Theorem aks5lem5a 42676

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 736	. . . . . . 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 736	. . . . . . 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 736	. . . . . . 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 485	. . . . . . 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 13469	. . . . . . . . . 10 ⊢ (𝑎 ∈ (1...𝐴) → 𝑎 ∈ ℤ)
15	14	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝑎 ∈ ℤ)
16	15	adantr 481	. . . . . . . 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 481	. . . . . . 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 2739	. . . . . . . . . . . . . 14 ⊢ (algSc‘𝑆) = (algSc‘𝑆)
19		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (ℤRHom‘𝑆) = (ℤRHom‘𝑆)
20		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑁)) = (ℤRHom‘(ℤ/nℤ‘𝑁))
21	5	simp2d 1149	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℕ)
22	21	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝑁 ∈ ℕ)
23	22	nnnn0d 12489	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝑁 ∈ ℕ0)
24		eqid 2739	. . . . . . . . . . . . . . . 16 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
25	24	zncrng 21519	. . . . . . . . . . . . . . 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 42673	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)) = ((ℤRHom‘𝑆)‘𝑎))
28	27	oveq2d 7372	. . . . . . . . . . . 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 7372	. . . . . . . . . . 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 8674	. . . . . . . . . 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 481	. . . . . . . . 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 485	. . . . . . . . 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 2745	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((ℤRHom‘𝑆)‘𝑎) = ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑎)))
34	33	oveq2d 7372	. . . . . . . . . . 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 8674	. . . . . . . . . 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 481	. . . . . . . . 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 2778	. . . . . . . 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 481	. . . . . . 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 42675	. . . . . 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 3131	. . . . 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 2739	. . . . . . . . . 10 ⊢ (Poly1‘𝐾) = (Poly1‘𝐾)
42		eqid 2739	. . . . . . . . . 10 ⊢ (algSc‘(Poly1‘𝐾)) = (algSc‘(Poly1‘𝐾))
43		eqid 2739	. . . . . . . . . 10 ⊢ (ℤRHom‘(Poly1‘𝐾)) = (ℤRHom‘(Poly1‘𝐾))
44		eqid 2739	. . . . . . . . . 10 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
45	2	fldcrngd 20714	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ CRing)
46	45	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝐾 ∈ CRing)
47	41, 42, 43, 44, 46, 15	ply1asclzrhval 42673	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎)) = ((ℤRHom‘(Poly1‘𝐾))‘𝑎))
48	47	oveq2d 7372	. . . . . . . 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 2739	. . . . . . . . 9 ⊢ (Base‘(Poly1‘𝐾)) = (Base‘(Poly1‘𝐾))
50		eqid 2739	. . . . . . . . 9 ⊢ (+g‘(Poly1‘𝐾)) = (+g‘(Poly1‘𝐾))
51	41	ply1crng 22183	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → (Poly1‘𝐾) ∈ CRing)
52	45, 51	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Poly1‘𝐾) ∈ CRing)
53	52	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (Poly1‘𝐾) ∈ CRing)
54		crngring 20217	. . . . . . . . . . 11 ⊢ ((Poly1‘𝐾) ∈ CRing → (Poly1‘𝐾) ∈ Ring)
55	53, 54	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (Poly1‘𝐾) ∈ Ring)
56	55	ringgrpd 20214	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → (Poly1‘𝐾) ∈ Grp)
57	46	crngringd 20218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → 𝐾 ∈ Ring)
58		eqid 2739	. . . . . . . . . . 11 ⊢ (var1‘𝐾) = (var1‘𝐾)
59	58, 41, 49	vr1cl 22202	. . . . . . . . . 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 21486	. . . . . . . . . . . 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 21428	. . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤring)
64	63, 49	rhmf 20455	. . . . . . . . . . 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 7026	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((ℤRHom‘(Poly1‘𝐾))‘𝑎) ∈ (Base‘(Poly1‘𝐾)))
67	49, 50, 56, 60, 66	grpcld 18914	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((var1‘𝐾)(+g‘(Poly1‘𝐾))((ℤRHom‘(Poly1‘𝐾))‘𝑎)) ∈ (Base‘(Poly1‘𝐾)))
68	48, 67	eqeltrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝐴)) → ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))) ∈ (Base‘(Poly1‘𝐾)))
69	68	adantr 481	. . . . . 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 481	. . . . . 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 42592	. . . . 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 258	. . . 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 413	. . 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 3151	. 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 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {csn 4555   class class class wbr 5072  {copab 5134  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  [cec 8631  1c1 11030  ℕcn 12165  ℕ₀cn0 12428  ℤcz 12515  ...cfz 13452   ∥ cdvds 16212  ℙcprime 16631  Basecbs 17170  +_gcplusg 17211   /_s cqus 17460  -_gcsg 18902  ._gcmg 19034   ~_QG cqg 19089  mulGrpcmgp 20112  1_rcur 20153  Ringcrg 20205  CRingccrg 20206   RingHom crh 20440  Fieldcfield 20702  RSpancrsp 21200  ℤ_ringczring 21421  ℤRHomczrh 21474  chrcchr 21476  ℤ/nℤczn 21477  algSccascl 21827  var₁cv1 22161  Poly₁cpl1 22162  eval₁ce1 22300   PrimRoots cprimroots 42576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107  ax-addf 11108  ax-mulf 11109

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-ofr 7621  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-ec 8635  df-qs 8639  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-sup 9345  df-inf 9346  df-oi 9415  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-rp 12934  df-fz 13453  df-fzo 13600  df-fl 13742  df-mod 13820  df-seq 13955  df-exp 14015  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-dvds 16213  df-prm 16632  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-starv 17226  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-unif 17234  df-hom 17235  df-cco 17236  df-0g 17395  df-gsum 17396  df-prds 17401  df-pws 17403  df-imas 17463  df-qus 17464  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-submnd 18743  df-grp 18903  df-minusg 18904  df-sbg 18905  df-mulg 19035  df-subg 19090  df-nsg 19091  df-eqg 19092  df-ghm 19179  df-cntz 19283  df-od 19494  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-srg 20159  df-ring 20207  df-cring 20208  df-oppr 20308  df-dvdsr 20328  df-rhm 20443  df-subrng 20518  df-subrg 20542  df-field 20704  df-lmod 20852  df-lss 20922  df-lsp 20962  df-sra 21163  df-rgmod 21164  df-lidl 21201  df-rsp 21202  df-2idl 21243  df-cnfld 21348  df-zring 21422  df-zrh 21478  df-chr 21480  df-zn 21481  df-assa 21828  df-asp 21829  df-ascl 21830  df-psr 21884  df-mvr 21885  df-mpl 21886  df-opsr 21888  df-evls 22050  df-evl 22051  df-psr1 22165  df-vr1 22166  df-ply1 22167  df-coe1 22168  df-evls1 22301  df-evl1 22302  df-primroots 42577

This theorem is referenced by:  aks5lem6  42677