| Step | Hyp | Ref
| Expression |
| 1 | | aks6d1c1p3.18 |
. . . . . . . . 9
⊢ 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) |
| 2 | 1 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))) |
| 3 | 2 | fveq2d 6910 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘𝐹) = (𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))) |
| 4 | 3 | fveq1d 6908 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘((𝑁 / 𝑃) ↑ 𝑦))) |
| 5 | | aks6d1c1p3.11 |
. . . . . . . 8
⊢ 𝑂 = (eval1‘𝐾) |
| 6 | | aks6d1c1p3.2 |
. . . . . . . 8
⊢ 𝑆 = (Poly1‘𝐾) |
| 7 | | eqid 2737 |
. . . . . . . 8
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 8 | | aks6d1c1p3.3 |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝑆) |
| 9 | | aks6d1c1p3.13 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ Field) |
| 10 | 9 | fldcrngd 20742 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ CRing) |
| 11 | 10 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ CRing) |
| 12 | | eqid 2737 |
. . . . . . . . . 10
⊢
(Base‘𝑉) =
(Base‘𝑉) |
| 13 | | aks6d1c1p3.7 |
. . . . . . . . . 10
⊢ ↑ =
(.g‘𝑉) |
| 14 | | aks6d1c1p3.6 |
. . . . . . . . . . . . . 14
⊢ 𝑉 = (mulGrp‘𝐾) |
| 15 | 14 | crngmgp 20238 |
. . . . . . . . . . . . 13
⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd) |
| 16 | 10, 15 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑉 ∈ CMnd) |
| 17 | 16 | cmnmndd 19822 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑉 ∈ Mnd) |
| 18 | 17 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑉 ∈ Mnd) |
| 19 | | aks6d1c1p3.17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑃 ∥ 𝑁) |
| 20 | | aks6d1c1p3.1 |
. . . . . . . . . . . . . . . 16
⊢ ∼ =
{〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))} |
| 21 | | aks6d1c1p3.20 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∼ 𝐹) |
| 22 | 20, 21 | aks6d1c1p1rcl 42109 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐹 ∈ 𝐵)) |
| 23 | 22 | simpld 494 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 24 | | aks6d1c1p3.14 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑃 ∈ ℙ) |
| 25 | | prmnn 16711 |
. . . . . . . . . . . . . . 15
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
| 26 | 24, 25 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑃 ∈ ℕ) |
| 27 | | nndivdvds 16299 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℕ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℕ)) |
| 28 | 23, 26, 27 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℕ)) |
| 29 | 19, 28 | mpbid 232 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℕ) |
| 30 | 29 | nnnn0d 12587 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁 / 𝑃) ∈
ℕ0) |
| 31 | 30 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 / 𝑃) ∈
ℕ0) |
| 32 | | aks6d1c1p3.15 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑅 ∈ ℕ) |
| 33 | 32 | nnnn0d 12587 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑅 ∈
ℕ0) |
| 34 | 16, 33, 13 | isprimroot 42094 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) ↔ (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0g‘𝑉) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 ↑ 𝑦) = (0g‘𝑉) → 𝑅 ∥ 𝑙)))) |
| 35 | 34 | biimpd 229 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0g‘𝑉) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 ↑ 𝑦) = (0g‘𝑉) → 𝑅 ∥ 𝑙)))) |
| 36 | 35 | imp 406 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0g‘𝑉) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 ↑ 𝑦) = (0g‘𝑉) → 𝑅 ∥ 𝑙))) |
| 37 | 36 | simp1d 1143 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉)) |
| 38 | 12, 13, 18, 31, 37 | mulgnn0cld 19113 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ 𝑦) ∈ (Base‘𝑉)) |
| 39 | 14, 7 | mgpbas 20142 |
. . . . . . . . . . . 12
⊢
(Base‘𝐾) =
(Base‘𝑉) |
| 40 | 39 | eqcomi 2746 |
. . . . . . . . . . 11
⊢
(Base‘𝑉) =
(Base‘𝐾) |
| 41 | 40 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → (Base‘𝑉) = (Base‘𝐾)) |
| 42 | 41 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (Base‘𝑉) = (Base‘𝐾)) |
| 43 | 38, 42 | eleqtrd 2843 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ 𝑦) ∈ (Base‘𝐾)) |
| 44 | | aks6d1c1p3.4 |
