Theorem aks5lem5a 43507

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 743	. . . . . . 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 743	. . . . . . 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 743	. . . . . . 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 490	. . . . . . 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 13696	. . . . . . . . . 10 ⊢ (𝑎 ∈ (1...𝐴) → 𝑎 ∈ ℤ)
15	14	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝑎 ∈ ℤ)
16	15	adantr 486	. . . . . . . 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 486	. . . . . . 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 2763	. . . . . . . . . . . . . 14 ⊢ (algSc‘𝑆) = (algSc‘𝑆)
19		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (ℤRHom‘𝑆) = (ℤRHom‘𝑆)
20		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑁)) = (ℤRHom‘(ℤ/nℤ‘𝑁))
21	5	simp2d 1161	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℕ)
22	21	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝑁 ∈ ℕ)
23	22	nnnn0d 12707	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝑁 ∈ ℕ0)
24		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
25	24	zncrng 21905	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) ∈ CRing)
26	23, 25	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (ℤ/nℤ‘𝑁) ∈ CRing)
27	12, 18, 19, 20, 26, 15	ply1asclzrhval 43504	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)) = ((ℤRHom‘𝑆)‘𝑎))
28	27	oveq2d 7467	. . . . . . . . . . . 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 7467	. . . . . . . . . . 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 8796	. . . . . . . . . 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 486	. . . . . . . . 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 490	. . . . . . . . 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 2769	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((ℤRHom‘𝑆)‘𝑎) = ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))
34	33	oveq2d 7467	. . . . . . . . . . 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 8796	. . . . . . . . . 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 486	. . . . . . . . 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 2802	. . . . . . . 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 486	. . . . . . 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 43506	. . . . . 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 3158	. . . . 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 2763	. . . . . . . . . 10 ⊢ (Poly1‘𝐾) = (Poly1‘𝐾)
42		eqid 2763	. . . . . . . . . 10 ⊢ (algSc‘(Poly1‘𝐾)) = (algSc‘(Poly1‘𝐾))
43		eqid 2763	. . . . . . . . . 10 ⊢ (ℤRHom‘(Poly1‘𝐾)) = (ℤRHom‘(Poly1‘𝐾))
44		eqid 2763	. . . . . . . . . 10 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
45	2	fldcrngd 21046	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ CRing)
46	45	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝐾 ∈ CRing)
47	41, 42, 43, 44, 46, 15	ply1asclzrhval 43504	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎)) = ((ℤRHom‘(Poly1‘𝐾))‘𝑎))
48	47	oveq2d 7467	. . . . . . . 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 2763	. . . . . . . . 9 ⊢ (Base‘(Poly1‘𝐾)) = (Base‘(Poly1‘𝐾))
50		eqid 2763	. . . . . . . . 9 ⊢ (+g‘(Poly1‘𝐾)) = (+g‘(Poly1‘𝐾))
51	41	ply1crng 22571	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → (Poly1‘𝐾) ∈ CRing)
52	45, 51	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (Poly1‘𝐾) ∈ CRing)
53	52	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (Poly1‘𝐾) ∈ CRing)
54		crngring 20523	. . . . . . . . . . 11 ⊢ ((Poly1‘𝐾) ∈ CRing → (Poly1‘𝐾) ∈ Ring)
55	53, 54	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (Poly1‘𝐾) ∈ Ring)
56	55	ringgrpd 20520	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (Poly1‘𝐾) ∈ Grp)
57	46	crngringd 20524	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝐾 ∈ Ring)
58		eqid 2763	. . . . . . . . . . 11 ⊢ (var1‘𝐾) = (var1‘𝐾)
59	58, 41, 49	vr1cl 22590	. . . . . . . . . 10 ⊢ (𝐾 ∈ Ring → (var1‘𝐾) ∈ (Base‘(Poly1‘𝐾)))
60	57, 59	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (var1‘𝐾) ∈ (Base‘(Poly1‘𝐾)))
61	43	zrhrhm 21872	. . . . . . . . . . . 12 ⊢ ((Poly1‘𝐾) ∈ Ring → (ℤRHom‘(Poly1‘𝐾)) ∈ (ℤring RingHom (Poly1‘𝐾)))
62	55, 61	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (ℤRHom‘(Poly1‘𝐾)) ∈ (ℤring RingHom (Poly1‘𝐾)))
63		zringbas 21813	. . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤring)
64	63, 49	rhmf 20766	. . . . . . . . . . 11 ⊢ ((ℤRHom‘(Poly1‘𝐾)) ∈ (ℤring RingHom (Poly1‘𝐾)) → (ℤRHom‘(Poly1‘𝐾)):ℤ⟶(Base‘(Poly1‘𝐾)))
65	62, 64	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (ℤRHom‘(Poly1‘𝐾)):ℤ⟶(Base‘(Poly1‘𝐾)))
66	65, 15	ffvelcdmd 7117	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((ℤRHom‘(Poly1‘𝐾))‘𝑎) ∈ (Base‘(Poly1‘𝐾)))
67	49, 50, 56, 60, 66	grpcld 19208	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((var1‘𝐾)(+g‘(Poly1‘𝐾))((ℤRHom‘(Poly1‘𝐾))‘𝑎)) ∈ (Base‘(Poly1‘𝐾)))
68	48, 67	eqeltrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))) ∈ (Base‘(Poly1‘𝐾)))
69	68	adantr 486	. . . . . 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 486	. . . . . 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 43423	. . . . 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 260	. . . 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 418	. . 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 3178	. 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 16	1 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {csn 4588   class class class wbr 5108  {copab 5173  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  [cec 8753  1c1 11240  ℕcn 12375  ℕ₀cn0 12646  ℤcz 12733  ...cfz 13679   ∥ cdvds 16466  ℙcprime 16890  Basecbs 17431  +_gcplusg 17472   /_s cqus 17727  -_gcsg 19196  ._gcmg 19327   ~_QG cqg 19382  mulGrpcmgp 20411  1_rcur 20458  Ringcrg 20510  CRingccrg 20511   RingHom crh 20750  Fieldcfield 21032  RSpancrsp 21536  ℤ_ringczring 21806  ℤRHomczrh 21860  chrcchr 21862  ℤ/nℤczn 21863  algSccascl 22215  var₁cv1 22549  Poly₁cpl1 22550  eval₁ce1 22687   PrimRoots cprimroots 43407

