑌) |
| 6 | | eqid 2737 |
. . . . . . . 8
⊢
(Base‘𝑌) =
(Base‘𝑌) |
| 7 | 5, 6 | mgpbas 20142 |
. . . . . . 7
⊢
(Base‘𝑌) =
(Base‘𝐺) |
| 8 | | eqid 2737 |
. . . . . . 7
⊢
(0g‘𝐺) = (0g‘𝐺) |
| 9 | | lgseisen.1 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖
{2})) |
| 10 | 9 | eldifad 3963 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ ℙ) |
| 11 | | lgseisen.7 |
. . . . . . . . . . 11
⊢ 𝑌 =
(ℤ/nℤ‘𝑃) |
| 12 | 11 | znfld 21579 |
. . . . . . . . . 10
⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field) |
| 13 | 10, 12 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑌 ∈ Field) |
| 14 | | isfld 20740 |
. . . . . . . . . 10
⊢ (𝑌 ∈ Field ↔ (𝑌 ∈ DivRing ∧ 𝑌 ∈ CRing)) |
| 15 | 14 | simprbi 496 |
. . . . . . . . 9
⊢ (𝑌 ∈ Field → 𝑌 ∈ CRing) |
| 16 | 13, 15 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ∈ CRing) |
| 17 | 5 | crngmgp 20238 |
. . . . . . . 8
⊢ (𝑌 ∈ CRing → 𝐺 ∈ CMnd) |
| 18 | 16, 17 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ CMnd) |
| 19 | | fzfid 14014 |
. . . . . . 7
⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈
Fin) |
| 20 | | crngring 20242 |
. . . . . . . . . . . 12
⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) |
| 21 | 16, 20 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑌 ∈ Ring) |
| 22 | | lgseisen.9 |
. . . . . . . . . . . 12
⊢ 𝐿 = (ℤRHom‘𝑌) |
| 23 | 22 | zrhrhm 21522 |
. . . . . . . . . . 11
⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring
RingHom 𝑌)) |
| 24 | 21, 23 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐿 ∈ (ℤring RingHom
𝑌)) |
| 25 | | zringbas 21464 |
. . . . . . . . . . 11
⊢ ℤ =
(Base‘ℤring) |
| 26 | 25, 6 | rhmf 20485 |
. . . . . . . . . 10
⊢ (𝐿 ∈ (ℤring
RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌)) |
| 27 | 24, 26 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌)) |
| 28 | | 2z 12649 |
. . . . . . . . . 10
⊢ 2 ∈
ℤ |
| 29 | | elfzelz 13564 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) → 𝑘 ∈ ℤ) |
| 30 | | zmulcl 12666 |
. . . . . . . . . 10
⊢ ((2
∈ ℤ ∧ 𝑘
∈ ℤ) → (2 · 𝑘) ∈ ℤ) |
| 31 | 28, 29, 30 | sylancr 587 |
. . . . . . . . 9
⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) → (2 · 𝑘) ∈
ℤ) |
| 32 | | ffvelcdm 7101 |
. . . . . . . . 9
⊢ ((𝐿:ℤ⟶(Base‘𝑌) ∧ (2 · 𝑘) ∈ ℤ) → (𝐿‘(2 · 𝑘)) ∈ (Base‘𝑌)) |
| 33 | 27, 31, 32 | syl2an 596 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑘)) ∈ (Base‘𝑌)) |
| 34 | 33 | fmpttd 7135 |
. . . . . . 7
⊢ (𝜑 → (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌)) |
| 35 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) = (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) |
| 36 | | fvexd 6921 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑘)) ∈ V) |
| 37 | | fvexd 6921 |
. . . . . . . 8
⊢ (𝜑 → (0g‘𝐺) ∈ V) |
| 38 | 35, 19, 36, 37 | fsuppmptdm 9416 |
. . . . . . 7
⊢ (𝜑 → (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) finSupp (0g‘𝐺)) |
| 39 | | lgseisen.2 |
. . . . . . . 8
⊢ (𝜑 → 𝑄 ∈ (ℙ ∖
{2})) |
| 40 | | lgseisen.3 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ≠ 𝑄) |
| 41 | | lgseisen.4 |
. . . . . . . 8
⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃) |
| 42 | | lgseisen.5 |
. . . . . . . 8
⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)) |
| 43 | | lgseisen.6 |
. . . . . . . 8
⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃) |
| 44 | 9, 39, 40, 41, 42, 43 | lgseisenlem2 27420 |
