aks6d1c1p3 - Mathbox for metakunt

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Theorem aks6d1c1p3 42939

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 6889	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘𝐹) = (𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))))
4	3	fveq1d 6887	. . . . . 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 2765	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
8		aks6d1c1p3.3	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑆)
9		aks6d1c1p3.13	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Field)
10	9	fldcrngd 20896	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ CRing)
11	10	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ CRing)
12		eqid 2765	. . . . . . . . . 10 ⊢ (Base‘𝑉) = (Base‘𝑉)
13		aks6d1c1p3.7	. . . . . . . . . 10 ⊢ ↑ = (._g‘𝑉)
14		aks6d1c1p3.6	. . . . . . . . . . . . . 14 ⊢ 𝑉 = (mulGrp‘𝐾)
15	14	crngmgp 20371	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
16	10, 15	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ∈ CMnd)
17	16	cmnmndd 19922	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ Mnd)
18	17	adantr 486	. . . . . . . . . 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 42937	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
23	22	simpld 500	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
24		aks6d1c1p3.14	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℙ)
25		prmnn 16758	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
26	24, 25	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ)
27		nndivdvds 16345	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℕ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℕ))
28	23, 26, 27	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℕ))
29	19, 28	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℕ)
30	29	nnnn0d 12584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℕ₀)
31	30	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 / 𝑃) ∈ ℕ₀)
32		aks6d1c1p3.15	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ ℕ)
33	32	nnnn0d 12584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
34	16, 33, 13	isprimroot 42922	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 1160	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
38	12, 13, 18, 31, 37	mulgnn0cld 19209	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ 𝑦) ∈ (Base‘𝑉))
39	14, 7	mgpbas 20269	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝑉)
40	39	eqcomi 2774	. . . . . . . . . . 11 ⊢ (Base‘𝑉) = (Base‘𝐾)
41	40	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑉) = (Base‘𝐾))
42	41	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (Base‘𝑉) = (Base‘𝐾))
43	38, 42	eleqtrd 2867	. . . . . . . 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 22551	. . . . . . . 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 20376	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Ring)
48		eqid 2765	. . . . . . . . . . . . 13 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
49	48	zrhrhm 21715	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
50		rhmghm 20616	. . . . . . . . . . . 12 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
51		zringbas 21657	. . . . . . . . . . . . 13 ⊢ ℤ = (Base‘ℤ_ring)
52	51, 7	ghmf 19338	. . . . . . . . . . . 12 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
53	47, 49, 50, 52	4syl 20	. . . . . . . . . . 11 ⊢ (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
54		aks6d1c1p3.19	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℤ)
55	53, 54	ffvelcdmd 7084	. . . . . . . . . 10 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾))
56	55	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾))
57	5, 6, 7, 46, 8, 11, 56, 43	evl1scad 22549	. . . . . . . 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 2765	. . . . . . . 8 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
60	5, 6, 7, 8, 11, 43, 45, 57, 58, 59	evl1addd 22555	. . . . . . 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 501	. . . . . 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 2800	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)) = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
63	3	fveq1d 6887	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
64	63	oveq2d 7435	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑁 / 𝑃) ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)))
65	42	eleq2d 2851	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦 ∈ (Base‘𝑉) ↔ 𝑦 ∈ (Base‘𝐾)))
66	37, 65	mpbid 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
67	5, 44, 40, 6, 8, 11, 37	evl1vard 22551	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘𝑦) = 𝑦))
68	5, 6, 7, 46, 8, 11, 56, 66	evl1scad 22549	. . . . . . . . . 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 22555	. . . . . . . . 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 501	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
71	70	oveq2d 7435	. . . . . . 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 2800	. . . . . 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 20630	. . . . . . . . . . . . 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 221	. . . . . . . . . . . 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 501	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