. . . . . . . . 9
⊢ 𝑋 = (var1‘𝐾) |
| 45 | 5, 44, 7, 6, 8, 11,
43 | evl1vard 22341 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘((𝑁 / 𝑃) ↑ 𝑦)) = ((𝑁 / 𝑃) ↑ 𝑦))) |
| 46 | | aks6d1c1p3.8 |
. . . . . . . . 9
⊢ 𝐶 = (algSc‘𝑆) |
| 47 | 10 | crngringd 20243 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐾 ∈ Ring) |
| 48 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(ℤRHom‘𝐾) = (ℤRHom‘𝐾) |
| 49 | 48 | zrhrhm 21522 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ Ring →
(ℤRHom‘𝐾)
∈ (ℤring RingHom 𝐾)) |
| 50 | | rhmghm 20484 |
. . . . . . . . . . . 12
⊢
((ℤRHom‘𝐾) ∈ (ℤring RingHom
𝐾) →
(ℤRHom‘𝐾)
∈ (ℤring GrpHom 𝐾)) |
| 51 | | zringbas 21464 |
. . . . . . . . . . . . 13
⊢ ℤ =
(Base‘ℤring) |
| 52 | 51, 7 | ghmf 19238 |
. . . . . . . . . . . 12
⊢
((ℤRHom‘𝐾) ∈ (ℤring GrpHom
𝐾) →
(ℤRHom‘𝐾):ℤ⟶(Base‘𝐾)) |
| 53 | 47, 49, 50, 52 | 4syl 19 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾)) |
| 54 | | aks6d1c1p3.19 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ ℤ) |
| 55 | 53, 54 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾)) |
| 56 | 55 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾)) |
| 57 | 5, 6, 7, 46, 8, 11, 56, 43 | evl1scad 22339 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘((𝑁 / 𝑃) ↑ 𝑦)) = ((ℤRHom‘𝐾)‘𝐴))) |
| 58 | | aks6d1c1p3.12 |
. . . . . . . 8
⊢ + =
(+g‘𝑆) |
| 59 | | eqid 2737 |
. . . . . . . 8
⊢
(+g‘𝐾) = (+g‘𝐾) |
| 60 | 5, 6, 7, 8, 11, 43, 45, 57, 58, 59 | evl1addd 22345 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵 ∧ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘((𝑁 / 𝑃) ↑ 𝑦)) = (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 61 | 60 | simprd 495 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘((𝑁 / 𝑃) ↑ 𝑦)) = (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) |
| 62 | 4, 61 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)) = (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) |
| 63 | 3 | fveq1d 6908 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)) |
| 64 | 63 | oveq2d 7447 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑁 / 𝑃) ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))) |
| 65 | 42 | eleq2d 2827 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦 ∈ (Base‘𝑉) ↔ 𝑦 ∈ (Base‘𝐾))) |
| 66 | 37, 65 | mpbid 232 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾)) |
| 67 | 5, 44, 40, 6, 8, 11, 37 | evl1vard 22341 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘𝑦) = 𝑦)) |
| 68 | 5, 6, 7, 46, 8, 11, 56, 66 | evl1scad 22339 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘𝑦) = ((ℤRHom‘𝐾)‘𝐴))) |
| 69 | 5, 6, 7, 8, 11, 66, 67, 68, 58, 59 | evl1addd 22345 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵 ∧ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 70 | 69 | simprd 495 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) |
| 71 | 70 | oveq2d 7447 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)) = ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 72 | 64, 71 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 73 | | aks6d1c1p3.21 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingIso 𝐾)) |
| 74 | 7, 7 | isrim 20492 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingIso 𝐾) ↔ ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingHom 𝐾) ∧ (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾))) |
| 75 | 73, 74 | sylib 218 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingHom 𝐾) ∧ (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾))) |
| 76 | 75 | simprd 495 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾)) |
| 77 | 76 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾)) |
| 78 | 11 | crnggrpd 20244 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ Grp) |
| 79 | 7, 59, 78, 43, 56 | grpcld 18965 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾)) |
| 80 | | f1ocnvfv1 7296 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾)) → (◡(𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) = (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) |
