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Theorem aks5lem5a 42192
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 730 . . . . . . 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 730 . . . . . . 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 730 . . . . . . 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 484 . . . . . . 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 13564 . . . . . . . . . 10 (𝑎 ∈ (1...𝐴) → 𝑎 ∈ ℤ)
1514adantl 481 . . . . . . . . 9 ((𝜑𝑎 ∈ (1...𝐴)) → 𝑎 ∈ ℤ)
1615adantr 480 . . . . . . . 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 480 . . . . . . 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 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))
215simp2d 1144 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℕ)
2221adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑎 ∈ (1...𝐴)) → 𝑁 ∈ ℕ)
2322nnnn0d 12587 . . . . . . . . . . . . . . 15 ((𝜑𝑎 ∈ (1...𝐴)) → 𝑁 ∈ ℕ0)
24 eqid 2737 . . . . . . . . . . . . . . . 16 (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
2524zncrng 21563 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) ∈ CRing)
2623, 25syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (1...𝐴)) → (ℤ/nℤ‘𝑁) ∈ CRing)
2712, 18, 19, 20, 26, 15ply1asclzrhval 42189 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (1...𝐴)) → ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)) = ((ℤRHom‘𝑆)‘𝑎))
2827oveq2d 7447 . . . . . . . . . . . 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 7447 . . . . . . . . . . 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 8785 . . . . . . . . . 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 480 . . . . . . . . 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 484 . . . . . . . . 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 2743 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (1...𝐴)) → ((ℤRHom‘𝑆)‘𝑎) = ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))
3433oveq2d 7447 . . . . . . . . . . 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 8785 . . . . . . . . . 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 480 . . . . . . . . 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 2781 . . . . . . . 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 480 . . . . . . 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 42191 . . . . . 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 3146 . . . . 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 2737 . . . . . . . . . 10 (Poly1𝐾) = (Poly1𝐾)
42 eqid 2737 . . . . . . . . . 10 (algSc‘(Poly1𝐾)) = (algSc‘(Poly1𝐾))
43 eqid 2737 . . . . . . . . . 10 (ℤRHom‘(Poly1𝐾)) = (ℤRHom‘(Poly1𝐾))
44 eqid 2737 . . . . . . . . . 10 (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
452fldcrngd 20742 . . . . . . . . . . 11 (𝜑𝐾 ∈ CRing)
4645adantr 480 . . . . . . . . . 10 ((𝜑𝑎 ∈ (1...𝐴)) → 𝐾 ∈ CRing)
4741, 42, 43, 44, 46, 15ply1asclzrhval 42189 . . . . . . . . 9 ((𝜑𝑎 ∈ (1...𝐴)) → ((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎)) = ((ℤRHom‘(Poly1𝐾))‘𝑎))
4847oveq2d 7447 . . . . . . . 8 ((𝜑𝑎 ∈ (1...𝐴)) → ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))) = ((var1𝐾)(+g‘(Poly1𝐾))((ℤRHom‘(Poly1𝐾))‘𝑎)))
49 eqid 2737 . . . . . . . . 9 (Base‘(Poly1𝐾)) = (Base‘(Poly1𝐾))
50 eqid 2737 . . . . . . . . 9 (+g‘(Poly1𝐾)) = (+g‘(Poly1𝐾))
5141ply1crng 22200 . . . . . . . . . . . . 13 (𝐾 ∈ CRing → (Poly1𝐾) ∈ CRing)