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317  ax-addf 11318  ax-mulf 11319

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725  df-ofr 7726  df-om 7912  df-1st 8035  df-2nd 8036  df-supp 8215  df-tpos 8281  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-ec 8757  df-qs 8761  df-map 8887  df-pm 8888  df-ixp 8964  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-fsupp 9392  df-sup 9471  df-inf 9472  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-rp 13161  df-fz 13680  df-fzo 13829  df-fl 13972  df-mod 14050  df-seq 14185  df-exp 14245  df-hash 14516  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-dvds 16467  df-prm 16891  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-mulr 17487  df-starv 17488  df-sca 17489  df-vsca 17490  df-ip 17491  df-tset 17492  df-ple 17493  df-ds 17495  df-unif 17496  df-hom 17497  df-cco 17498  df-euf 17499  df-0g 17662  df-gsum 17663  df-prds 17668  df-pws 17670  df-imas 17730  df-qus 17731  df-mre 17806  df-mrc 17807  df-acs 17809  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-mhm 19028  df-submnd 19029  df-grp 19197  df-minusg 19198  df-sbg 19199  df-mulg 19328  df-subg 19383  df-nsg 19384  df-eqg 19385  df-ghm 19478  df-cntz 19582  df-od 19793  df-cmn 20047  df-abl 20048  df-mgp 20412  df-rng 20426  df-ur 20459  df-srg 20464  df-ring 20512  df-cring 20513  df-oppr 20618  df-dvdsr 20638  df-rhm 20753  df-subrng 20849  df-subrg 20873  df-field 21034  df-lmod 21188  df-lss 21258  df-lsp 21298  df-sra 21499  df-rgmod 21500  df-lidl 21537  df-rsp 21538  df-2idl 21594  df-cnfld 21730  df-zring 21807  df-zrh 21864  df-chr 21866  df-zn 21867  df-assa 22216  df-asp 22217  df-ascl 22218  df-psr 22272  df-mvr 22273  df-mpl 22274  df-opsr 22276  df-evls 22438  df-evl 22439  df-psr1 22553  df-vr1 22554  df-ply1 22555  df-coe1 22556  df-evls1 22688  df-evl1 22689  df-primroots 43408

This theorem is used by:  aks5lem6  43508