Theorem aks5lem5a 42647

Description: Lemma for AKS, section 5, connect to Theorem 6.1. (Contributed by metakunt, 17-Jun-2025.)

Hypotheses

Ref	Expression
aks5lema.1	⊢ (𝜑 → 𝐾 ∈ Field)
aks5lema.2	⊢ 𝑃 = (chr‘𝐾)
aks5lema.3	⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))
aks5lema.9	⊢ 𝐵 = (𝑆 /s (𝑆 ~QG 𝐿))
aks5lema.10	⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(-g‘𝑆)(1r‘𝑆))})
aks5lema.11	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks5lema.14	⊢ ∼ = {⟨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)))}
aks5lema.15	⊢ 𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))
aks5lem5a.13	⊢ (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))

Assertion

Ref	Expression
aks5lem5a	⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((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

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 731	. . . . . . 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 731	. . . . . . 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 731	. . . . . . 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 13472	. . . . . . . . . 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 12492	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝑁 ∈ ℕ0)
24		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
25	24	zncrng 21537	. . . . . . . . . . . . . . 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 42644	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)) = ((ℤRHom‘𝑆)‘𝑎))
28	27	oveq2d 7377	. . . . . . . . . . . 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 7377	. . . . . . . . . . 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 8678	. . . . . . . . . 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 7377	. . . . . . . . . . 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 8678	. . . . . . . . . 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 2776	. . . . . . . 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 42646	. . . . . 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 3130	. . . . 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 20713	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ CRing)
46	45	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝐾 ∈ CRing)
47	41, 42, 43, 44, 46, 15	ply1asclzrhval 42644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎)) = ((ℤRHom‘(Poly1‘𝐾))‘𝑎))
48	47	oveq2d 7377	. . . . . . . 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 22175	. . . . . . . . . . . . 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 20220	. . . . . . . . . . 11 ⊢ ((Poly1‘𝐾) ∈ CRing → (Poly1‘𝐾) ∈ Ring)
55	53, 54	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (Poly1‘𝐾) ∈ Ring)
56	55	ringgrpd 20217	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (Poly1‘𝐾) ∈ Grp)
57	46	crngringd 20221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝐾 ∈ Ring)
58		eqid 2737	. . . . . . . . . . 11 ⊢ (var1‘𝐾) = (var1‘𝐾)
59	58, 41, 49	vr1cl 22194	. . . . . . . . . 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 21504	. . . . . . . . . . . 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 21446	. . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤring)
64	63, 49	rhmf 20458	. . . . . . . . . . 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 7032	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((ℤRHom‘(Poly1‘𝐾))‘𝑎) ∈ (Base‘(Poly1‘𝐾)))
67	49, 50, 56, 60, 66	grpcld 18917	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((var1‘𝐾)(+g‘(Poly1‘𝐾))((ℤRHom‘(Poly1‘𝐾))‘𝑎)) ∈ (Base‘(Poly1‘𝐾)))
68	48, 67	eqeltrd 2837	. . . . . . 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 42563	. . . . 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 3150	. 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))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {csn 4568   class class class wbr 5086  {copab 5148  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  [cec 8635  1c1 11033  ℕcn 12168  ℕ₀cn0 12431  ℤcz 12518  ...cfz 13455   ∥ cdvds 16215  ℙcprime 16634  Basecbs 17173  +_gcplusg 17214   /_s cqus 17463  -_gcsg 18905  ._gcmg 19037   ~_QG cqg 19092  mulGrpcmgp 20115  1_rcur 20156  Ringcrg 20208  CRingccrg 20209   RingHom crh 20443  Fieldcfield 20701  RSpancrsp 21200  ℤ_ringczring 21439  ℤRHomczrh 21492  chrcchr 21494  ℤ/nℤczn 21495  algSccascl 21845  var₁cv1 22152  Poly₁cpl1 22153  eval₁ce1 22292   PrimRoots cprimroots 42547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110  ax-addf 11111  ax-mulf 11112

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7625  df-ofr 7626  df-om 7812  df-1st 7936  df-2nd 7937  df-supp 8105  df-tpos 8170  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-ec 8639  df-qs 8643  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-sup 9349  df-inf 9350  df-oi 9419  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-div 11802  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-uz 12783  df-rp 12937  df-fz 13456  df-fzo 13603  df-fl 13745  df-mod 13823  df-seq 13958  df-exp 14018  df-hash 14287  df-cj 15055  df-re 15056  df-im 15057  df-sqrt 15191  df-abs 15192  df-dvds 16216  df-prm 16635  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-starv 17229  df-sca 17230  df-vsca 17231  df-ip 17232  df-tset 17233  df-ple 17234  df-ds 17236  df-unif 17237  df-hom 17238  df-cco 17239  df-0g 17398  df-gsum 17399  df-prds 17404  df-pws 17406  df-imas 17466  df-qus 17467  df-mre 17542  df-mrc 17543  df-acs 17545  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-submnd 18746  df-grp 18906  df-minusg 18907  df-sbg 18908  df-mulg 19038  df-subg 19093  df-nsg 19094  df-eqg 19095  df-ghm 19182  df-cntz 19286  df-od 19497  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-srg 20162  df-ring 20210  df-cring 20211  df-oppr 20311  df-dvdsr 20331  df-rhm 20446  df-subrng 20517  df-subrg 20541  df-field 20703  df-lmod 20851  df-lss 20921  df-lsp 20961  df-sra 21163  df-rgmod 21164  df-lidl 21201  df-rsp 21202  df-2idl 21243  df-cnfld 21348  df-zring 21440  df-zrh 21496  df-chr 21498  df-zn 21499  df-assa 21846  df-asp 21847  df-ascl 21848  df-psr 21902  df-mvr 21903  df-mpl 21904  df-opsr 21906  df-evls 22065  df-evl 22066  df-psr1 22156  df-vr1 22157  df-ply1 22158  df-coe1 22159  df-evls1 22293  df-evl1 22294  df-primroots 42548

This theorem is referenced by:  aks5lem6  42648