Theorem aks6d1c1p6 42109

Description: If a polynomials 𝐹 is introspective to 𝐸, then so are its powers. (Contributed by metakunt, 30-Apr-2025.)

Hypotheses

Ref	Expression
aks6d1c1.1	⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}
aks6d1c1.2	⊢ 𝑆 = (Poly1‘𝐾)
aks6d1c1.3	⊢ 𝐵 = (Base‘𝑆)
aks6d1c1.4	⊢ 𝑋 = (var1‘𝐾)
aks6d1c1.5	⊢ 𝑊 = (mulGrp‘𝑆)
aks6d1c1.6	⊢ 𝑉 = (mulGrp‘𝐾)
aks6d1c1.7	⊢ ↑ = (.g‘𝑉)
aks6d1c1.8	⊢ 𝐶 = (algSc‘𝑆)
aks6d1c1.9	⊢ 𝐷 = (.g‘𝑊)
aks6d1c1.10	⊢ 𝑃 = (chr‘𝐾)
aks6d1c1.11	⊢ 𝑂 = (eval1‘𝐾)
aks6d1c1.12	⊢ + = (+g‘𝑆)
aks6d1c1.13	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c1.14	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c1.15	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c1.16	⊢ (𝜑 → 𝑁 ∈ ℕ)
aks6d1c1.17	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c1.18	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c1p6.1	⊢ (𝜑 → 𝐸 ∼ 𝐹)
aks6d1c1p6.2	⊢ (𝜑 → 𝐿 ∈ ℕ0)

Assertion

Ref	Expression
aks6d1c1p6	⊢ (𝜑 → 𝐸 ∼ (𝐿𝐷𝐹))

Distinct variable groups:   ↑ ,𝑒,𝑓,𝑦   𝑦, ∼   𝐵,𝑒,𝑓   𝐷,𝑒,𝑓,𝑦   𝑒,𝐸,𝑓,𝑦   𝑒,𝐹,𝑓,𝑦   𝑒,𝑂,𝑓,𝑦   𝑅,𝑒,𝑓,𝑦   𝑒,𝑉,𝑓,𝑦   𝑒,𝑊,𝑓,𝑦   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑒,𝑓)   𝐵(𝑦)   𝐶(𝑦,𝑒,𝑓)   𝑃(𝑦,𝑒,𝑓)   + (𝑦,𝑒,𝑓)   ∼ (𝑒,𝑓)   𝑆(𝑦,𝑒,𝑓)   𝐾(𝑦,𝑒,𝑓)   𝐿(𝑦,𝑒,𝑓)   𝑁(𝑦,𝑒,𝑓)   𝑋(𝑦,𝑒,𝑓)

