aks6d1c1p3 - Mathbox for metakunt

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Theorem aks6d1c1p3 42509

Description: In a field with a Frobenius isomorphism (read: algebraic closure or finite field), 𝑁 and linear factors are introspective. (Contributed by metakunt, 25-Apr-2025.)

**Hypotheses**
Ref	Expression
aks6d1c1p3.1	⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}
aks6d1c1p3.2	⊢ 𝑆 = (Poly₁‘𝐾)
aks6d1c1p3.3	⊢ 𝐵 = (Base‘𝑆)
aks6d1c1p3.4	⊢ 𝑋 = (var₁‘𝐾)
aks6d1c1p3.5	⊢ 𝑊 = (mulGrp‘𝑆)
aks6d1c1p3.6	⊢ 𝑉 = (mulGrp‘𝐾)
aks6d1c1p3.7	⊢ ↑ = (._g‘𝑉)
aks6d1c1p3.8	⊢ 𝐶 = (algSc‘𝑆)
aks6d1c1p3.9	⊢ 𝐷 = (._g‘𝑊)
aks6d1c1p3.10	⊢ 𝑃 = (chr‘𝐾)
aks6d1c1p3.11	⊢ 𝑂 = (eval₁‘𝐾)
aks6d1c1p3.12	⊢ + = (+_g‘𝑆)
aks6d1c1p3.13	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c1p3.14	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c1p3.15	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c1p3.16	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c1p3.17	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c1p3.18	⊢ 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))
aks6d1c1p3.19	⊢ (𝜑 → 𝐴 ∈ ℤ)
aks6d1c1p3.20	⊢ (𝜑 → 𝑁 ∼ 𝐹)
aks6d1c1p3.21	⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingIso 𝐾))

**Assertion**
Ref	Expression
aks6d1c1p3	⊢ (𝜑 → (𝑁 / 𝑃) ∼ 𝐹)

Distinct variable groups: ↑ ,𝑒,𝑓,𝑦 𝑥, ↑ ,𝑦 𝑥,𝐴 𝐵,𝑒,𝑓 𝑒,𝐹,𝑓,𝑦 𝑥,𝐾 𝑒,𝑁,𝑓,𝑦 𝑥,𝑁 𝑒,𝑂,𝑓,𝑦 𝑃,𝑒,𝑓,𝑦 𝑥,𝑃 𝑅,𝑒,𝑓,𝑦 𝑥,𝑅 𝑒,𝑉,𝑓,𝑦 𝑥,𝑉 𝜑,𝑦,𝑥 Allowed substitution hints: 𝜑(𝑒,𝑓) 𝐴(𝑦,𝑒,𝑓) 𝐵(𝑥,𝑦) 𝐶(𝑥,𝑦,𝑒,𝑓) 𝐷(𝑥,𝑦,𝑒,𝑓) + (𝑥,𝑦,𝑒,𝑓) ∼ (𝑥,𝑦,𝑒,𝑓) 𝑆(𝑥,𝑦,𝑒,𝑓) 𝐹(𝑥) 𝐾(𝑦,𝑒,𝑓) 𝑂(𝑥) 𝑊(𝑥,𝑦,𝑒,𝑓) 𝑋(𝑥,𝑦,𝑒,𝑓)

Proof of Theorem aks6d1c1p3 Dummy variables 𝑧 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aks6d1c1p3.18	. . . . . . . . 9 ⊢ 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))
2	1	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))
3	2	fveq2d 6848	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘𝐹) = (𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))))
4	3	fveq1d 6846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘((𝑁 / 𝑃) ↑ 𝑦)))
5		aks6d1c1p3.11	. . . . . . . 8 ⊢ 𝑂 = (eval₁‘𝐾)
6		aks6d1c1p3.2	. . . . . . . 8 ⊢ 𝑆 = (Poly₁‘𝐾)
7		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
8		aks6d1c1p3.3	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑆)
9		aks6d1c1p3.13	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Field)
10	9	fldcrngd 20692	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ CRing)
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ CRing)
12		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑉) = (Base‘𝑉)
13		aks6d1c1p3.7	. . . . . . . . . 10 ⊢ ↑ = (._g‘𝑉)
14		aks6d1c1p3.6	. . . . . . . . . . . . . 14 ⊢ 𝑉 = (mulGrp‘𝐾)
15	14	crngmgp 20193	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
16	10, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ∈ CMnd)
17	16	cmnmndd 19750	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ Mnd)
18	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑉 ∈ Mnd)
19		aks6d1c1p3.17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
20		aks6d1c1p3.1	. . . . . . . . . . . . . . . 16 ⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}
21		aks6d1c1p3.20	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∼ 𝐹)
22	20, 21	aks6d1c1p1rcl 42507	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
23	22	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
24		aks6d1c1p3.14	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℙ)
25		prmnn 16615	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
26	24, 25	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ)
27		nndivdvds 16202	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℕ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℕ))
28	23, 26, 27	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℕ))
29	19, 28	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℕ)
30	29	nnnn0d 12476	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℕ₀)
31	30	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 / 𝑃) ∈ ℕ₀)
32		aks6d1c1p3.15	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ ℕ)
33	32	nnnn0d 12476	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
34	16, 33, 13	isprimroot 42492	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) ↔ (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0_g‘𝑉) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙 ↑ 𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑙))))
35	34	biimpd 229	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0_g‘𝑉) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙 ↑ 𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑙))))
