Users' Mathboxes Mathbox for metakunt < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aks5lem5a Structured version   Visualization version   GIF version

Theorem aks5lem5a 41903
Description: Lemma for AKS, section 5, connect to Theorem 6.1. (Contributed by metakunt, 17-Jun-2025.)
Hypotheses
Ref Expression
aks5lema.1 (𝜑𝐾 ∈ Field)
aks5lema.2 𝑃 = (chr‘𝐾)
aks5lema.3 (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃𝑁))
aks5lema.9 𝐵 = (𝑆 /s (𝑆 ~QG 𝐿))
aks5lema.10 𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(-g𝑆)(1r𝑆))})
aks5lema.11 (𝜑𝑅 ∈ ℕ)
aks5lema.14 = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘𝑓)‘𝑦)) = (((eval1𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}
aks5lema.15 𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))
aks5lem5a.13 (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))
Assertion
Ref Expression
aks5lem5a (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
Distinct variable groups:   𝑦,𝐴   𝐵,𝑒   𝑒,𝐾,𝑓,𝑦   𝑦,𝐿   𝑒,𝑁,𝑓,𝑦   𝑅,𝑒,𝑓   𝑦,𝑆   𝑒,𝑎,𝑓,𝑦   𝜑,𝑎,𝑦
Allowed substitution hints:   𝜑(𝑒,𝑓)   𝐴(𝑒,𝑓,𝑎)   𝐵(𝑦,𝑓,𝑎)   𝑃(𝑦,𝑒,𝑓,𝑎)   (𝑦,𝑒,𝑓,𝑎)   𝑅(𝑦,𝑎)   𝑆(𝑒,𝑓,𝑎)   𝐾(𝑎)   𝐿(𝑒,𝑓,𝑎)   𝑁(𝑎)

Proof of Theorem aks5lem5a
StepHypRef Expression
1 aks5lem5a.13 . 2 (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))
2 aks5lema.1 . . . . . . . 8 (𝜑𝐾 ∈ Field)
32ad3antrrr 728 . . . . . . 7 ((((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((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 ∧ 𝑃𝑁))
65ad3antrrr 728 . . . . . . 7 ((((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) ∧ 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃𝑁))
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)))(-g𝑆)(1r𝑆))})
9 aks5lema.11 . . . . . . . 8 (𝜑𝑅 ∈ ℕ)
109ad3antrrr 728 . . . . . . 7 ((((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) ∧ 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) → 𝑅 ∈ ℕ)
11 aks5lema.14 . . . . . . 7 = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘𝑓)‘𝑦)) = (((eval1𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}
12 aks5lema.15 . . . . . . 7 𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))
13 simpr 483 . . . . . . 7 ((((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) ∧ 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) → 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
14 elfzelz 13549 . . . . . . . . . 10 (𝑎 ∈ (1...𝐴) → 𝑎 ∈ ℤ)
1514adantl 480 . . . . . . . . 9 ((𝜑𝑎 ∈ (1...𝐴)) → 𝑎 ∈ ℤ)
1615adantr 479 . . . . . . . 8 (((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) → 𝑎 ∈ ℤ)
1716adantr 479 . . . . . . 7 ((((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) ∧ 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) → 𝑎 ∈ ℤ)
18 eqid 2726 . . . . . . . . . . . . . 14 (algSc‘𝑆) = (algSc‘𝑆)
19 eqid 2726 . . . . . . . . . . . . . 14 (ℤRHom‘𝑆) = (ℤRHom‘𝑆)
20 eqid 2726 . . . . . . . . . . . . . 14 (ℤRHom‘(ℤ/nℤ‘𝑁)) = (ℤRHom‘(ℤ/nℤ‘𝑁))
215simp2d 1140 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℕ)
2221adantr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝑎 ∈ (1...𝐴)) → 𝑁 ∈ ℕ)
2322nnnn0d 12578 . . . . . . . . . . . . . . 15 ((𝜑𝑎 ∈ (1...𝐴)) → 𝑁 ∈ ℕ0)
24 eqid 2726 . . . . . . . . . . . . . . . 16 (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