Proof of Theorem aks6d1c1p6

Dummy variables ℎ 𝑖 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aks6d1c1p6.2	. 2 ⊢ (𝜑 → 𝐿 ∈ ℕ0)
2		oveq1 7397	. . . 4 ⊢ (ℎ = 0 → (ℎ𝐷𝐹) = (0𝐷𝐹))
3	2	breq2d 5122	. . 3 ⊢ (ℎ = 0 → (𝐸 ∼ (ℎ𝐷𝐹) ↔ 𝐸 ∼ (0𝐷𝐹)))
4		oveq1 7397	. . . 4 ⊢ (ℎ = 𝑖 → (ℎ𝐷𝐹) = (𝑖𝐷𝐹))
5	4	breq2d 5122	. . 3 ⊢ (ℎ = 𝑖 → (𝐸 ∼ (ℎ𝐷𝐹) ↔ 𝐸 ∼ (𝑖𝐷𝐹)))
6		oveq1 7397	. . . 4 ⊢ (ℎ = (𝑖 + 1) → (ℎ𝐷𝐹) = ((𝑖 + 1)𝐷𝐹))
7	6	breq2d 5122	. . 3 ⊢ (ℎ = (𝑖 + 1) → (𝐸 ∼ (ℎ𝐷𝐹) ↔ 𝐸 ∼ ((𝑖 + 1)𝐷𝐹)))
8		oveq1 7397	. . . 4 ⊢ (ℎ = 𝐿 → (ℎ𝐷𝐹) = (𝐿𝐷𝐹))
9	8	breq2d 5122	. . 3 ⊢ (ℎ = 𝐿 → (𝐸 ∼ (ℎ𝐷𝐹) ↔ 𝐸 ∼ (𝐿𝐷𝐹)))
10		aks6d1c1.1	. . . . . . . . . . . . . . . 16 ⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}
11		aks6d1c1p6.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 ∼ 𝐹)
12	10, 11	aks6d1c1p1rcl 42103	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
13	12	simprd 495	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
14		aks6d1c1.3	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝑆)
15	13, 14	eleqtrdi 2839	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑆))
16		aks6d1c1.5	. . . . . . . . . . . . . 14 ⊢ 𝑊 = (mulGrp‘𝑆)
17		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑆) = (Base‘𝑆)
18	16, 17	mgpbas 20061	. . . . . . . . . . . . 13 ⊢ (Base‘𝑆) = (Base‘𝑊)
19	15, 18	eleqtrdi 2839	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑊))
20		eqid 2730	. . . . . . . . . . . . 13 ⊢ (Base‘𝑊) = (Base‘𝑊)
21		eqid 2730	. . . . . . . . . . . . 13 ⊢ (0g‘𝑊) = (0g‘𝑊)
22		aks6d1c1.9	. . . . . . . . . . . . 13 ⊢ 𝐷 = (.g‘𝑊)
23	20, 21, 22	mulg0 19013	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Base‘𝑊) → (0𝐷𝐹) = (0g‘𝑊))
24	19, 23	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (0𝐷𝐹) = (0g‘𝑊))
25		eqid 2730	. . . . . . . . . . . . 13 ⊢ (1r‘𝑆) = (1r‘𝑆)
26	16, 25	ringidval 20099	. . . . . . . . . . . 12 ⊢ (1r‘𝑆) = (0g‘𝑊)
27	26	eqcomi 2739	. . . . . . . . . . 11 ⊢ (0g‘𝑊) = (1r‘𝑆)
28	24, 27	eqtrdi 2781	. . . . . . . . . 10 ⊢ (𝜑 → (0𝐷𝐹) = (1r‘𝑆))
29	28	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (0𝐷𝐹) = (1r‘𝑆))
30	29	fveq2d 6865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘(0𝐷𝐹)) = (𝑂‘(1r‘𝑆)))
31	30	fveq1d 6863	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(0𝐷𝐹))‘𝑦) = ((𝑂‘(1r‘𝑆))‘𝑦))
32	31	oveq2d 7406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(0𝐷𝐹))‘𝑦)) = (𝐸 ↑ ((𝑂‘(1r‘𝑆))‘𝑦)))
33		aks6d1c1.13	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ Field)
34	33	fldcrngd 20658	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ CRing)
35		crngring 20161	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
36	34, 35	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Ring)
37		aks6d1c1.2	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (Poly1‘𝐾)
38		aks6d1c1.8	. . . . . . . . . . . . . 14 ⊢ 𝐶 = (algSc‘𝑆)
39		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (1r‘𝐾) = (1r‘𝐾)
40	37, 38, 39, 25	ply1scl1 22186	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Ring → (𝐶‘(1r‘𝐾)) = (1r‘𝑆))
41	36, 40	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶‘(1r‘𝐾)) = (1r‘𝑆))
42	41	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐶‘(1r‘𝐾)) = (1r‘𝑆))
43	42	eqcomd 2736	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (1r‘𝑆) = (𝐶‘(1r‘𝐾)))