77	76	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
78	11	crnggrpd 20377	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ Grp)
79	7, 59, 78, 43, 56	grpcld 19062	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾))
80		f1ocnvfv1 7283	. . . . . . . . . 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 596	. . . . . . . . 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 2771	. . . . . . . 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 2766	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)))
84		id 23	. . . . . . . . . . . . 13 ⊢ (𝑥 = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) → 𝑥 = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
85	84	adantl 487	. . . . . . . . . . . 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 7435	. . . . . . . . . . 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 2765	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
88	87, 7	mgpbas 20269	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
89	14	fveq2i 6888	. . . . . . . . . . . . 13 ⊢ (._g‘𝑉) = (._g‘(mulGrp‘𝐾))
90	13, 89	eqtri 2788	. . . . . . . . . . . 12 ⊢ ↑ = (._g‘(mulGrp‘𝐾))
91	87	ringmgp 20369	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Ring → (mulGrp‘𝐾) ∈ Mnd)
92	47, 91	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
93	92	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (mulGrp‘𝐾) ∈ Mnd)
94	26	nnnn0d 12584	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
95	94	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ∈ ℕ₀)
96	88, 90, 93, 95, 79	mulgnn0cld 19209	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘𝐾))
97	83, 86, 79, 96	fvmptd 7001	. . . . . . . . . 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 2771	. . . . . . . . . . . 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 500	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingHom 𝐾))
100		rhmghm 20616	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingHom 𝐾) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 GrpHom 𝐾))
101	99, 100	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 GrpHom 𝐾))
102	101	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 GrpHom 𝐾))
103	7, 59, 59	ghmlin 19339	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ((𝑁 / 𝑃) ↑ 𝑦) → 𝑥 = ((𝑁 / 𝑃) ↑ 𝑦))
106	105	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((𝑁 / 𝑃) ↑ 𝑦)) → 𝑥 = ((𝑁 / 𝑃) ↑ 𝑦))
107	106	oveq2d 7435	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((𝑁 / 𝑃) ↑ 𝑦)) → (𝑃 ↑ 𝑥) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)))
108	88, 90, 93, 95, 43	mulgnn0cld 19209	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)) ∈ (Base‘𝐾))
109	83, 107, 43, 108	fvmptd 7001	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((𝑁 / 𝑃) ↑ 𝑦)) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)))
110		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ((ℤRHom‘𝐾)‘𝐴) → 𝑥 = ((ℤRHom‘𝐾)‘𝐴))
111	110	oveq2d 7435	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ((ℤRHom‘𝐾)‘𝐴) → (𝑃 ↑ 𝑥) = (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)))
112	111	adantl 487	. . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℤRHom‘𝐾)‘𝐴) = ((ℤRHom‘𝐾)‘𝐴)
115	113, 7, 90, 114, 24, 54, 10	fermltlchr 21733	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)) = ((ℤRHom‘𝐾)‘𝐴))
116	115	eqcomd 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) = (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)))
117	116	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((ℤRHom‘𝐾)‘𝐴) = (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)))
118	117, 56	eqeltrrd 2866	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾))
119	83, 112, 56, 118	fvmptd 7001	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((ℤRHom‘𝐾)‘𝐴)) = (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)))
120	109, 119	oveq12d 7437	. . . . . . . . . . . 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 2804	. . . . . . . . . . 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 12268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℂ)
123	122	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 ∈ ℂ)
124	26	nncnd 12268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℂ)
125	124	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ∈ ℂ)
126	26	nnne0d 12305	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ≠ 0)
127	126	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ≠ 0)
128	123, 125, 127	divcan2d 12012	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 · (𝑁 / 𝑃)) = 𝑁)
129	128	oveq1d 7434	. . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑁 ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)))
131	70	oveq2d 7435	. . . . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑁 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
133	132	eqcomd 2771	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)))
134		fveq2 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘𝑧) = ((𝑂‘𝐹)‘𝑦))
135	134	oveq2d 7435	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑦 → (𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)))