36	35	imp 406	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0_g‘𝑉) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙 ↑ 𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑙)))
37	36	simp1d 1143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
38	12, 13, 18, 31, 37	mulgnn0cld 19042	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ 𝑦) ∈ (Base‘𝑉))
39	14, 7	mgpbas 20097	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝑉)
40	39	eqcomi 2746	. . . . . . . . . . 11 ⊢ (Base‘𝑉) = (Base‘𝐾)
41	40	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑉) = (Base‘𝐾))
42	41	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (Base‘𝑉) = (Base‘𝐾))
43	38, 42	eleqtrd 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ 𝑦) ∈ (Base‘𝐾))
44		aks6d1c1p3.4	. . . . . . . . 9 ⊢ 𝑋 = (var₁‘𝐾)
45	5, 44, 7, 6, 8, 11, 43	evl1vard 22298	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘((𝑁 / 𝑃) ↑ 𝑦)) = ((𝑁 / 𝑃) ↑ 𝑦)))
46		aks6d1c1p3.8	. . . . . . . . 9 ⊢ 𝐶 = (algSc‘𝑆)
47	10	crngringd 20198	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Ring)
48		eqid 2737	. . . . . . . . . . . . 13 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
49	48	zrhrhm 21483	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
50		rhmghm 20436	. . . . . . . . . . . 12 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
51		zringbas 21425	. . . . . . . . . . . . 13 ⊢ ℤ = (Base‘ℤ_ring)
52	51, 7	ghmf 19166	. . . . . . . . . . . 12 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
53	47, 49, 50, 52	4syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
54		aks6d1c1p3.19	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℤ)
55	53, 54	ffvelcdmd 7041	. . . . . . . . . 10 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾))
56	55	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾))
57	5, 6, 7, 46, 8, 11, 56, 43	evl1scad 22296	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘((𝑁 / 𝑃) ↑ 𝑦)) = ((ℤRHom‘𝐾)‘𝐴)))
58		aks6d1c1p3.12	. . . . . . . 8 ⊢ + = (+_g‘𝑆)
59		eqid 2737	. . . . . . . 8 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
60	5, 6, 7, 8, 11, 43, 45, 57, 58, 59	evl1addd 22302	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵 ∧ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘((𝑁 / 𝑃) ↑ 𝑦)) = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
61	60	simprd 495	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘((𝑁 / 𝑃) ↑ 𝑦)) = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
62	4, 61	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)) = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
63	3	fveq1d 6846	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
64	63	oveq2d 7386	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑁 / 𝑃) ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)))
65	42	eleq2d 2823	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦 ∈ (Base‘𝑉) ↔ 𝑦 ∈ (Base‘𝐾)))
66	37, 65	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
67	5, 44, 40, 6, 8, 11, 37	evl1vard 22298	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘𝑦) = 𝑦))
68	5, 6, 7, 46, 8, 11, 56, 66	evl1scad 22296	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘𝑦) = ((ℤRHom‘𝐾)‘𝐴)))
69	5, 6, 7, 8, 11, 66, 67, 68, 58, 59	evl1addd 22302	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵 ∧ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
70	69	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
71	70	oveq2d 7386	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)) = ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
72	64, 71	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
73		aks6d1c1p3.21	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingIso 𝐾))
74	7, 7	isrim 20444	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingIso 𝐾) ↔ ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingHom 𝐾) ∧ (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾)))
75	73, 74	sylib 218	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingHom 𝐾) ∧ (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾)))
76	75	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
77	76	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
78	11	crnggrpd 20199	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ Grp)
79	7, 59, 78, 43, 56	grpcld 18894	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾))
80		f1ocnvfv1 7234	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾)) → (^◡(𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
81	77, 79, 80	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (^◡(𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