44	43	fveq2d 6865	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘(1r‘𝑆)) = (𝑂‘(𝐶‘(1r‘𝐾))))
45	44	fveq1d 6863	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(1r‘𝑆))‘𝑦) = ((𝑂‘(𝐶‘(1r‘𝐾)))‘𝑦))
46	45	oveq2d 7406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(1r‘𝑆))‘𝑦)) = (𝐸 ↑ ((𝑂‘(𝐶‘(1r‘𝐾)))‘𝑦)))
47		aks6d1c1.11	. . . . . . . . . . 11 ⊢ 𝑂 = (eval1‘𝐾)
48		eqid 2730	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
49	34	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ CRing)
50	48, 39	ringidcl 20181	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Ring → (1r‘𝐾) ∈ (Base‘𝐾))
51	36, 50	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1r‘𝐾) ∈ (Base‘𝐾))
52	51	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (1r‘𝐾) ∈ (Base‘𝐾))
53		aks6d1c1.6	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑉 = (mulGrp‘𝐾)
54	53	crngmgp 20157	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
55	34, 54	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑉 ∈ CMnd)
56		aks6d1c1.15	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ∈ ℕ)
57	56	nnnn0d 12510	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
58		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (.g‘𝑉) = (.g‘𝑉)
59	55, 57, 58	isprimroot 42088	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) ↔ (𝑦 ∈ (Base‘𝑉) ∧ (𝑅(.g‘𝑉)𝑦) = (0g‘𝑉) ∧ ∀𝑧 ∈ ℕ0 ((𝑧(.g‘𝑉)𝑦) = (0g‘𝑉) → 𝑅 ∥ 𝑧))))
60	59	biimpd 229	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅(.g‘𝑉)𝑦) = (0g‘𝑉) ∧ ∀𝑧 ∈ ℕ0 ((𝑧(.g‘𝑉)𝑦) = (0g‘𝑉) → 𝑅 ∥ 𝑧))))
61	60	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅(.g‘𝑉)𝑦) = (0g‘𝑉) ∧ ∀𝑧 ∈ ℕ0 ((𝑧(.g‘𝑉)𝑦) = (0g‘𝑉) → 𝑅 ∥ 𝑧)))
62	61	simp1d 1142	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
63	53, 48	mgpbas 20061	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘𝑉)
64	63	eqcomi 2739	. . . . . . . . . . . 12 ⊢ (Base‘𝑉) = (Base‘𝐾)
65	62, 64	eleqtrdi 2839	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
66	47, 37, 48, 38, 14, 49, 52, 65	evl1scad 22229	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘(1r‘𝐾)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘(1r‘𝐾)))‘𝑦) = (1r‘𝐾)))
67	66	simprd 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝐶‘(1r‘𝐾)))‘𝑦) = (1r‘𝐾))
68	67	oveq2d 7406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(𝐶‘(1r‘𝐾)))‘𝑦)) = (𝐸 ↑ (1r‘𝐾)))
69	55	cmnmndd 19741	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ Mnd)
70	12	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℕ)
71	70	nnnn0d 12510	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℕ0)
72		eqid 2730	. . . . . . . . . . 11 ⊢ (Base‘𝑉) = (Base‘𝑉)
73		aks6d1c1.7	. . . . . . . . . . 11 ⊢ ↑ = (.g‘𝑉)
74	53, 39	ringidval 20099	. . . . . . . . . . 11 ⊢ (1r‘𝐾) = (0g‘𝑉)
75	72, 73, 74	mulgnn0z 19040	. . . . . . . . . 10 ⊢ ((𝑉 ∈ Mnd ∧ 𝐸 ∈ ℕ0) → (𝐸 ↑ (1r‘𝐾)) = (1r‘𝐾))
76	69, 71, 75	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↑ (1r‘𝐾)) = (1r‘𝐾))
77	76	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ (1r‘𝐾)) = (1r‘𝐾))
78	69	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑉 ∈ Mnd)
79	71	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐸 ∈ ℕ0)
