aks6d1c1p3 - Mathbox for metakunt

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Theorem aks6d1c1p3 42213

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 6826	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘𝐹) = (𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))))
4	3	fveq1d 6824	. . . . . 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 2731	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
8		aks6d1c1p3.3	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑆)
9		aks6d1c1p3.13	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Field)
10	9	fldcrngd 20657	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ CRing)
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ CRing)
12		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘𝑉) = (Base‘𝑉)
13		aks6d1c1p3.7	. . . . . . . . . 10 ⊢ ↑ = (._g‘𝑉)
14		aks6d1c1p3.6	. . . . . . . . . . . . . 14 ⊢ 𝑉 = (mulGrp‘𝐾)
15	14	crngmgp 20159	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
16	10, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ∈ CMnd)
17	16	cmnmndd 19716	. . . . . . . . . . 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 42211	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
23	22	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
24		aks6d1c1p3.14	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℙ)
25		prmnn 16585	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
26	24, 25	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ)
27		nndivdvds 16172	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℕ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℕ))
28	23, 26, 27	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℕ))
29	19, 28	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℕ)
30	29	nnnn0d 12442	. . . . . . . . . . 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 12442	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
34	16, 33, 13	isprimroot 42196	. . . . . . . . . . . . 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 1142	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
38	12, 13, 18, 31, 37	mulgnn0cld 19008	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ 𝑦) ∈ (Base‘𝑉))
39	14, 7	mgpbas 20063	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝑉)
40	39	eqcomi 2740	. . . . . . . . . . 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 2833	. . . . . . . 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 22252	. . . . . . . 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 20164	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Ring)
48		eqid 2731	. . . . . . . . . . . . 13 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
49	48	zrhrhm 21448	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
50		rhmghm 20401	. . . . . . . . . . . 12 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
51		zringbas 21390	. . . . . . . . . . . . 13 ⊢ ℤ = (Base‘ℤ_ring)
52	51, 7	ghmf 19132	. . . . . . . . . . . 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 7018	. . . . . . . . . 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 22250	. . . . . . . 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 2731	. . . . . . . 8 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
60	5, 6, 7, 8, 11, 43, 45, 57, 58, 59	evl1addd 22256	. . . . . . 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 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)) = (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
63	3	fveq1d 6824	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
64	63	oveq2d 7362	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑁 / 𝑃) ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)))
65	42	eleq2d 2817	. . . . . . . . . . 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 22252	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘𝑦) = 𝑦))
68	5, 6, 7, 46, 8, 11, 56, 66	evl1scad 22250	. . . . . . . . . 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 22256	. . . . . . . . 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 7362	. . . . . . 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 2766	. . . . . 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 20409	. . . . . . . . . . . . 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 20165	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ Grp)
79	7, 59, 78, 43, 56	grpcld 18860	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾))
80		f1ocnvfv1 7210	. . . . . . . . . 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 584	. . . . . . . . 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 2737	. . . . . . . 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 2732	. . . . . . . . . . 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 7362	. . . . . . . . . . 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 2731	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
88	87, 7	mgpbas 20063	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
89	14	fveq2i 6825	. . . . . . . . . . . . 13 ⊢ (._g‘𝑉) = (._g‘(mulGrp‘𝐾))
90	13, 89	eqtri 2754	. . . . . . . . . . . 12 ⊢ ↑ = (._g‘(mulGrp‘𝐾))
91	87	ringmgp 20157	. . . . . . . . . . . . . 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 12442	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
95	94	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ∈ ℕ₀)
96	88, 90, 93, 95, 79	mulgnn0cld 19008	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘𝐾))
97	83, 86, 79, 96	fvmptd 6936	. . . . . . . . . 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 2737	. . . . . . . . . . . 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 20401	. . . . . . . . . . . . . . 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 19133	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 7362	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((𝑁 / 𝑃) ↑ 𝑦)) → (𝑃 ↑ 𝑥) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)))
108	88, 90, 93, 95, 43	mulgnn0cld 19008	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)) ∈ (Base‘𝐾))
109	83, 107, 43, 108	fvmptd 6936	. . . . . . . . . . . . 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 7362	. . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℤRHom‘𝐾)‘𝐴) = ((ℤRHom‘𝐾)‘𝐴)
115	113, 7, 90, 114, 24, 54, 10	fermltlchr 21466	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)) = ((ℤRHom‘𝐾)‘𝐴))
116	115	eqcomd 2737	. . . . . . . . . . . . . . . 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 2832	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾))
119	83, 112, 56, 118	fvmptd 6936	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((ℤRHom‘𝐾)‘𝐴)) = (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)))
120	109, 119	oveq12d 7364	. . . . . . . . . . . 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 2770	. . . . . . . . . . 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 12141	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℂ)
123	122	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 ∈ ℂ)
124	26	nncnd 12141	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℂ)
125	124	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ∈ ℂ)
126	26	nnne0d 12175	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ≠ 0)
127	126	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ≠ 0)
128	123, 125, 127	divcan2d 11899	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 · (𝑁 / 𝑃)) = 𝑁)
129	128	oveq1d 7361	. . . . . . . . . . . . . . 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 7362	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑁 ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)))
131	70	oveq2d 7362	. . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑁 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
133	132	eqcomd 2737	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)))
134		fveq2 6822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘𝑧) = ((𝑂‘𝐹)‘𝑦))