. . . . . . 7
⊢ (𝜑 → 𝑀:(1...((𝑃 − 1) / 2))–1-1-onto→(1...((𝑃 − 1) / 2))) |
| 45 | 7, 8, 18, 19, 34, 38, 44 | gsumf1o 19934 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘)))) = (𝐺 Σg ((𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) ∘ 𝑀))) |
| 46 | 4, 45 | eqtr3id 2791 |
. . . . 5
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) = (𝐺 Σg ((𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) ∘ 𝑀))) |
| 47 | 9, 39, 40, 41, 42 | lgseisenlem1 27419 |
. . . . . . . 8
⊢ (𝜑 → 𝑀:(1...((𝑃 − 1) / 2))⟶(1...((𝑃 − 1) /
2))) |
| 48 | 42 | fmpt 7130 |
. . . . . . . 8
⊢
(∀𝑥 ∈
(1...((𝑃 − 1) /
2))((((-1↑𝑅) ·
𝑅) mod 𝑃) / 2) ∈ (1...((𝑃 − 1) / 2)) ↔ 𝑀:(1...((𝑃 − 1) / 2))⟶(1...((𝑃 − 1) /
2))) |
| 49 | 47, 48 | sylibr 234 |
. . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ (1...((𝑃 − 1) / 2))((((-1↑𝑅) · 𝑅) mod 𝑃) / 2) ∈ (1...((𝑃 − 1) / 2))) |
| 50 | 42 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))) |
| 51 | | eqidd 2738 |
. . . . . . 7
⊢ (𝜑 → (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) = (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘)))) |
| 52 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑘 = ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2) → (2 · 𝑘) = (2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))) |
| 53 | 52 | fveq2d 6910 |
. . . . . . 7
⊢ (𝑘 = ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2) → (𝐿‘(2 · 𝑘)) = (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)))) |
| 54 | 49, 50, 51, 53 | fmptcof 7150 |
. . . . . 6
⊢ (𝜑 → ((𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) ∘ 𝑀) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))))) |
| 55 | 54 | oveq2d 7447 |
. . . . 5
⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) ∘ 𝑀)) = (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)))))) |
| 56 | 39 | eldifad 3963 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑄 ∈ ℙ) |
| 57 | 56 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℙ) |
| 58 | | prmz 16712 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑄 ∈ ℙ → 𝑄 ∈
ℤ) |
| 59 | 57, 58 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℤ) |
| 60 | | 2nn 12339 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 2 ∈
ℕ |
| 61 | | elfznn 13593 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ) |
| 62 | 61 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ) |
| 63 | | nnmulcl 12290 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((2
∈ ℕ ∧ 𝑥
∈ ℕ) → (2 · 𝑥) ∈ ℕ) |
| 64 | 60, 62, 63 | sylancr 587 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈
ℕ) |
| 65 | 64 | nnzd 12640 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈
ℤ) |
| 66 | 59, 65 | zmulcld 12728 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℤ) |
| 67 | 10 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℙ) |
| 68 | | prmnn 16711 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
| 69 | 67, 68 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ) |
| 70 | 66, 69 | zmodcld 13932 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) ∈
ℕ0) |
| 71 | 41, 70 | eqeltrid 2845 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈
ℕ0) |
| 72 | 71 | nn0zd 12639 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℤ) |
| 73 | | m1expcl 14127 |
. . . . . . . . . . . . . . 15
⊢ (𝑅 ∈ ℤ →
(-1↑𝑅) ∈
ℤ) |
| 74 | 72, 73 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈
