aks6d1c1p3 - Mathbox for metakunt

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Theorem aks6d1c1p3 42567

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 6840	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘𝐹) = (𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))))
4	3	fveq1d 6838	. . . . . 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 20714	. . . . . . . . 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 20217	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
16	10, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ∈ CMnd)
17	16	cmnmndd 19774	. . . . . . . . . . 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 42565	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
23	22	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
24		aks6d1c1p3.14	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℙ)
25		prmnn 16638	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
26	24, 25	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ)
27		nndivdvds 16225	. . . . . . . . . . . . . 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 12493	. . . . . . . . . . 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 12493	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
34	16, 33, 13	isprimroot 42550	. . . . . . . . . . . . 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 19066	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ↑ 𝑦) ∈ (Base‘𝑉))
39	14, 7	mgpbas 20121	. . . . . . . . . . . 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 22316	. . . . . . . 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 20222	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Ring)
48		eqid 2737	. . . . . . . . . . . . 13 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
49	48	zrhrhm 21505	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
50		rhmghm 20458	. . . . . . . . . . . 12 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
51		zringbas 21447	. . . . . . . . . . . . 13 ⊢ ℤ = (Base‘ℤ_ring)
52	51, 7	ghmf 19190	. . . . . . . . . . . 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 7033	. . . . . . . . . 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 22314	. . . . . . . 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 22320	. . . . . . 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 6838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
64	63	oveq2d 7378	. . . . . . 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 22316	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘𝑦) = 𝑦))
68	5, 6, 7, 46, 8, 11, 56, 66	evl1scad 22314	. . . . . . . . . 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 22320	. . . . . . . . 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 7378	. . . . . . 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 20466	. . . . . . . . . . . . 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 20223	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ Grp)
79	7, 59, 78, 43, 56	grpcld 18918	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾))
80		f1ocnvfv1 7226	. . . . . . . . . 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 7378	. . . . . . . . . . 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 20121	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
89	14	fveq2i 6839	. . . . . . . . . . . . 13 ⊢ (._g‘𝑉) = (._g‘(mulGrp‘𝐾))
90	13, 89	eqtri 2760	. . . . . . . . . . . 12 ⊢ ↑ = (._g‘(mulGrp‘𝐾))
91	87	ringmgp 20215	. . . . . . . . . . . . . 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 12493	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
95	94	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ∈ ℕ₀)
96	88, 90, 93, 95, 79	mulgnn0cld 19066	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ (((𝑁 / 𝑃) ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘𝐾))
97	83, 86, 79, 96	fvmptd 6951	. . . . . . . . . 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 20458	. . . . . . . . . . . . . . 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 19191	. . . . . . . . . . . . 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 7378	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((𝑁 / 𝑃) ↑ 𝑦)) → (𝑃 ↑ 𝑥) = (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)))
108	88, 90, 93, 95, 43	mulgnn0cld 19066	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((𝑁 / 𝑃) ↑ 𝑦)) ∈ (Base‘𝐾))
109	83, 107, 43, 108	fvmptd 6951	. . . . . . . . . . . . 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 7378	. . . . . . . . . . . . . . 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 21523	. . . . . . . . . . . . . . . . 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 6951	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥))‘((ℤRHom‘𝐾)‘𝐴)) = (𝑃 ↑ ((ℤRHom‘𝐾)‘𝐴)))
120	109, 119	oveq12d 7380	. . . . . . . . . . . 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 12185	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℂ)
123	122	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 ∈ ℂ)
124	26	nncnd 12185	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℂ)
125	124	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ∈ ℂ)
126	26	nnne0d 12222	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ≠ 0)
127	126	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ≠ 0)
128	123, 125, 127	divcan2d 11928	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 · (𝑁 / 𝑃)) = 𝑁)
129	128	oveq1d 7377	. . . . . . . . . . . . . . 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 7378	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑁 ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)))
131	70	oveq2d 7378	. . . . . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘𝑧) = ((𝑂‘𝐹)‘𝑦))
135	134	oveq2d 7378	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑦 → (𝑁 ↑ ((𝑂‘𝐹)‘𝑧)) = (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)))
136		oveq2 7370	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑦 → (𝑁 ↑ 𝑧) = (𝑁 ↑ 𝑦))
137	136	fveq2d 6840	. . . . . . . . . . . . . . . . . . 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 22176	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐾 ∈ CRing → 𝑆 ∈ CRing)
140	10, 139	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑆 ∈ CRing)
141	140	crnggrpd 20223	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑆 ∈ Grp)
142	44, 6, 8	vr1cl 22195	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ 𝐵)
143	47, 142	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
144	6, 46, 7, 8	ply1sclcl 22265	. . . . . . . . . . . . . . . . . . . . . . . . . 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 18912	. . . . . . . . . . . . . . . . . . . . . . . 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 42564	. . . . . . . . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘𝐹)‘𝑧))
155	154	oveq2d 7378	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑧 → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑁 ↑ ((𝑂‘𝐹)‘𝑧)))
156		oveq2 7370	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → (𝑁 ↑ 𝑦) = (𝑁 ↑ 𝑧))
157	156	fveq2d 6840	. . . . . . . . . . . . . . . . . . . . . 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 3566	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)))
164	3	fveq1d 6838	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝑁 ↑ 𝑦)) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁 ↑ 𝑦)))
165	23	nnnn0d 12493	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
166	165	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 ∈ ℕ₀)
167	12, 13, 18, 166, 37	mulgnn0cld 19066	. . . . . . . . . . . . . . . . . . . . 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 22316	. . . . . . . . . . . . . . . . . . . . . 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 22314	. . . . . . . . . . . . . . . . . . . . . 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 22320	. . . . . . . . . . . . . . . . . . 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 7377	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ↑ 𝑦) = ((𝑃 · (𝑁 / 𝑃)) ↑ 𝑦))
185	184, 117	oveq12d 7380	. . . . . . . . . . . . . 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 19081	. . . . . . . . . . . . . . 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 7377	. . . . . . . . . . . . 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 18918	. . . . . . . . . . . . . . 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 19081	. . . . . . . . . . . . 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 7378	. . . . . . . . . . . . 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 19066	. . . . . . . . . . . 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 6951	. . . . . . . . . . 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 6840	. . . . . . . 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 7226	. . . . . . . . 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 42564	. 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 5086 {copab 5148 ↦ cmpt 5167 ^◡ccnv 5625 ⟶wf 6490 –1-1-onto→wf1o 6493 ‘cfv 6494 (class class class)co 7362 ℂcc 11031 0cc0 11033 1c1 11034 · cmul 11038 / cdiv 11802 ℕcn 12169 ℕ₀cn0 12432 ℤcz 12519 ∥ cdvds 16216 gcd cgcd 16458 ℙcprime 16635 Basecbs 17174 +_gcplusg 17215 0_gc0g 17397 Mndcmnd 18697 Grpcgrp 18904 ._gcmg 19038 GrpHom cghm 19182 CMndccmn 19750 mulGrpcmgp 20116 Ringcrg 20209 CRingccrg 20210 RingHom crh 20444 RingIso crs 20445 Fieldcfield 20702 ℤ_ringczring 21440 ℤRHomczrh 21493 chrcchr 21495 algSccascl 21846 var₁cv1 22153 Poly₁cpl1 22154 eval₁ce1 22293 PrimRoots cprimroots 42548