82	81	eqcomd 2743	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = (^◡(𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))))
83		eqidd 2738	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)))
84		id 22	. . . . . . . . . . . . 13 ⊢ (𝑥 = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) → 𝑥 = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
85	84	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) → 𝑥 = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
86	85	oveq2d 7386	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) → (𝑃 ↑ 𝑥) = (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
87		eqid 2737	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
88	87, 7	mgpbas 20097	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
89	14	fveq2i 6847	. . . . . . . . . . . . 13 ⊢ (._g‘𝑉) = (._g‘(mulGrp‘𝐾))
90	13, 89	eqtri 2760	. . . . . . . . . . . 12 ⊢ ↑ = (._g‘(mulGrp‘𝐾))
91	87	ringmgp 20191	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Ring → (mulGrp‘𝐾) ∈ Mnd)
92	47, 91	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
93	92	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (mulGrp‘𝐾) ∈ Mnd)
94	26	nnnn0d 12476	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
95	94	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ∈ ℕ₀)
96	88, 90, 93, 95, 79	mulgnn0cld 19042	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘𝐾))
97	83, 86, 79, 96	fvmptd 6959	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
98	97	eqcomd 2743	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
99	75	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingHom 𝐾))
100		rhmghm 20436	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingHom 𝐾) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 GrpHom 𝐾))
101	99, 100	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 GrpHom 𝐾))
102	101	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 GrpHom 𝐾))
103	7, 59, 59	ghmlin 19167	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 GrpHom 𝐾) ∧ ((𝑁 / 𝑃) ↑ 𝑦) ∈ (Base‘𝐾) ∧ ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ 𝑦))(+_g‘𝐾)((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((ℤRHom‘𝐾)‘𝐴))))
104	102, 43, 56, 103	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ 𝑦))(+_g‘𝐾)((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((ℤRHom‘𝐾)‘𝐴))))
105		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ((𝑁 / 𝑃) ↑ 𝑦) → 𝑥 = ((𝑁 / 𝑃) ↑ 𝑦))
106	105	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((𝑁 / 𝑃) ↑ 𝑦)) → 𝑥 = ((𝑁 / 𝑃) ↑ 𝑦))
107	106	oveq2d 7386	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((𝑁 / 𝑃) ↑ 𝑦)) → (𝑃 ↑ 𝑥) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)))
108	88, 90, 93, 95, 43	mulgnn0cld 19042	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)) ∈ (Base‘𝐾))
109	83, 107, 43, 108	fvmptd 6959	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ 𝑦)) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)))
110		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ((ℤRHom‘𝐾)‘𝐴) → 𝑥 = ((ℤRHom‘𝐾)‘𝐴))
111	110	oveq2d 7386	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ((ℤRHom‘𝐾)‘𝐴) → (𝑃 ↑ 𝑥) = (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)))
112	111	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((ℤRHom‘𝐾)‘𝐴)) → (𝑃 ↑ 𝑥) = (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)))
113		aks6d1c1p3.10	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑃 = (chr‘𝐾)
114		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℤRHom‘𝐾)‘𝐴) = ((ℤRHom‘𝐾)‘𝐴)
115	113, 7, 90, 114, 24, 54, 10	fermltlchr 21501	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)) = ((ℤRHom‘𝐾)‘𝐴))
116	115	eqcomd 2743	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) = (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)))
117	116	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((ℤRHom‘𝐾)‘𝐴) = (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)))
118	117, 56	eqeltrrd 2838	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾))
119	83, 112, 56, 118	fvmptd 6959	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((ℤRHom‘𝐾)‘𝐴)) = (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)))
120	109, 119	oveq12d 7388	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ 𝑦))(+_g‘𝐾)((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦))(+_g‘𝐾)(𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴))))
121	98, 104, 120	3eqtrd 2776	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦))(+_g‘𝐾)(𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴))))
122	23	nncnd 12175	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℂ)