5245, 51syl 17 . . . . . . . . . . . 12 (𝜑 → (Poly1𝐾) ∈ CRing)
5352adantr 480 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (1...𝐴)) → (Poly1𝐾) ∈ CRing)
54 crngring 20242 . . . . . . . . . . 11 ((Poly1𝐾) ∈ CRing → (Poly1𝐾) ∈ Ring)
5553, 54syl 17 . . . . . . . . . 10 ((𝜑𝑎 ∈ (1...𝐴)) → (Poly1𝐾) ∈ Ring)
5655ringgrpd 20239 . . . . . . . . 9 ((𝜑𝑎 ∈ (1...𝐴)) → (Poly1𝐾) ∈ Grp)
5746crngringd 20243 . . . . . . . . . 10 ((𝜑𝑎 ∈ (1...𝐴)) → 𝐾 ∈ Ring)
58 eqid 2737 . . . . . . . . . . 11 (var1𝐾) = (var1𝐾)
5958, 41, 49vr1cl 22219 . . . . . . . . . 10 (𝐾 ∈ Ring → (var1𝐾) ∈ (Base‘(Poly1𝐾)))
6057, 59syl 17 . . . . . . . . 9 ((𝜑𝑎 ∈ (1...𝐴)) → (var1𝐾) ∈ (Base‘(Poly1𝐾)))
6143zrhrhm 21522 . . . . . . . . . . . 12 ((Poly1𝐾) ∈ Ring → (ℤRHom‘(Poly1𝐾)) ∈ (ℤring RingHom (Poly1𝐾)))
6255, 61syl 17 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (1...𝐴)) → (ℤRHom‘(Poly1𝐾)) ∈ (ℤring RingHom (Poly1𝐾)))
63 zringbas 21464 . . . . . . . . . . . 12 ℤ = (Base‘ℤring)
6463, 49rhmf 20485 . . . . . . . . . . 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 7105 . . . . . . . . 9 ((𝜑𝑎 ∈ (1...𝐴)) → ((ℤRHom‘(Poly1𝐾))‘𝑎) ∈ (Base‘(Poly1𝐾)))
6749, 50, 56, 60, 66grpcld 18965 . . . . . . . 8 ((𝜑𝑎 ∈ (1...𝐴)) → ((var1𝐾)(+g‘(Poly1𝐾))((ℤRHom‘(Poly1𝐾))‘𝑎)) ∈ (Base‘(Poly1𝐾)))
6848, 67eqeltrd 2841 . . . . . . 7 ((𝜑𝑎 ∈ (1...𝐴)) → ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))) ∈ (Base‘(Poly1𝐾)))
6968adantr 480 . . . . . 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 480 . . . . . 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 42108 . . . . 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 257 . . . 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 412 . . 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 3167 . 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 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  {csn 4626   class class class wbr 5143  {copab 5205  wf 6557  cfv 6561  (class class class)co 7431  [cec 8743  1c1 11156  cn 12266  0cn0 12526  cz 12613  ...cfz 13547  cdvds 16290  cprime 16708  Basecbs 17247  +gcplusg 17297   /s cqus 17550  -gcsg 18953  .gcmg 19085   ~QG cqg 19140  mulGrpcmgp 20137  1rcur 20178  Ringcrg 20230  CRingccrg 20231   RingHom crh 20469  Fieldcfield 20730  RSpancrsp 21217  ringczring 21457  ℤRHomczrh 21510  chrcchr 21512  ℤ/nczn 21513  algSccascl 21872  var1cv1 22177  Poly1cpl1 22178  eval1ce1 22318   PrimRoots cprimroots 42092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234  ax-mulf 11235
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-dvds 16291  df-prm 16709  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-imas 17553  df-qus 17554  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-nsg 19142  df-eqg 19143  df-ghm 19231  df-cntz 19335  df-od 19546  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-oppr 20334  df-dvdsr 20357  df-rhm 20472  df-subrng 20546  df-subrg 20570  df-field 20732  df-lmod 20860  df-lss 20930  df-lsp 20970  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-rsp 21219  df-2idl 21260  df-cnfld 21365  df-zring 21458  df-zrh 21514  df-chr 21516  df-zn 21517  df-assa 21873  df-asp 21874  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-evls 22098  df-evl 22099  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-evls1 22319  df-evl1 22320  df-primroots 42093
This theorem is referenced by:  aks5lem6  42193
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