136		oveq2 7427	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑦 → (𝑁 ↑ 𝑧) = (𝑁 ↑ 𝑦))
137	136	fveq2d 6889	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)))
138	135, 137	eqeq12d 2781	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑦 → ((𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)) ↔ (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦))))
139	6	ply1crng 22412	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐾 ∈ CRing → 𝑆 ∈ CRing)
140	10, 139	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑆 ∈ CRing)
141	140	crnggrpd 20377	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑆 ∈ Grp)
142	44, 6, 8	vr1cl 22431	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ 𝐵)
143	47, 142	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
144	6, 46, 7, 8	ply1sclcl 22501	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾)) → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
145	47, 55, 144	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
146	141, 143, 145	3jca 1146	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵))
147	8, 58	grpcl 19056	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵)
148	146, 147	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵)
149	1	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))
150	149	eleq1d 2850	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐹 ∈ 𝐵 ↔ (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵))
151	148, 150	mpbird 260	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
152	20, 151, 23	aks6d1c1p1 42936	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑁 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦))))
153	21, 152	mpbid 235	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)))
154		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘𝐹)‘𝑧))
155	154	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑧 → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑁 ↑ ((𝑂‘𝐹)‘𝑧)))
156		oveq2 7427	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → (𝑁 ↑ 𝑦) = (𝑁 ↑ 𝑧))
157	156	fveq2d 6889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑧 → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)))
158	155, 157	eqeq12d 2781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑧 → ((𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) ↔ (𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧))))
159	158	cbvralvw 3245	. . . . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)))
161	160	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)))
162		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (𝑉 PrimRoots 𝑅))
163	138, 161, 162	rspcdva 3584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)))
164	3	fveq1d 6887	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁 ↑ 𝑦)))
165	23	nnnn0d 12584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
166	165	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 ∈ ℕ₀)
167	12, 13, 18, 166, 37	mulgnn0cld 19209	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ 𝑦) ∈ (Base‘𝑉))
168	167, 42	eleqtrd 2867	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ 𝑦) ∈ (Base‘𝐾))
169	143	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑋 ∈ 𝐵)
170	5, 44, 7, 6, 8, 11, 168	evl1vard 22551	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝑁 ↑ 𝑦)) = (𝑁 ↑ 𝑦)))
171	170	simprd 501	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝑋)‘(𝑁 ↑ 𝑦)) = (𝑁 ↑ 𝑦))
172	169, 171	jca 521	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝑁 ↑ 𝑦)) = (𝑁 ↑ 𝑦)))
173	145	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
174	5, 6, 7, 46, 8, 11, 56, 168	evl1scad 22549	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑁 ↑ 𝑦)) = ((ℤRHom‘𝐾)‘𝐴)))
175	174	simprd 501	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑁 ↑ 𝑦)) = ((ℤRHom‘𝐾)‘𝐴))
176	173, 175	jca 521	. . . . . . . . . . . . . . . . . . . 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 22555	. . . . . . . . . . . . . . . . . . 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 501	. . . . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) = ((𝑁 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
180	163, 179	eqtrd 2800	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑁 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
181	133, 180	eqtrd 2800	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑁 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
182	129, 181	eqtrd 2800	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 = (𝑃 · (𝑁 / 𝑃)))
184	183	oveq1d 7434	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ 𝑦) = ((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦))
185	184, 117	oveq12d 7437	. . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . 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 2875	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘(mulGrp‘𝐾)))
188	95, 31, 187	3jca 1146	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ∈ ℕ₀ ∧ (𝑁 / 𝑃) ∈ ℕ₀ ∧ 𝑦 ∈ (Base‘(mulGrp‘𝐾))))
189		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘(mulGrp‘𝐾))
190	189, 90	mulgnn0ass 19224	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)))