2524zncrng 21538 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) ∈ CRing)
2623, 25syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (1...𝐴)) → (ℤ/nℤ‘𝑁) ∈ CRing)
2712, 18, 19, 20, 26, 15ply1asclzrhval 41900 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (1...𝐴)) → ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)) = ((ℤRHom‘𝑆)‘𝑎))
2827oveq2d 7432 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (1...𝐴)) → ((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎))) = ((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))
2928oveq2d 7432 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (1...𝐴)) → (𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))) = (𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))))
3029eceq1d 8766 . . . . . . . . . 10 ((𝜑𝑎 ∈ (1...𝐴)) → [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎))))](𝑆 ~QG 𝐿) = [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿))
3130adantr 479 . . . . . . . . 9 (((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) → [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎))))](𝑆 ~QG 𝐿) = [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿))
32 simpr 483 . . . . . . . . 9 (((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) → [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))
3327eqcomd 2732 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (1...𝐴)) → ((ℤRHom‘𝑆)‘𝑎) = ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))
3433oveq2d 7432 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (1...𝐴)) → ((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)) = ((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎))))
3534eceq1d 8766 . . . . . . . . . 10 ((𝜑𝑎 ∈ (1...𝐴)) → [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))](𝑆 ~QG 𝐿))
3635adantr 479 . . . . . . . . 9 (((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) → [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))](𝑆 ~QG 𝐿))
3731, 32, 363eqtrd 2770 . . . . . . . 8 (((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) → [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎))))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))](𝑆 ~QG 𝐿))
3837adantr 479 . . . . . . 7 ((((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) ∧ 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) → [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎))))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))](𝑆 ~QG 𝐿))
393, 4, 6, 7, 8, 10, 11, 12, 13, 17, 38aks5lem4a 41902 . . . . . 6 ((((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) ∧ 𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) → (𝑁(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))))‘𝑦)) = (((eval1𝐾)‘((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))))‘(𝑁(.g‘(mulGrp‘𝐾))𝑦)))
4039ralrimiva 3136 . . . . 5 (((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) → ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑁(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))))‘𝑦)) = (((eval1𝐾)‘((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))))‘(𝑁(.g‘(mulGrp‘𝐾))𝑦)))
41 eqid 2726 . . . . . . . . . 10 (Poly1𝐾) = (Poly1𝐾)
42 eqid 2726 . . . . . . . . . 10 (algSc‘(Poly1𝐾)) = (algSc‘(Poly1𝐾))