123	122	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 ∈ ℂ)
124	26	nncnd 12175	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℂ)
125	124	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ∈ ℂ)
126	26	nnne0d 12209	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ≠ 0)
127	126	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ≠ 0)
128	123, 125, 127	divcan2d 11933	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 · (𝑁 / 𝑃)) = 𝑁)
129	128	oveq1d 7385	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑁 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
130	63	oveq2d 7386	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑁 ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)))
131	70	oveq2d 7386	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)) = (𝑁 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
132	130, 131	eqtrd 2772	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑁 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
133	132	eqcomd 2743	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)))
134		fveq2 6844	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘𝑧) = ((𝑂‘𝐹)‘𝑦))
135	134	oveq2d 7386	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑦 → (𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)))
136		oveq2 7378	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑦 → (𝑁 ↑ 𝑧) = (𝑁 ↑ 𝑦))
137	136	fveq2d 6848	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)))
138	135, 137	eqeq12d 2753	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑦 → ((𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)) ↔ (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦))))
139	6	ply1crng 22156	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐾 ∈ CRing → 𝑆 ∈ CRing)
140	10, 139	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑆 ∈ CRing)
141	140	crnggrpd 20199	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑆 ∈ Grp)
142	44, 6, 8	vr1cl 22175	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ 𝐵)
143	47, 142	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
144	6, 46, 7, 8	ply1sclcl 22245	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾)) → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
145	47, 55, 144	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
146	141, 143, 145	3jca 1129	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵))
147	8, 58	grpcl 18888	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵)
148	146, 147	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵)
149	1	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))
150	149	eleq1d 2822	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐹 ∈ 𝐵 ↔ (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵))
151	148, 150	mpbird 257	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
152	20, 151, 23	aks6d1c1p1 42506	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑁 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦))))
153	21, 152	mpbid 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)))
154		fveq2 6844	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘𝐹)‘𝑧))
155	154	oveq2d 7386	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑧 → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑁 ↑ ((𝑂‘𝐹)‘𝑧)))
156		oveq2 7378	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → (𝑁 ↑ 𝑦) = (𝑁 ↑ 𝑧))
157	156	fveq2d 6848	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑧 → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)))
158	155, 157	eqeq12d 2753	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑧 → ((𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) ↔ (𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧))))
159	158	cbvralvw 3216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) ↔ ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)))
160	153, 159	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)))
161	160	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)))
162		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (𝑉 PrimRoots 𝑅))
163	138, 161, 162	rspcdva 3579	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)))
164	3	fveq1d 6846	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁 ↑ 𝑦)))
165	23	nnnn0d 12476	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
166	165	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 ∈ ℕ₀)
167	12, 13, 18, 166, 37	mulgnn0cld 19042	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ 𝑦) ∈ (Base‘𝑉))
168	167, 42	eleqtrd 2839	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ 𝑦) ∈ (Base‘𝐾))
169	143	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑋 ∈ 𝐵)