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112 ax-mulf 11113

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 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-se 5580 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7626 df-ofr 7627 df-om 7813 df-1st 7937 df-2nd 7938 df-supp 8106 df-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-oadd 8404 df-er 8638 df-map 8770 df-pm 8771 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fsupp 9270 df-sup 9350 df-inf 9351 df-oi 9420 df-dju 9820 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-xnn0 12506 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-dvds 16217 df-gcd 16459 df-prm 16636 df-phi 16731 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-0g 17399 df-gsum 17400 df-prds 17405 df-pws 17407 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19039 df-subg 19094 df-ghm 19183 df-cntz 19287 df-od 19498 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-srg 20163 df-ring 20211 df-cring 20212 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-rhm 20447 df-rim 20448 df-subrng 20518 df-subrg 20542 df-drng 20703 df-field 20704 df-lmod 20852 df-lss 20922 df-lsp 20962 df-cnfld 21349 df-zring 21441 df-zrh 21497 df-chr 21499 df-assa 21847 df-asp 21848 df-ascl 21849 df-psr 21903 df-mvr 21904 df-mpl 21905 df-opsr 21907 df-evls 22066 df-evl 22067 df-psr1 22157 df-vr1 22158 df-ply1 22159 df-evl1 22295 df-primroots 42549

This theorem is referenced by: aks6d1c1 42573

W3C validator