| 81 | 77, 79, 80 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (◡(𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) = (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) |
| 82 | 81 | eqcomd 2743 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = (◡(𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))) |
| 83 | | eqidd 2738 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))) |
| 84 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) → 𝑥 = (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) |
| 85 | 84 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) → 𝑥 = (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) |
| 86 | 85 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) → (𝑃 ↑ 𝑥) = (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 87 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(mulGrp‘𝐾) =
(mulGrp‘𝐾) |
| 88 | 87, 7 | mgpbas 20142 |
. . . . . . . . . . . 12
⊢
(Base‘𝐾) =
(Base‘(mulGrp‘𝐾)) |
| 89 | 14 | fveq2i 6909 |
. . . . . . . . . . . . 13
⊢
(.g‘𝑉) = (.g‘(mulGrp‘𝐾)) |
| 90 | 13, 89 | eqtri 2765 |
. . . . . . . . . . . 12
⊢ ↑ =
(.g‘(mulGrp‘𝐾)) |
| 91 | 87 | ringmgp 20236 |
. . . . . . . . . . . . . 14
⊢ (𝐾 ∈ Ring →
(mulGrp‘𝐾) ∈
Mnd) |
| 92 | 47, 91 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd) |
| 93 | 92 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (mulGrp‘𝐾) ∈ Mnd) |
| 94 | 26 | nnnn0d 12587 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑃 ∈
ℕ0) |
| 95 | 94 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ∈
ℕ0) |
| 96 | 88, 90, 93, 95, 79 | mulgnn0cld 19113 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘𝐾)) |
| 97 | 83, 86, 79, 96 | fvmptd 7023 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 98 | 97 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 99 | 75 | simpld 494 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingHom 𝐾)) |
| 100 | | rhmghm 20484 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingHom 𝐾) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 GrpHom 𝐾)) |
| 101 | 99, 100 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 GrpHom 𝐾)) |
| 102 | 101 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 GrpHom 𝐾)) |
| 103 | 7, 59, 59 | ghmlin 19239 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 GrpHom 𝐾) ∧ ((𝑁 / 𝑃) ↑ 𝑦) ∈ (Base‘𝐾) ∧ ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ 𝑦))(+g‘𝐾)((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((ℤRHom‘𝐾)‘𝐴)))) |
| 104 | 102, 43, 56, 103 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ 𝑦))(+g‘𝐾)((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((ℤRHom‘𝐾)‘𝐴)))) |
| 105 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = ((𝑁 / 𝑃) ↑ 𝑦) → 𝑥 = ((𝑁 / 𝑃) ↑ 𝑦)) |
| 106 | 105 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((𝑁 / 𝑃) ↑ 𝑦)) → 𝑥 = ((𝑁 / 𝑃) ↑ 𝑦)) |
| 107 | 106 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((𝑁 / 𝑃) ↑ 𝑦)) → (𝑃 ↑ 𝑥) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦))) |
| 108 | 88, 90, 93, 95, 43 | mulgnn0cld 19113 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)) ∈ (Base‘𝐾)) |
| 109 | 83, 107, 43, 108 | fvmptd 7023 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ 𝑦)) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦))) |
| 110 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = ((ℤRHom‘𝐾)‘𝐴) → 𝑥 = ((ℤRHom‘𝐾)‘𝐴)) |
| 111 | 110 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = ((ℤRHom‘𝐾)‘𝐴) → (𝑃 ↑ 𝑥) = (𝑃 ↑
((ℤRHom‘𝐾)‘𝐴))) |
| 112 | 111 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((ℤRHom‘𝐾)‘𝐴)) → (𝑃 ↑ 𝑥) = (𝑃 ↑
((ℤRHom‘𝐾)‘𝐴))) |
| 113 | | aks6d1c1p3.10 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑃 = (chr‘𝐾) |
| 114 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢
((ℤRHom‘𝐾)‘𝐴) = ((ℤRHom‘𝐾)‘𝐴) |
| 115 | 113, 7, 90, 114, 24, 54, 10 | fermltlchr 21544 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑃 ↑
((ℤRHom‘𝐾)‘𝐴)) = ((ℤRHom‘𝐾)‘𝐴)) |
| 116 | 115 | eqcomd 2743 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) = (𝑃 ↑
((ℤRHom‘𝐾)‘𝐴))) |
| 117 | 116 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((ℤRHom‘𝐾)‘𝐴) = (𝑃 ↑
((ℤRHom‘𝐾)‘𝐴))) |
| 118 | 117, 56 | eqeltrrd 2842 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑
((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾)) |
| 119 | 83, 112, 56, 118 | fvmptd 7023 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((ℤRHom‘𝐾)‘𝐴)) = (𝑃 ↑
((ℤRHom‘𝐾)‘𝐴))) |
| 120 | 109, 119 | oveq12d 7449 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ 𝑦))(+g‘𝐾)((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦))(+g‘𝐾)(𝑃 ↑
((ℤRHom‘𝐾)‘𝐴)))) |
| 121 | 98, 104, 120 | 3eqtrd 2781 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦))(+g‘𝐾)(𝑃 ↑
((ℤRHom‘𝐾)‘𝐴)))) |
| 122 | 23 | nncnd 12282 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 123 | 122 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 ∈ ℂ) |
| 124 | 26 | nncnd 12282 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑃 ∈ ℂ) |
| 125 | 124 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ∈ ℂ) |
| 126 | 26 | nnne0d 12316 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑃 ≠ 0) |
| 127 | 126 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ≠ 0) |
| 128 | 123, 125,
127 | divcan2d 12045 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 · (𝑁 / 𝑃)) = 𝑁) |
| 129 | 128 | oveq1d 7446 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑁 ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 130 | 63 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑁 ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))) |
| 131 | 70 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)) = (𝑁 ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 132 | 130, 131 | eqtrd 2777 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑁 ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 133 | 132 | eqcomd 2743 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑁 ↑ ((𝑂‘𝐹)‘𝑦))) |
| 134 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘𝑧) = ((𝑂‘𝐹)‘𝑦)) |
| 135 | 134 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = 𝑦 → (𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = (𝑁 ↑ ((𝑂‘𝐹)‘𝑦))) |
| 136 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = 𝑦 → (𝑁 ↑ 𝑧) = (𝑁 ↑ 𝑦)) |
| 137 | 136 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦))) |
| 138 | 135, 137 | eqeq12d 2753 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 𝑦 → ((𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)) ↔ (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)))) |
| 139 | 6 | ply1crng 22200 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝐾 ∈ CRing → 𝑆 ∈ CRing) |
| 140 | 10, 139 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝑆 ∈ CRing) |
| 141 | 140 | crnggrpd 20244 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 𝑆 ∈ Grp) |
| 142 | 44, 6, 8 | vr1cl 22219 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝐾 ∈ Ring → 𝑋 ∈ 𝐵) |
| 143 | 47, 142 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| 144 | 6, 46, 7, 8 | ply1sclcl 22289 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝐾 ∈ Ring ∧
((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾)) → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵) |
| 145 | 47, 55, 144 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵) |
| 146 | 141, 143,
145 | 3jca 1129 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)) |
| 147 | 8, 58 | grpcl 18959 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵) |
| 148 | 146, 147 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵) |
| 149 | 1 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))) |
| 150 | 149 | eleq1d 2826 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝐹 ∈ 𝐵 ↔ (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵)) |
| 151 | 148, 150 | mpbird 257 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝐹 ∈ 𝐵) |
| 152 | 20, 151, 23 | aks6d1c1p1 42108 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑁 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)))) |
| 153 | 21, 152 | mpbid 232 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦))) |
| 154 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = 𝑧 → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘𝐹)‘𝑧)) |
| 155 | 154 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = 𝑧 → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑁 ↑ ((𝑂‘𝐹)‘𝑧))) |