170	5, 44, 7, 6, 8, 11, 168	evl1vard 22298	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝑁 ↑ 𝑦)) = (𝑁 ↑ 𝑦)))
171	170	simprd 495	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝑋)‘(𝑁 ↑ 𝑦)) = (𝑁 ↑ 𝑦))
172	169, 171	jca 511	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝑁 ↑ 𝑦)) = (𝑁 ↑ 𝑦)))
173	145	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
174	5, 6, 7, 46, 8, 11, 56, 168	evl1scad 22296	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑁 ↑ 𝑦)) = ((ℤRHom‘𝐾)‘𝐴)))
175	174	simprd 495	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑁 ↑ 𝑦)) = ((ℤRHom‘𝐾)‘𝐴))
176	173, 175	jca 511	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑁 ↑ 𝑦)) = ((ℤRHom‘𝐾)‘𝐴)))
177	5, 6, 7, 8, 11, 168, 172, 176, 58, 59	evl1addd 22302	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵 ∧ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁 ↑ 𝑦)) = ((𝑁 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
178	177	simprd 495	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁 ↑ 𝑦)) = ((𝑁 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
179	164, 178	eqtrd 2772	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) = ((𝑁 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
180	163, 179	eqtrd 2772	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑁 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
181	133, 180	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑁 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
182	129, 181	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑁 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
183	128	eqcomd 2743	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 = (𝑃 · (𝑁 / 𝑃)))
184	183	oveq1d 7385	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ 𝑦) = ((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦))
185	184, 117	oveq12d 7388	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = (((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦)(+_g‘𝐾)(𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴))))
186	182, 185	eqtr2d 2773	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦)(+_g‘𝐾)(𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 · (𝑁 / 𝑃)) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
187	66, 88	eleqtrdi 2847	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘(mulGrp‘𝐾)))
188	95, 31, 187	3jca 1129	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ∈ ℕ₀ ∧ (𝑁 / 𝑃) ∈ ℕ₀ ∧ 𝑦 ∈ (Base‘(mulGrp‘𝐾))))
189		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘(mulGrp‘𝐾))
190	189, 90	mulgnn0ass 19057	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ (𝑃 ∈ ℕ₀ ∧ (𝑁 / 𝑃) ∈ ℕ₀ ∧ 𝑦 ∈ (Base‘(mulGrp‘𝐾)))) → ((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)))
191	93, 188, 190	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)))
192	191	oveq1d 7385	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦)(+_g‘𝐾)(𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦))(+_g‘𝐾)(𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴))))
193	186, 192	eqtr3d 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦))(+_g‘𝐾)(𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴))))
194	7, 59, 78, 66, 56	grpcld 18894	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾))
195	194, 88	eleqtrdi 2847	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(mulGrp‘𝐾)))
196	95, 31, 195	3jca 1129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ∈ ℕ₀ ∧ (𝑁 / 𝑃) ∈ ℕ₀ ∧ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(mulGrp‘𝐾))))
197	189, 90	mulgnn0ass 19057	. . . . . . . . . . . . 13 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ (𝑃 ∈ ℕ₀ ∧ (𝑁 / 𝑃) ∈ ℕ₀ ∧ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(mulGrp‘𝐾)))) → ((𝑃 · (𝑁 / 𝑃)) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))))
198	93, 196, 197	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))))
199	193, 198	eqtr3d 2774	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦))(+_g‘𝐾)(𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴))) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))))
200	121, 199	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))))
201		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) → 𝑥 = ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
202	201	oveq2d 7386	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) → (𝑃 ↑ 𝑥) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))))
203	202	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) → (𝑃 ↑ 𝑥) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))))