ℤ) |
| 75 | 74, 72 | zmulcld 12728 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑅) ∈ ℤ) |
| 76 | 75, 69 | zmodcld 13932 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈
ℕ0) |
| 77 | 76 | nn0cnd 12589 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℂ) |
| 78 | | 2cnd 12344 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈
ℂ) |
| 79 | | 2ne0 12370 |
. . . . . . . . . . . 12
⊢ 2 ≠
0 |
| 80 | 79 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ≠
0) |
| 81 | 77, 78, 80 | divcan2d 12045 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 ·
((((-1↑𝑅) ·
𝑅) mod 𝑃) / 2)) = (((-1↑𝑅) · 𝑅) mod 𝑃)) |
| 82 | 81 | fveq2d 6910 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))) = (𝐿‘(((-1↑𝑅) · 𝑅) mod 𝑃))) |
| 83 | 69 | nnrpd 13075 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈
ℝ+) |
| 84 | | eqidd 2738 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) mod 𝑃) = ((-1↑𝑅) mod 𝑃)) |
| 85 | 41 | oveq1i 7441 |
. . . . . . . . . . . . . 14
⊢ (𝑅 mod 𝑃) = (((𝑄 · (2 · 𝑥)) mod 𝑃) mod 𝑃) |
| 86 | 66 | zred 12722 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℝ) |
| 87 | | modabs2 13945 |
. . . . . . . . . . . . . . 15
⊢ (((𝑄 · (2 · 𝑥)) ∈ ℝ ∧ 𝑃 ∈ ℝ+)
→ (((𝑄 · (2
· 𝑥)) mod 𝑃) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃)) |
| 88 | 86, 83, 87 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((𝑄 · (2 · 𝑥)) mod 𝑃) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃)) |
| 89 | 85, 88 | eqtrid 2789 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑅 mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃)) |
| 90 | 74, 74, 72, 66, 83, 84, 89 | modmul12d 13966 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) = (((-1↑𝑅) · (𝑄 · (2 · 𝑥))) mod 𝑃)) |
| 91 | 75 | zred 12722 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑅) ∈ ℝ) |
| 92 | | modabs2 13945 |
. . . . . . . . . . . . 13
⊢
((((-1↑𝑅)
· 𝑅) ∈ ℝ
∧ 𝑃 ∈
ℝ+) → ((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = (((-1↑𝑅) · 𝑅) mod 𝑃)) |
| 93 | 91, 83, 92 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = (((-1↑𝑅) · 𝑅) mod 𝑃)) |
| 94 | 74 | zcnd 12723 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈
ℂ) |
| 95 | 59 | zcnd 12723 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℂ) |
| 96 | 65 | zcnd 12723 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈
ℂ) |
| 97 | 94, 95, 96 | mulassd 11284 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑄) · (2 · 𝑥)) = ((-1↑𝑅) · (𝑄 · (2 · 𝑥)))) |
| 98 | 97 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((((-1↑𝑅) · 𝑄) · (2 · 𝑥)) mod 𝑃) = (((-1↑𝑅) · (𝑄 · (2 · 𝑥))) mod 𝑃)) |
| 99 | 90, 93, 98 | 3eqtr4d 2787 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = ((((-1↑𝑅) · 𝑄) · (2 · 𝑥)) mod 𝑃)) |
| 100 | 10, 68 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑃 ∈ ℕ) |
| 101 | 100 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ) |
| 102 | 76 | nn0zd 12639 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℤ) |
| 103 | 74, 59 | zmulcld 12728 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑄) ∈ ℤ) |
| 104 | 103, 65 | zmulcld 12728 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑄) · (2 · 𝑥)) ∈ ℤ) |
| 105 | | moddvds 16301 |
. . . . . . . . . . . 12
⊢ ((𝑃 ∈ ℕ ∧
(((-1↑𝑅) ·
𝑅) mod 𝑃) ∈ ℤ ∧ (((-1↑𝑅) · 𝑄) · (2 · 𝑥)) ∈ ℤ) → (((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = ((((-1↑𝑅) · 𝑄) · (2 · 𝑥)) mod 𝑃) ↔ 𝑃 ∥ ((((-1↑𝑅) · 𝑅) mod 𝑃) − (((-1↑𝑅) · 𝑄) · (2 · 𝑥))))) |