| 156 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = 𝑧 → (𝑁 ↑ 𝑦) = (𝑁 ↑ 𝑧)) |
| 157 | 156 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = 𝑧 → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧))) |
| 158 | 155, 157 | eqeq12d 2753 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑧 → ((𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) ↔ (𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)))) |
| 159 | 158 | cbvralvw 3237 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑦 ∈
(𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) ↔ ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧))) |
| 160 | 153, 159 | sylib 218 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧))) |
| 161 | 160 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧))) |
| 162 | | simpr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (𝑉 PrimRoots 𝑅)) |
| 163 | 138, 161,
162 | rspcdva 3623 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦))) |
| 164 | 3 | fveq1d 6908 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁 ↑ 𝑦))) |
| 165 | 23 | nnnn0d 12587 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 166 | 165 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 ∈
ℕ0) |
| 167 | 12, 13, 18, 166, 37 | mulgnn0cld 19113 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ 𝑦) ∈ (Base‘𝑉)) |
| 168 | 167, 42 | eleqtrd 2843 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ 𝑦) ∈ (Base‘𝐾)) |
| 169 | 143 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑋 ∈ 𝐵) |
| 170 | 5, 44, 7, 6, 8, 11,
168 | evl1vard 22341 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝑁 ↑ 𝑦)) = (𝑁 ↑ 𝑦))) |
| 171 | 170 | simprd 495 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝑋)‘(𝑁 ↑ 𝑦)) = (𝑁 ↑ 𝑦)) |
| 172 | 169, 171 | jca 511 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝑁 ↑ 𝑦)) = (𝑁 ↑ 𝑦))) |
| 173 | 145 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵) |
| 174 | 5, 6, 7, 46, 8, 11, 56, 168 | evl1scad 22339 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑁 ↑ 𝑦)) = ((ℤRHom‘𝐾)‘𝐴))) |
| 175 | 174 | simprd 495 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑁 ↑ 𝑦)) = ((ℤRHom‘𝐾)‘𝐴)) |
| 176 | 173, 175 | jca 511 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑁 ↑ 𝑦)) = ((ℤRHom‘𝐾)‘𝐴))) |
| 177 | 5, 6, 7, 8, 11, 168, 172, 176, 58, 59 | evl1addd 22345 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵 ∧ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁 ↑ 𝑦)) = ((𝑁 ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 178 | 177 | simprd 495 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁 ↑ 𝑦)) = ((𝑁 ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) |
| 179 | 164, 178 | eqtrd 2777 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) = ((𝑁 ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) |
| 180 | 163, 179 | eqtrd 2777 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑁 ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) |
| 181 | 133, 180 | eqtrd 2777 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑁 ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) |
| 182 | 129, 181 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑁 ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) |
| 183 | 128 | eqcomd 2743 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 = (𝑃 · (𝑁 / 𝑃))) |
| 184 | 183 | oveq1d 7446 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ 𝑦) = ((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦)) |
| 185 | 184, 117 | oveq12d 7449 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = (((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦)(+g‘𝐾)(𝑃 ↑
((ℤRHom‘𝐾)‘𝐴)))) |
| 186 | 182, 185 | eqtr2d 2778 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦)(+g‘𝐾)(𝑃 ↑
((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 · (𝑁 / 𝑃)) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 187 | 66, 88 | eleqtrdi 2851 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘(mulGrp‘𝐾))) |
| 188 | 95, 31, 187 | 3jca 1129 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ∈ ℕ0 ∧ (𝑁 / 𝑃) ∈ ℕ0 ∧ 𝑦 ∈
(Base‘(mulGrp‘𝐾)))) |