43 eqid 2726 . . . . . . . . . 10 (ℤRHom‘(Poly1𝐾)) = (ℤRHom‘(Poly1𝐾))
44 eqid 2726 . . . . . . . . . 10 (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
452fldcrngd 20716 . . . . . . . . . . 11 (𝜑𝐾 ∈ CRing)
4645adantr 479 . . . . . . . . . 10 ((𝜑𝑎 ∈ (1...𝐴)) → 𝐾 ∈ CRing)
4741, 42, 43, 44, 46, 15ply1asclzrhval 41900 . . . . . . . . 9 ((𝜑𝑎 ∈ (1...𝐴)) → ((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎)) = ((ℤRHom‘(Poly1𝐾))‘𝑎))
4847oveq2d 7432 . . . . . . . 8 ((𝜑𝑎 ∈ (1...𝐴)) → ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))) = ((var1𝐾)(+g‘(Poly1𝐾))((ℤRHom‘(Poly1𝐾))‘𝑎)))
49 eqid 2726 . . . . . . . . 9 (Base‘(Poly1𝐾)) = (Base‘(Poly1𝐾))
50 eqid 2726 . . . . . . . . 9 (+g‘(Poly1𝐾)) = (+g‘(Poly1𝐾))
5141ply1crng 22184 . . . . . . . . . . . . 13 (𝐾 ∈ CRing → (Poly1𝐾) ∈ CRing)
5245, 51syl 17 . . . . . . . . . . . 12 (𝜑 → (Poly1𝐾) ∈ CRing)
5352adantr 479 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (1...𝐴)) → (Poly1𝐾) ∈ CRing)
54 crngring 20224 . . . . . . . . . . 11 ((Poly1𝐾) ∈ CRing → (Poly1𝐾) ∈ Ring)
5553, 54syl 17 . . . . . . . . . 10 ((𝜑𝑎 ∈ (1...𝐴)) → (Poly1𝐾) ∈ Ring)
5655ringgrpd 20221 . . . . . . . . 9 ((𝜑𝑎 ∈ (1...𝐴)) → (Poly1𝐾) ∈ Grp)
5746crngringd 20225 . . . . . . . . . 10 ((𝜑𝑎 ∈ (1...𝐴)) → 𝐾 ∈ Ring)
58 eqid 2726 . . . . . . . . . . 11 (var1𝐾) = (var1𝐾)
5958, 41, 49vr1cl 22203 . . . . . . . . . 10 (𝐾 ∈ Ring → (var1𝐾) ∈ (Base‘(Poly1𝐾)))
6057, 59syl 17 . . . . . . . . 9 ((𝜑𝑎 ∈ (1...𝐴)) → (var1𝐾) ∈ (Base‘(Poly1𝐾)))
6143zrhrhm 21497 . . . . . . . . . . . 12 ((Poly1𝐾) ∈ Ring → (ℤRHom‘(Poly1𝐾)) ∈ (ℤring RingHom (Poly1𝐾)))
6255, 61syl 17 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (1...𝐴)) → (ℤRHom‘(Poly1𝐾)) ∈ (ℤring RingHom (Poly1𝐾)))
63 zringbas 21439 . . . . . . . . . . . 12 ℤ = (Base‘ℤring)
6463, 49rhmf 20463 . . . . . . . . . . 11 ((ℤRHom‘(Poly1𝐾)) ∈ (ℤring RingHom (Poly1𝐾)) → (ℤRHom‘(Poly1𝐾)):ℤ⟶(Base‘(Poly1𝐾)))
6562, 64syl 17 . . . . . . . . . 10 ((𝜑𝑎 ∈ (1...𝐴)) → (ℤRHom‘(Poly1𝐾)):ℤ⟶(Base‘(Poly1𝐾)))
6665, 15ffvelcdmd 7091 . . . . . . . . 9 ((𝜑𝑎 ∈ (1...𝐴)) → ((ℤRHom‘(Poly1𝐾))‘𝑎) ∈ (Base‘(Poly1𝐾)))
6749, 50, 56, 60, 66grpcld 18937 . . . . . . . 8 ((𝜑𝑎 ∈ (1...𝐴)) → ((var1𝐾)(+g‘(Poly1𝐾))((ℤRHom‘(Poly1𝐾))‘𝑎)) ∈ (Base‘(Poly1𝐾)))
6848, 67eqeltrd 2826 . . . . . . 7 ((𝜑𝑎 ∈ (1...𝐴)) → ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))) ∈ (Base‘(Poly1𝐾)))
6968adantr 479 . . . . . 6 (((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) → ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))) ∈ (Base‘(Poly1𝐾)))
7022adantr 479 . . . . . 6 (((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) → 𝑁 ∈ ℕ)
7111, 69, 70aks6d1c1p1 41819 . . . . 5 (((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) → (𝑁 ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))) ↔ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑁(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))))‘𝑦)) = (((eval1𝐾)‘((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))))‘(𝑁(.g‘(mulGrp‘𝐾))𝑦))))
7240, 71mpbird 256 . . . 4 (((𝜑𝑎 ∈ (1...𝐴)) ∧ [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿)) → 𝑁 ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