| 106 | 101, 102,
104, 105 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = ((((-1↑𝑅) · 𝑄) · (2 · 𝑥)) mod 𝑃) ↔ 𝑃 ∥ ((((-1↑𝑅) · 𝑅) mod 𝑃) − (((-1↑𝑅) · 𝑄) · (2 · 𝑥))))) |
| 107 | 99, 106 | mpbid 232 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∥ ((((-1↑𝑅) · 𝑅) mod 𝑃) − (((-1↑𝑅) · 𝑄) · (2 · 𝑥)))) |
| 108 | 69 | nnnn0d 12587 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈
ℕ0) |
| 109 | 11, 22 | zndvds 21568 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ ℕ0
∧ (((-1↑𝑅)
· 𝑅) mod 𝑃) ∈ ℤ ∧
(((-1↑𝑅) ·
𝑄) · (2 ·
𝑥)) ∈ ℤ) →
((𝐿‘(((-1↑𝑅) · 𝑅) mod 𝑃)) = (𝐿‘(((-1↑𝑅) · 𝑄) · (2 · 𝑥))) ↔ 𝑃 ∥ ((((-1↑𝑅) · 𝑅) mod 𝑃) − (((-1↑𝑅) · 𝑄) · (2 · 𝑥))))) |
| 110 | 108, 102,
104, 109 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(((-1↑𝑅) · 𝑅) mod 𝑃)) = (𝐿‘(((-1↑𝑅) · 𝑄) · (2 · 𝑥))) ↔ 𝑃 ∥ ((((-1↑𝑅) · 𝑅) mod 𝑃) − (((-1↑𝑅) · 𝑄) · (2 · 𝑥))))) |
| 111 | 107, 110 | mpbird 257 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(((-1↑𝑅) · 𝑅) mod 𝑃)) = (𝐿‘(((-1↑𝑅) · 𝑄) · (2 · 𝑥)))) |
| 112 | 24 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿 ∈ (ℤring RingHom
𝑌)) |
| 113 | | zringmulr 21468 |
. . . . . . . . . . 11
⊢ ·
= (.r‘ℤring) |
| 114 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(.r‘𝑌) = (.r‘𝑌) |
| 115 | 25, 113, 114 | rhmmul 20486 |
. . . . . . . . . 10
⊢ ((𝐿 ∈ (ℤring
RingHom 𝑌) ∧
((-1↑𝑅) · 𝑄) ∈ ℤ ∧ (2
· 𝑥) ∈ ℤ)
→ (𝐿‘(((-1↑𝑅) · 𝑄) · (2 · 𝑥))) = ((𝐿‘((-1↑𝑅) · 𝑄))(.r‘𝑌)(𝐿‘(2 · 𝑥)))) |
| 116 | 112, 103,
65, 115 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(((-1↑𝑅) · 𝑄) · (2 · 𝑥))) = ((𝐿‘((-1↑𝑅) · 𝑄))(.r‘𝑌)(𝐿‘(2 · 𝑥)))) |
| 117 | 82, 111, 116 | 3eqtrd 2781 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))) = ((𝐿‘((-1↑𝑅) · 𝑄))(.r‘𝑌)(𝐿‘(2 · 𝑥)))) |
| 118 | 117 | mpteq2dva 5242 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((𝐿‘((-1↑𝑅) · 𝑄))(.r‘𝑌)(𝐿‘(2 · 𝑥))))) |
| 119 | 27 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿:ℤ⟶(Base‘𝑌)) |
| 120 | 119, 103 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝑌)) |
| 121 | 119, 65 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ∈ (Base‘𝑌)) |
| 122 | | eqidd 2738 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) |
| 123 | | eqidd 2738 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) |
| 124 | 19, 120, 121, 122, 123 | offval2 7717 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘f
(.r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((𝐿‘((-1↑𝑅) · 𝑄))(.r‘𝑌)(𝐿‘(2 · 𝑥))))) |
| 125 | 118, 124 | eqtr4d 2780 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)))) = ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘f
(.r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) |
| 126 | 125 | oveq2d 7447 |
. . . . 5
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))))) = (𝐺 Σg ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘f
(.r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))))) |
| 127 | 46, 55, 126 | 3eqtrd 2781 |
. . . 4
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) = (𝐺 Σg ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘f
(.r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))))) |
| 128 | 5, 114 | mgpplusg 20141 |
. . . . 5
⊢
(.r‘𝑌) = (+g‘𝐺) |
| 129 | | eqid 2737 |
. . . . 5
⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) |
| 130 | | eqid 2737 |
. . . . 5
⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))) |
| 131 | 7, 128, 18, 19, 120, 121, 129, 130 | gsummptfidmadd2 19944 |
. . . 4
⊢ (𝜑 → (𝐺 Σg ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘f
(.r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = ((𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))(.r‘𝑌)(𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))))) |
| 132 | 127, 131 | eqtrd 2777 |
. . 3
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) = ((𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))(.r‘𝑌)(𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))))) |
| 133 | 132 | oveq1d 7446 |
. 2
⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))(/r‘𝑌)(𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = (((𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))(.r‘𝑌)(𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))))(/r‘𝑌)(𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))))) |
| 134 | | eqid 2737 |
. . . . . 6
⊢
(Unit‘𝑌) =
(Unit‘𝑌) |
| 135 | 134, 5 | unitsubm 20386 |
. . . . 5
⊢ (𝑌 ∈ Ring →
(Unit‘𝑌) ∈
(SubMnd‘𝐺)) |
| 136 | 21, 135 | syl 17 |
. . . 4
⊢ (𝜑 → (Unit‘𝑌) ∈ (SubMnd‘𝐺)) |
| 137 | | elfzle2 13568 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ≤ ((𝑃 − 1) / 2)) |
| 138 | 137 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ≤ ((𝑃 − 1) / 2)) |
| 139 | 62 | nnred 12281 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℝ) |
| 140 | | prmuz2 16733 |
. . . . . . . . . . . . 13
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
(ℤ≥‘2)) |
| 141 | | uz2m1nn 12965 |
. . . . . . . . . . . . 13
⊢ (𝑃 ∈
(ℤ≥‘2) → (𝑃 − 1) ∈ ℕ) |
| 142 | 67, 140, 141 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 − 1) ∈ ℕ) |
| 143 | 142 | nnred 12281 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 − 1) ∈ ℝ) |
| 144 | | 2re 12340 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℝ |
| 145 | 144 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈
ℝ) |
| 146 | | 2pos 12369 |
. . . . . . . . . . . 12
⊢ 0 <
2 |
| 147 | 146 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 0 <
2) |
| 148 | | lemuldiv2 12149 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ (𝑃 − 1) ∈ ℝ ∧
(2 ∈ ℝ ∧ 0 < 2)) → ((2 · 𝑥) ≤ (𝑃 − 1) ↔ 𝑥 ≤ ((𝑃 − 1) / 2))) |
| 149 | 139, 143,
145, 147, 148 | syl112anc 1376 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((2 · 𝑥) ≤ (𝑃 − 1) ↔ 𝑥 ≤ ((𝑃 − 1) / 2))) |
| 150 | 138, 149 | mpbird 257 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ≤ (𝑃 − 1)) |
| 151 | | prmz 16712 |
. . . . . . . . . . . 12
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℤ) |
| 152 | 67, 151 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℤ) |
| 153 | | peano2zm 12660 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ ℤ → (𝑃 − 1) ∈
ℤ) |
| 154 | 152, 153 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 − 1) ∈ ℤ) |
| 155 | | fznn 13632 |
. . . . . . . . . 10
⊢ ((𝑃 − 1) ∈ ℤ
→ ((2 · 𝑥)
∈ (1...(𝑃 − 1))
↔ ((2 · 𝑥)
∈ ℕ ∧ (2 · 𝑥) ≤ (𝑃 − 1)))) |
| 156 | 154, 155 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((2 · 𝑥) ∈ (1...(𝑃 − 1)) ↔ ((2 · 𝑥) ∈ ℕ ∧ (2
· 𝑥) ≤ (𝑃 − 1)))) |
| 157 | 64, 150, 156 | mpbir2and 713 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ (1...(𝑃 − 1))) |
| 158 | | fzm1ndvds 16359 |
. . . . . . . 8
⊢ ((𝑃 ∈ ℕ ∧ (2
· 𝑥) ∈
(1...(𝑃 − 1))) →
¬ 𝑃 ∥ (2 ·
𝑥)) |