204	88, 90, 93, 31, 194	mulgnn0cld 19042	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘𝐾))
205	200, 96	eqeltrrd 2838	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) ∈ (Base‘𝐾))
206	83, 203, 204, 205	fvmptd 6959	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))))
207	206	eqcomd 2743	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) = ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))))
208	97, 200, 207	3eqtrd 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))))
209	208	fveq2d 6848	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (^◡(𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘(((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) = (^◡(𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))))
210		f1ocnvfv1 7234	. . . . . . . . 9 ⊢ (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘𝐾)) → (^◡(𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))) = ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
211	77, 204, 210	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (^◡(𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))) = ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
212	82, 209, 211	3eqtrd 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
213	212	eqcomd 2743	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
214	72, 213	eqtr2d 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)))
215	62, 214	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)) = ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)))
216	215	eqcomd 2743	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)))
217	216	ralrimiva 3130	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)))
218	20, 151, 29	aks6d1c1p1 42506	. 2 ⊢ (𝜑 → ((𝑁 / 𝑃) ∼ 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦))))
219	217, 218	mpbird 257	1 ⊢ (𝜑 → (𝑁 / 𝑃) ∼ 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 class class class wbr 5100 {copab 5162 ↦ cmpt 5181 ^◡ccnv 5633 ⟶wf 6498 –1-1-onto→wf1o 6501 ‘cfv 6502 (class class class)co 7370 ℂcc 11038 0cc0 11040 1c1 11041 · cmul 11045 / cdiv 11808 ℕcn 12159 ℕ₀cn0 12415 ℤcz 12502 ∥ cdvds 16193 gcd cgcd 16435 ℙcprime 16612 Basecbs 17150 +_gcplusg 17191 0_gc0g 17373 Mndcmnd 18673 Grpcgrp 18880 ._gcmg 19014 GrpHom cghm 19158 CMndccmn 19726 mulGrpcmgp 20092 Ringcrg 20185 CRingccrg 20186 RingHom crh 20422 RingIso crs 20423 Fieldcfield 20680 ℤ_ringczring 21418 ℤRHomczrh 21471 chrcchr 21473 algSccascl 21824 var₁cv1 22133 Poly₁cpl1 22134 eval₁ce1 22275 PrimRoots cprimroots 42490

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 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 11118 ax-addf 11119 ax-mulf 11120

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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-ofr 7635 df-om 7821 df-1st 7945 df-2nd 7946 df-supp 8115 df-tpos 8180 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-oadd 8413 df-er 8647 df-map 8779 df-pm 8780 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fsupp 9279 df-sup 9359 df-inf 9360 df-oi 9429 df-dju 9827 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-div 11809 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-7 12227 df-8 12228 df-9 12229 df-n0 12416 df-xnn0 12489 df-z 12503 df-dec 12622 df-uz 12766 df-rp 12920 df-fz 13438 df-fzo 13585 df-fl 13726 df-mod 13804 df-seq 13939 df-exp 13999 df-hash 14268 df-cj 15036 df-re 15037 df-im 15038 df-sqrt 15172 df-abs 15173 df-dvds 16194 df-gcd 16436 df-prm 16613 df-phi 16707 df-struct 17088 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-ress 17172 df-plusg 17204 df-mulr 17205 df-starv 17206 df-sca 17207 df-vsca 17208 df-ip 17209 df-tset 17210 df-ple 17211 df-ds 17213 df-unif 17214 df-hom 17215 df-cco 17216 df-0g 17375 df-gsum 17376 df-prds 17381 df-pws 17383 df-mre 17519 df-mrc 17520 df-acs 17522 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-mhm 18722 df-submnd 18723 df-grp 18883 df-minusg 18884 df-sbg 18885 df-mulg 19015 df-subg 19070 df-ghm 19159 df-cntz 19263 df-od 19474 df-cmn 19728 df-abl 19729 df-mgp 20093 df-rng 20105 df-ur 20134 df-srg 20139 df-ring 20187 df-cring 20188 df-oppr 20290 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-dvr 20354 df-rhm 20425 df-rim 20426 df-subrng 20496 df-subrg 20520 df-drng 20681 df-field 20682 df-lmod 20830 df-lss 20900 df-lsp 20940 df-cnfld 21327 df-zring 21419 df-zrh 21475 df-chr 21477 df-assa 21825 df-asp 21826 df-ascl 21827 df-psr 21882 df-mvr 21883 df-mpl 21884 df-opsr 21886 df-evls 22046 df-evl 22047 df-psr1 22137 df-vr1 22138 df-ply1 22139 df-evl1 22277 df-primroots 42491

This theorem is referenced by: aks6d1c1 42515

W3C validator