| 189 | | eqid 2737 |
. . . . . . . . . . . . . . . 16
⊢
(Base‘(mulGrp‘𝐾)) = (Base‘(mulGrp‘𝐾)) |
| 190 | 189, 90 | mulgnn0ass 19128 |
. . . . . . . . . . . . . . 15
⊢
(((mulGrp‘𝐾)
∈ Mnd ∧ (𝑃 ∈
ℕ0 ∧ (𝑁 / 𝑃) ∈ ℕ0 ∧ 𝑦 ∈
(Base‘(mulGrp‘𝐾)))) → ((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦))) |
| 191 | 93, 188, 190 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦))) |
| 192 | 191 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦)(+g‘𝐾)(𝑃 ↑
((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦))(+g‘𝐾)(𝑃 ↑
((ℤRHom‘𝐾)‘𝐴)))) |
| 193 | 186, 192 | eqtr3d 2779 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦))(+g‘𝐾)(𝑃 ↑
((ℤRHom‘𝐾)‘𝐴)))) |
| 194 | 7, 59, 78, 66, 56 | grpcld 18965 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾)) |
| 195 | 194, 88 | eleqtrdi 2851 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(mulGrp‘𝐾))) |
| 196 | 95, 31, 195 | 3jca 1129 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ∈ ℕ0 ∧ (𝑁 / 𝑃) ∈ ℕ0 ∧ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(mulGrp‘𝐾)))) |
| 197 | 189, 90 | mulgnn0ass 19128 |
. . . . . . . . . . . . 13
⊢
(((mulGrp‘𝐾)
∈ Mnd ∧ (𝑃 ∈
ℕ0 ∧ (𝑁 / 𝑃) ∈ ℕ0 ∧ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(mulGrp‘𝐾)))) → ((𝑃 · (𝑁 / 𝑃)) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))) |
| 198 | 93, 196, 197 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))) |
| 199 | 193, 198 | eqtr3d 2779 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦))(+g‘𝐾)(𝑃 ↑
((ℤRHom‘𝐾)‘𝐴))) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))) |
| 200 | 121, 199 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))) |
| 201 | | id 22 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) → 𝑥 = ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 202 | 201 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ (𝑥 = ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) → (𝑃 ↑ 𝑥) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))) |
| 203 | 202 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) → (𝑃 ↑ 𝑥) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))) |
| 204 | 88, 90, 93, 31, 194 | mulgnn0cld 19113 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘𝐾)) |
| 205 | 200, 96 | eqeltrrd 2842 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) ∈ (Base‘𝐾)) |
| 206 | 83, 203, 204, 205 | fvmptd 7023 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))) |
| 207 | 206 | eqcomd 2743 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) = ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))) |
| 208 | 97, 200, 207 | 3eqtrd 2781 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))) |
| 209 | 208 | fveq2d 6910 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (◡(𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) = (◡(𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))))) |
| 210 | | f1ocnvfv1 7296 |
. . . . . . . . 9
⊢ (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘𝐾)) → (◡(𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))) = ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 211 | 77, 204, 210 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (◡(𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))) = ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 212 | 82, 209, 211 | 3eqtrd 2781 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) |
| 213 | 212 | eqcomd 2743 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ (𝑦(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) |
| 214 | 72, 213 | eqtr2d 2778 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) ↑ 𝑦)(+g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦))) |
| 215 | 62, 214 | eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)) = ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦))) |
| 216 | 215 | eqcomd 2743 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦))) |
| 217 | 216 | ralrimiva 3146 |
. 2
⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦))) |
| 218 | 20, 151, 29 | aks6d1c1p1 42108 |
. 2
⊢ (𝜑 → ((𝑁 / 𝑃) ∼ 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)))) |
| 219 | 217, 218 | mpbird 257 |
1
⊢ (𝜑 → (𝑁 / 𝑃) ∼ 𝐹) |