| 159 | 69, 157, 158 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 𝑃 ∥ (2 · 𝑥)) |
| 160 | | eqid 2737 |
. . . . . . . . . 10
⊢
(0g‘𝑌) = (0g‘𝑌) |
| 161 | 11, 22, 160 | zndvds0 21569 |
. . . . . . . . 9
⊢ ((𝑃 ∈ ℕ0
∧ (2 · 𝑥) ∈
ℤ) → ((𝐿‘(2 · 𝑥)) = (0g‘𝑌) ↔ 𝑃 ∥ (2 · 𝑥))) |
| 162 | 108, 65, 161 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(2 · 𝑥)) = (0g‘𝑌) ↔ 𝑃 ∥ (2 · 𝑥))) |
| 163 | 162 | necon3abid 2977 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(2 · 𝑥)) ≠
(0g‘𝑌)
↔ ¬ 𝑃 ∥ (2
· 𝑥))) |
| 164 | 159, 163 | mpbird 257 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ≠ (0g‘𝑌)) |
| 165 | 14 | simplbi 497 |
. . . . . . . . 9
⊢ (𝑌 ∈ Field → 𝑌 ∈
DivRing) |
| 166 | 13, 165 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ∈ DivRing) |
| 167 | 166 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑌 ∈ DivRing) |
| 168 | 6, 134, 160 | drngunit 20734 |
. . . . . . 7
⊢ (𝑌 ∈ DivRing → ((𝐿‘(2 · 𝑥)) ∈ (Unit‘𝑌) ↔ ((𝐿‘(2 · 𝑥)) ∈ (Base‘𝑌) ∧ (𝐿‘(2 · 𝑥)) ≠ (0g‘𝑌)))) |
| 169 | 167, 168 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(2 · 𝑥)) ∈ (Unit‘𝑌) ↔ ((𝐿‘(2 · 𝑥)) ∈ (Base‘𝑌) ∧ (𝐿‘(2 · 𝑥)) ≠ (0g‘𝑌)))) |
| 170 | 121, 164,
169 | mpbir2and 713 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ∈ (Unit‘𝑌)) |
| 171 | 170 | fmpttd 7135 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))):(1...((𝑃 − 1) / 2))⟶(Unit‘𝑌)) |
| 172 | | fvexd 6921 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ∈ V) |
| 173 | 130, 19, 172, 37 | fsuppmptdm 9416 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))) finSupp (0g‘𝐺)) |
| 174 | 8, 18, 19, 136, 171, 173 | gsumsubmcl 19937 |
. . 3
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) ∈ (Unit‘𝑌)) |
| 175 | | eqid 2737 |
. . . 4
⊢
(/r‘𝑌) = (/r‘𝑌) |
| 176 | | eqid 2737 |
. . . 4
⊢
(1r‘𝑌) = (1r‘𝑌) |
| 177 | 134, 175,
176 | dvrid 20406 |
. . 3
⊢ ((𝑌 ∈ Ring ∧ (𝐺 Σg
(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) ∈ (Unit‘𝑌)) → ((𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))(/r‘𝑌)(𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = (1r‘𝑌)) |
| 178 | 21, 174, 177 | syl2anc 584 |
. 2
⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))(/r‘𝑌)(𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = (1r‘𝑌)) |
| 179 | 120 | fmpttd 7135 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌)) |
| 180 | | fvexd 6921 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑𝑅) · 𝑄)) ∈ V) |
| 181 | 129, 19, 180, 37 | fsuppmptdm 9416 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) finSupp (0g‘𝐺)) |
| 182 | 7, 8, 18, 19, 179, 181 | gsumcl 19933 |
. . 3
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) ∈ (Base‘𝑌)) |
| 183 | 6, 134, 175, 114 | dvrcan3 20410 |
. . 3
⊢ ((𝑌 ∈ Ring ∧ (𝐺 Σg
(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) ∈ (Base‘𝑌) ∧ (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) ∈ (Unit‘𝑌)) → (((𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))(.r‘𝑌)(𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))))(/r‘𝑌)(𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))) |
| 184 | 21, 182, 174, 183 | syl3anc 1373 |
. 2
⊢ (𝜑 → (((𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))(.r‘𝑌)(𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))))(/r‘𝑌)(𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))) |
| 185 | 133, 178,
184 | 3eqtr3rd 2786 |
1
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) = (1r‘𝑌)) |