Proof of Theorem aks5lem5a
| Step | Hyp | Ref
| Expression |
| 1 | | aks5lem5a.13 |
. 2
⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) |
| 2 | | aks5lema.1 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ Field) |
| 3 | 2 | ad3antrrr 730 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) ∧ 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) → 𝐾 ∈ Field) |
| 4 | | aks5lema.2 |
. . . . . . 7
⊢ 𝑃 = (chr‘𝐾) |
| 5 | | aks5lema.3 |
. . . . . . . 8
⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁)) |
| 6 | 5 | ad3antrrr 730 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) ∧ 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁)) |
| 7 | | aks5lema.9 |
. . . . . . 7
⊢ 𝐵 = (𝑆 /s (𝑆 ~QG 𝐿)) |
| 8 | | aks5lema.10 |
. . . . . . 7
⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(-g‘𝑆)(1r‘𝑆))}) |
| 9 | | aks5lema.11 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ ℕ) |
| 10 | 9 | ad3antrrr 730 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) ∧ 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) → 𝑅 ∈ ℕ) |
| 11 | | aks5lema.14 |
. . . . . . 7
⊢ ∼ =
{〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈
(Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))} |
| 12 | | aks5lema.15 |
. . . . . . 7
⊢ 𝑆 =
(Poly1‘(ℤ/nℤ‘𝑁)) |
| 13 | | simpr 484 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) ∧ 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) → 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) |
| 14 | | elfzelz 13564 |
. . . . . . . . . 10
⊢ (𝑎 ∈ (1...𝐴) → 𝑎 ∈ ℤ) |
| 15 | 14 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝑎 ∈ ℤ) |
| 16 | 15 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) → 𝑎 ∈ ℤ) |
| 17 | 16 | adantr 480 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) ∧ 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) → 𝑎 ∈ ℤ) |
| 18 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(algSc‘𝑆) =
(algSc‘𝑆) |
| 19 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(ℤRHom‘𝑆) = (ℤRHom‘𝑆) |
| 20 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(ℤRHom‘(ℤ/nℤ‘𝑁)) =
(ℤRHom‘(ℤ/nℤ‘𝑁)) |
| 21 | 5 | simp2d 1144 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 22 | 21 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝑁 ∈ ℕ) |
| 23 | 22 | nnnn0d 12587 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝑁 ∈
ℕ0) |
| 24 | | eqid 2737 |
. . . . . . . . . . . . . . . 16
⊢
(ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁) |
| 25 | 24 | zncrng 21563 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℕ0
→ (ℤ/nℤ‘𝑁) ∈ CRing) |
| 26 | 23, 25 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) →
(ℤ/nℤ‘𝑁) ∈ CRing) |
| 27 | 12, 18, 19, 20, 26, 15 | ply1asclzrhval 42189 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)) = ((ℤRHom‘𝑆)‘𝑎)) |
| 28 | 27 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) →
((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎))) =
((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))) |
| 29 | 28 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))) = (𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))) |
| 30 | 29 | eceq1d 8785 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎))))](𝑆
~QG 𝐿) = [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿)) |
| 31 | 30 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) → [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎))))](𝑆
~QG 𝐿) = [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿)) |
| 32 | | simpr 484 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) → [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) |
| 33 | 27 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((ℤRHom‘𝑆)‘𝑎) = ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎))) |
| 34 | 33 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)) = ((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))) |
| 35 | 34 | eceq1d 8785 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))](𝑆
~QG 𝐿)) |
| 36 | 35 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) → [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))](𝑆
~QG 𝐿)) |
| 37 | 31, 32, 36 | 3eqtrd 2781 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) → [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎))))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))](𝑆
~QG 𝐿)) |
| 38 | 37 | adantr 480 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) ∧ 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) → [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎))))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))](𝑆
~QG 𝐿)) |
| 39 | 3, 4, 6, 7, 8, 10,