192	191	oveq1d 7434	. . . . . . . . . . . . 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 2802	. . . . . . . . . . . 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 19062	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾))
195	194, 88	eleqtrdi 2875	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(mulGrp‘𝐾)))
196	95, 31, 195	3jca 1146	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ∈ ℕ₀ ∧ (𝑁 / 𝑃) ∈ ℕ₀ ∧ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(mulGrp‘𝐾))))
197	189, 90	mulgnn0ass 19224	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 2802	. . . . . . . . . . 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 2800	. . . . . . . . . 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 23	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) → 𝑥 = ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
202	201	oveq2d 7435	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) → (𝑃 ↑ 𝑥) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))))
203	202	adantl 487	. . . . . . . . . . . 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 19209	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘𝐾))
205	200, 96	eqeltrrd 2866	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) ∈ (Base‘𝐾))
206	83, 203, 204, 205	fvmptd 7001	. . . . . . . . . . 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 2771	. . . . . . . . . 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 2804	. . . . . . . . 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 6889	. . . . . . . 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 7283	. . . . . . . . 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 596	. . . . . . . 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 2804	. . . . . . 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 2771	. . . . . 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 2801	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)))
215	62, 214	eqtrd 2800	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)) = ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)))
216	215	eqcomd 2771	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)))
217	216	ralrimiva 3159	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)))
218	20, 151, 29	aks6d1c1p1 42936	. 2 ⊢ (𝜑 → ((𝑁 / 𝑃) ∼ 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦))))
219	217, 218	mpbird 260	1 ⊢ (𝜑 → (𝑁 / 𝑃) ∼ 𝐹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 class class class wbr 5111 {copab 5175 ↦ cmpt 5194 ^◡ccnv 5662 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7419 ℂcc 11117 0cc0 11119 1c1 11120 · cmul 11124 / cdiv 11890 ℕcn 12252 ℕ₀cn0 12523 ℤcz 12610 ∥ cdvds 16336 gcd cgcd 16578 ℙcprime 16755 Basecbs 17295 +_gcplusg 17336 0_gc0g 17518 Mndcmnd 18828 Grpcgrp 19048 ._gcmg 19181 GrpHom cghm 19331 CMndccmn 19898 mulGrpcmgp 20264 Ringcrg 20363 CRingccrg 20364 RingHom crh 20601 RingIso crs 20602 Fieldcfield 20882 ℤ_ringczring 21650 ℤRHomczrh 21703 chrcchr 21705 algSccascl 22056 var₁cv1 22390 Poly₁cpl1 22391 eval₁ce1 22528 PrimRoots cprimroots 42920

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11175 ax-resscn 11176 ax-1cn 11177 ax-icn 11178 ax-addcl 11179 ax-addrcl 11180 ax-mulcl 11181 ax-mulrcl 11182 ax-mulcom 11183 ax-addass 11184 ax-mulass 11185 ax-distr 11186 ax-i2m1 11187 ax-1ne0 11188 ax-1rid 11189 ax-rnegex 11190 ax-rrecex 11191 ax-cnre 11192 ax-pre-lttri 11193 ax-pre-lttrn 11194 ax-pre-ltadd 11195 ax-pre-mulgt0 11196 ax-pre-sup 11197 ax-addf 11198 ax-mulf 11199

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 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-ofr 7685 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-sup 9409 df-inf 9410 df-oi 9479 df-dju 9903 df-card 9941 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11462 df-neg 11463 df-div 11891 df-nn 12253 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12732 df-uz 12883 df-rp 13037 df-fz 13556 df-fzo 13704 df-fl 13847 df-mod 13925 df-seq 14060 df-exp 14120 df-hash 14389 df-cj 15178 df-re 15179 df-im 15180 df-sqrt 15314 df-abs 15315 df-dvds 16337 df-gcd 16579 df-prm 16756 df-phi 16851 df-struct 17233 df-sets 17250 df-slot 17268 df-ndx 17280 df-base 17296 df-ress 17317 df-plusg 17349 df-mulr 17350 df-starv 17351 df-sca 17352 df-vsca 17353 df-ip 17354 df-tset 17355 df-ple 17356 df-ds 17358 df-unif 17359 df-hom 17360 df-cco 17361 df-0g 17520 df-gsum 17521 df-prds 17526 df-pws 17528 df-mre 17664 df-mrc 17665 df-acs 17667 df-mgm 18724 df-sgrp 18813 df-mnd 18829 df-mhm 18882 df-submnd 18883 df-grp 19051 df-minusg 19052 df-sbg 19053 df-mulg 19182 df-subg 19237 df-ghm 19332 df-cntz 19435 df-od 19646 df-cmn 19900 df-abl 19901 df-mgp 20265 df-rng 20279 df-ur 20312 df-srg 20317 df-ring 20365 df-cring 20366 df-oppr 20469 df-dvdsr 20489 df-unit 20490 df-invr 20520 df-dvr 20533 df-rhm 20604 df-rim 20605 df-subrng 20699 df-subrg 20723 df-drng 20883 df-field 20884 df-lmod 21037 df-lss 21107 df-lsp 21147 df-cnfld 21577 df-zring 21651 df-zrh 21707 df-chr 21709 df-assa 22057 df-asp 22058 df-ascl 22059 df-psr 22113 df-mvr 22114 df-mpl 22115 df-opsr 22117 df-evls 22279 df-evl 22280 df-psr1 22394 df-vr1 22395 df-ply1 22396 df-evl1 22530 df-primroots 42921

This theorem is used by: aks6d1c1 42945

W3C validator