7372ex 411 . . 3 ((𝜑𝑎 ∈ (1...𝐴)) → ([(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿) → 𝑁 ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎)))))
7473ralimdva 3157 . 2 (𝜑 → (∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿) → ∀𝑎 ∈ (1...𝐴)𝑁 ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎)))))
751, 74mpd 15 1 (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  {csn 4623   class class class wbr 5145  {copab 5207  wf 6542  cfv 6546  (class class class)co 7416  [cec 8724  1c1 11150  cn 12258  0cn0 12518  cz 12604  ...cfz 13532  cdvds 16251  cprime 16667  Basecbs 17208  +gcplusg 17261   /s cqus 17515  -gcsg 18925  .gcmg 19057   ~QG cqg 19112  mulGrpcmgp 20113  1rcur 20160  Ringcrg 20212  CRingccrg 20213   RingHom crh 20447  Fieldcfield 20704  RSpancrsp 21192  ringczring 21432  ℤRHomczrh 21485  chrcchr 21487  ℤ/nczn 21488  algSccascl 21846  var1cv1 22161  Poly1cpl1 22162  eval1ce1 22302   PrimRoots cprimroots 41803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226  ax-pre-sup 11227  ax-addf 11228  ax-mulf 11229
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-iin 4996  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-isom 6555  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-ofr 7683  df-om 7869  df-1st 7995  df-2nd 7996  df-supp 8167  df-tpos 8233  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-er 8726  df-ec 8728  df-qs 8732  df-map 8849  df-pm 8850  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9399  df-sup 9478  df-inf 9479  df-oi 9546  df-card 9975  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-div 11913  df-nn 12259  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12519  df-z 12605  df-dec 12724  df-uz 12869  df-rp 13023  df-fz 13533  df-fzo 13676  df-fl 13806  df-mod 13884  df-seq 14016  df-exp 14076  df-hash 14343  df-cj 15099  df-re 15100  df-im 15101  df-sqrt 15235  df-abs 15236  df-dvds 16252  df-prm 16668  df-struct 17144  df-sets 17161  df-slot 17179  df-ndx 17191  df-base 17209  df-ress 17238  df-plusg 17274  df-mulr 17275  df-starv 17276  df-sca 17277  df-vsca 17278  df-ip 17279  df-tset 17280  df-ple 17281  df-ds 17283  df-unif 17284  df-hom 17285  df-cco 17286  df-0g 17451  df-gsum 17452  df-prds 17457  df-pws 17459  df-imas 17518  df-qus 17519  df-mre 17594  df-mrc 17595  df-acs 17597  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-mhm 18768  df-submnd 18769  df-grp 18926  df-minusg 18927  df-sbg 18928  df-mulg 19058  df-subg 19113  df-nsg 19114  df-eqg 19115  df-ghm 19203  df-cntz 19307  df-od 19522  df-cmn 19776  df-abl 19777  df-mgp 20114  df-rng 20132  df-ur 20161  df-srg 20166  df-ring 20214  df-cring 20215  df-oppr 20312  df-dvdsr 20335  df-rhm 20450  df-subrng 20524  df-subrg 20549  df-field 20706  df-lmod 20834  df-lss 20905  df-lsp 20945  df-sra 21147  df-rgmod 21148  df-lidl 21193  df-rsp 21194  df-2idl 21235  df-cnfld 21340  df-zring 21433  df-zrh 21489  df-chr 21491  df-zn 21492  df-assa 21847  df-asp 21848  df-ascl 21849  df-psr 21902  df-mvr 21903  df-mpl 21904  df-opsr 21906  df-evls 22083  df-evl 22084  df-psr1 22165  df-vr1 22166  df-ply1 22167  df-coe1 22168  df-evls1 22303  df-evl1 22304  df-primroots 41804
This theorem is referenced by:  aks5lem6  41904
  Copyright terms: Public domain W3C validator