11, 12, 13, 17, 38 | aks5lem4a 42191 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) ∧ 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) → (𝑁(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))‘𝑦)) = (((eval1‘𝐾)‘((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))‘(𝑁(.g‘(mulGrp‘𝐾))𝑦))) |
| 40 | 39 | ralrimiva 3146 |
. . . . 5
⊢ (((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) →
∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑁(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))‘𝑦)) = (((eval1‘𝐾)‘((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))‘(𝑁(.g‘(mulGrp‘𝐾))𝑦))) |
| 41 | | eqid 2737 |
. . . . . . . . . 10
⊢
(Poly1‘𝐾) = (Poly1‘𝐾) |
| 42 | | eqid 2737 |
. . . . . . . . . 10
⊢
(algSc‘(Poly1‘𝐾)) =
(algSc‘(Poly1‘𝐾)) |
| 43 | | eqid 2737 |
. . . . . . . . . 10
⊢
(ℤRHom‘(Poly1‘𝐾)) =
(ℤRHom‘(Poly1‘𝐾)) |
| 44 | | eqid 2737 |
. . . . . . . . . 10
⊢
(ℤRHom‘𝐾) = (ℤRHom‘𝐾) |
| 45 | 2 | fldcrngd 20742 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐾 ∈ CRing) |
| 46 | 45 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝐾 ∈ CRing) |
| 47 | 41, 42, 43, 44, 46, 15 | ply1asclzrhval 42189 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) →
((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎)) =
((ℤRHom‘(Poly1‘𝐾))‘𝑎)) |
| 48 | 47 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))) = ((var1‘𝐾)(+g‘(Poly1‘𝐾))((ℤRHom‘(Poly1‘𝐾))‘𝑎))) |
| 49 | | eqid 2737 |
. . . . . . . . 9
⊢
(Base‘(Poly1‘𝐾)) =
(Base‘(Poly1‘𝐾)) |
| 50 | | eqid 2737 |
. . . . . . . . 9
⊢
(+g‘(Poly1‘𝐾)) =
(+g‘(Poly1‘𝐾)) |
| 51 | 41 | ply1crng 22200 |
. . . . . . . . . . . . 13
⊢ (𝐾 ∈ CRing →
(Poly1‘𝐾)
∈ CRing) |
| 52 | 45, 51 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 →
(Poly1‘𝐾)
∈ CRing) |
| 53 | 52 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (Poly1‘𝐾) ∈ CRing) |
| 54 | | crngring 20242 |
. . . . . . . . . . 11
⊢
((Poly1‘𝐾) ∈ CRing →
(Poly1‘𝐾)
∈ Ring) |
| 55 | 53, 54 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (Poly1‘𝐾) ∈ Ring) |
| 56 | 55 | ringgrpd 20239 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (Poly1‘𝐾) ∈ Grp) |
| 57 | 46 | crngringd 20243 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝐾 ∈ Ring) |
| 58 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(var1‘𝐾) = (var1‘𝐾) |
| 59 | 58, 41, 49 | vr1cl 22219 |
. . . . . . . . . 10
⊢ (𝐾 ∈ Ring →
(var1‘𝐾)
∈ (Base‘(Poly1‘𝐾))) |
| 60 | 57, 59 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (var1‘𝐾) ∈
(Base‘(Poly1‘𝐾))) |
| 61 | 43 | zrhrhm 21522 |
. . . . . . . . . . . 12
⊢
((Poly1‘𝐾) ∈ Ring →
(ℤRHom‘(Poly1‘𝐾)) ∈ (ℤring RingHom
(Poly1‘𝐾))) |
| 62 | 55, 61 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) →
(ℤRHom‘(Poly1‘𝐾)) ∈ (ℤring RingHom
(Poly1‘𝐾))) |
| 63 | | zringbas 21464 |
. . . . . . . . . . . 12
⊢ ℤ =
(Base‘ℤring) |
| 64 | 63, 49 | rhmf 20485 |
. . . . . . . . . . 11
⊢
((ℤRHom‘(Poly1‘𝐾)) ∈ (ℤring RingHom
(Poly1‘𝐾))
→ (ℤRHom‘(Poly1‘𝐾)):ℤ⟶(Base‘(Poly1‘𝐾))) |
| 65 | 62, 64 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) →
(ℤRHom‘(Poly1‘𝐾)):ℤ⟶(Base‘(Poly1‘𝐾))) |
| 66 | 65, 15 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) →
((ℤRHom‘(Poly1‘𝐾))‘𝑎) ∈
(Base‘(Poly1‘𝐾))) |
| 67 | 49, 50, 56, 60, 66 | grpcld 18965 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((var1‘𝐾)(+g‘(Poly1‘𝐾))((ℤRHom‘(Poly1‘𝐾))‘𝑎)) ∈ (Base‘(Poly1‘𝐾))) |
| 68 | 48, 67 | eqeltrd 2841 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))) ∈ (Base‘(Poly1‘𝐾))) |
| 69 | 68 | adantr 480 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) →
((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))) ∈ (Base‘(Poly1‘𝐾))) |
| 70 | 22 | adantr 480 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) → 𝑁 ∈ ℕ) |
| 71 | 11, 69, 70 | aks6d1c1p1 42108 |
. . . . 5
⊢ (((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) → (𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))) ↔ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑁(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))‘𝑦)) = (((eval1‘𝐾)‘((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))‘(𝑁(.g‘(mulGrp‘𝐾))𝑦)))) |
| 72 | 40, 71 | mpbird 257 |
. . . 4
⊢ (((𝜑 ∧ 𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿)) → 𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎)))) |
| 73 | 72 | ex 412 |
. . 3
⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ([(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿) → 𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))) |
| 74 | 73 | ralimdva 3167 |
. 2
⊢ (𝜑 → (∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆
~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆
~QG 𝐿) → ∀𝑎 ∈ (1...𝐴)𝑁 ∼
((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))) |
| 75 | 1, 74 | mpd 15 |
1
⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼
((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎)))) |