135	134	oveq2d 7362	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑦 → (𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)))
136		oveq2 7354	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑦 → (𝑁 ↑ 𝑧) = (𝑁 ↑ 𝑦))
137	136	fveq2d 6826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)))
138	135, 137	eqeq12d 2747	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑦 → ((𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)) ↔ (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦))))
139	6	ply1crng 22111	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐾 ∈ CRing → 𝑆 ∈ CRing)
140	10, 139	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑆 ∈ CRing)
141	140	crnggrpd 20165	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑆 ∈ Grp)
142	44, 6, 8	vr1cl 22130	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ 𝐵)
143	47, 142	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
144	6, 46, 7, 8	ply1sclcl 22200	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾)) → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
145	47, 55, 144	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
146	141, 143, 145	3jca 1128	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵))
147	8, 58	grpcl 18854	. . . . . . . . . . . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐹 ∈ 𝐵 ↔ (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵))
151	148, 150	mpbird 257	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
152	20, 151, 23	aks6d1c1p1 42210	. . . . . . . . . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘𝐹)‘𝑧))
155	154	oveq2d 7362	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑧 → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑁 ↑ ((𝑂‘𝐹)‘𝑧)))
156		oveq2 7354	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → (𝑁 ↑ 𝑦) = (𝑁 ↑ 𝑧))
157	156	fveq2d 6826	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑧 → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧)))
158	155, 157	eqeq12d 2747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑧 → ((𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) ↔ (𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑧))))
159	158	cbvralvw 3210	. . . . . . . . . . . . . . . . . . . 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 3573	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)))
164	3	fveq1d 6824	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁 ↑ 𝑦)))
165	23	nnnn0d 12442	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
166	165	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 ∈ ℕ₀)
167	12, 13, 18, 166, 37	mulgnn0cld 19008	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ 𝑦) ∈ (Base‘𝑉))
168	167, 42	eleqtrd 2833	. . . . . . . . . . . . . . . . . . . 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 22252	. . . . . . . . . . . . . . . . . . . . . 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 22250	. . . . . . . . . . . . . . . . . . . . . 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 22256	. . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) = ((𝑁 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
180	163, 179	eqtrd 2766	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑁 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
181	133, 180	eqtrd 2766	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑁 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
182	129, 181	eqtrd 2766	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 = (𝑃 · (𝑁 / 𝑃)))
184	183	oveq1d 7361	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ 𝑦) = ((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦))
185	184, 117	oveq12d 7364	. . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘(mulGrp‘𝐾)))
188	95, 31, 187	3jca 1128	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ∈ ℕ₀ ∧ (𝑁 / 𝑃) ∈ ℕ₀ ∧ 𝑦 ∈ (Base‘(mulGrp‘𝐾))))
189		eqid 2731	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘(mulGrp‘𝐾))
190	189, 90	mulgnn0ass 19023	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)))
192	191	oveq1d 7361	. . . . . . . . . . . . 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 2768	. . . . . . . . . . . 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 18860	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾))
195	194, 88	eleqtrdi 2841	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(mulGrp‘𝐾)))
196	95, 31, 195	3jca 1128	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ∈ ℕ₀ ∧ (𝑁 / 𝑃) ∈ ℕ₀ ∧ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(mulGrp‘𝐾))))
197	189, 90	mulgnn0ass 19023	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 2768	. . . . . . . . . . 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 2766	. . . . . . . . . 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 7362	. . . . . . . . . . . . 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 19008	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘𝐾))
205	200, 96	eqeltrrd 2832	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((𝑁 / 𝑃) ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))) ∈ (Base‘𝐾))
206	83, 203, 204, 205	fvmptd 6936	. . . . . . . . . . 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 2737	. . . . . . . . . 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 2770	. . . . . . . . 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 6826	. . . . . . . 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 7210	. . . . . . . . 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 584	. . . . . . . 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 2770	. . . . . . 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 2737	. . . . . 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 2767	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)))
215	62, 214	eqtrd 2766	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)) = ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)))
216	215	eqcomd 2737	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)))
217	216	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)((𝑁 / 𝑃) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝑁 / 𝑃) ↑ 𝑦)))
218	20, 151, 29	aks6d1c1p1 42210	. 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 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 class class class wbr 5089 {copab 5151 ↦ cmpt 5170 ^◡ccnv 5613 ⟶wf 6477 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 0cc0 11006 1c1 11007 · cmul 11011 / cdiv 11774 ℕcn 12125 ℕ₀cn0 12381 ℤcz 12468 ∥ cdvds 16163 gcd cgcd 16405 ℙcprime 16582 Basecbs 17120 +_gcplusg 17161 0_gc0g 17343 Mndcmnd 18642 Grpcgrp 18846 ._gcmg 18980 GrpHom cghm 19124 CMndccmn 19692 mulGrpcmgp 20058 Ringcrg 20151 CRingccrg 20152 RingHom crh 20387 RingIso crs 20388 Fieldcfield 20645 ℤ_ringczring 21383 ℤRHomczrh 21436 chrcchr 21438 algSccascl 21789 var₁cv1 22088 Poly₁cpl1 22089 eval₁ce1 22229 PrimRoots cprimroots 42194

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085 ax-mulf 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-xnn0 12455 df-z 12469 df-dec 12589 df-uz 12733 df-rp 12891 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-dvds 16164 df-gcd 16406 df-prm 16583 df-phi 16677 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19125 df-cntz 19229 df-od 19440 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-srg 20105 df-ring 20153 df-cring 20154 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-dvr 20319 df-rhm 20390 df-rim 20391 df-subrng 20461 df-subrg 20485 df-drng 20646 df-field 20647 df-lmod 20795 df-lss 20865 df-lsp 20905 df-cnfld 21292 df-zring 21384 df-zrh 21440 df-chr 21442 df-assa 21790 df-asp 21791 df-ascl 21792 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-evls 22009 df-evl 22010 df-psr1 22092 df-vr1 22093 df-ply1 22094 df-evl1 22231 df-primroots 42195

This theorem is referenced by: aks6d1c1 42219

W3C validator