80	63, 73	mulgnn0cl 19029	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ Mnd ∧ 𝐸 ∈ ℕ0 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝐸 ↑ 𝑦) ∈ (Base‘𝐾))
81	78, 79, 65, 80	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ 𝑦) ∈ (Base‘𝐾))
82	47, 37, 48, 38, 14, 49, 52, 81	evl1scad 22229	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘(1r‘𝐾)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘(1r‘𝐾)))‘(𝐸 ↑ 𝑦)) = (1r‘𝐾)))
83	82	simprd 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝐶‘(1r‘𝐾)))‘(𝐸 ↑ 𝑦)) = (1r‘𝐾))
84	83	eqcomd 2736	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (1r‘𝐾) = ((𝑂‘(𝐶‘(1r‘𝐾)))‘(𝐸 ↑ 𝑦)))
85	68, 77, 84	3eqtrd 2769	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(𝐶‘(1r‘𝐾)))‘𝑦)) = ((𝑂‘(𝐶‘(1r‘𝐾)))‘(𝐸 ↑ 𝑦)))
86	42	fveq2d 6865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘(𝐶‘(1r‘𝐾))) = (𝑂‘(1r‘𝑆)))
87	86	fveq1d 6863	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝐶‘(1r‘𝐾)))‘(𝐸 ↑ 𝑦)) = ((𝑂‘(1r‘𝑆))‘(𝐸 ↑ 𝑦)))
88	46, 85, 87	3eqtrd 2769	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(1r‘𝑆))‘𝑦)) = ((𝑂‘(1r‘𝑆))‘(𝐸 ↑ 𝑦)))
89	29	eqcomd 2736	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (1r‘𝑆) = (0𝐷𝐹))
90	89	fveq2d 6865	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘(1r‘𝑆)) = (𝑂‘(0𝐷𝐹)))
91	90	fveq1d 6863	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(1r‘𝑆))‘(𝐸 ↑ 𝑦)) = ((𝑂‘(0𝐷𝐹))‘(𝐸 ↑ 𝑦)))
92	32, 88, 91	3eqtrd 2769	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(0𝐷𝐹))‘𝑦)) = ((𝑂‘(0𝐷𝐹))‘(𝐸 ↑ 𝑦)))
93	92	ralrimiva 3126	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘(0𝐷𝐹))‘𝑦)) = ((𝑂‘(0𝐷𝐹))‘(𝐸 ↑ 𝑦)))
94	37	ply1ring 22139	. . . . . . . 8 ⊢ (𝐾 ∈ Ring → 𝑆 ∈ Ring)
95	36, 94	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring)
96	17, 25	ringidcl 20181	. . . . . . 7 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ (Base‘𝑆))
97	95, 96	syl 17	. . . . . 6 ⊢ (𝜑 → (1r‘𝑆) ∈ (Base‘𝑆))
98	28	eqcomd 2736	. . . . . . 7 ⊢ (𝜑 → (1r‘𝑆) = (0𝐷𝐹))
99	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝑆))
100	99	eqcomd 2736	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑆) = 𝐵)
101	98, 100	eleq12d 2823	. . . . . 6 ⊢ (𝜑 → ((1r‘𝑆) ∈ (Base‘𝑆) ↔ (0𝐷𝐹) ∈ 𝐵))
102	97, 101	mpbid 232	. . . . 5 ⊢ (𝜑 → (0𝐷𝐹) ∈ 𝐵)
103	10, 102, 70	aks6d1c1p1 42102	. . . 4 ⊢ (𝜑 → (𝐸 ∼ (0𝐷𝐹) ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘(0𝐷𝐹))‘𝑦)) = ((𝑂‘(0𝐷𝐹))‘(𝐸 ↑ 𝑦))))
104	93, 103	mpbird 257	. . 3 ⊢ (𝜑 → 𝐸 ∼ (0𝐷𝐹))
105		aks6d1c1.4	. . . . 5 ⊢ 𝑋 = (var1‘𝐾)
106		aks6d1c1.10	. . . . 5 ⊢ 𝑃 = (chr‘𝐾)
107		aks6d1c1.12	. . . . 5 ⊢ + = (+g‘𝑆)
108	33	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐾 ∈ Field)
109		aks6d1c1.14	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ)
110	109	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝑃 ∈ ℙ)
111	56	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝑅 ∈ ℕ)
112		aks6d1c1.18	. . . . . 6 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
113	112	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → (𝑁 gcd 𝑅) = 1)
114		aks6d1c1.17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
115	114	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝑃 ∥ 𝑁)
116		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐸 ∼ (𝑖𝐷𝐹))
117	11	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐸 ∼ 𝐹)
118	10, 37, 14, 105, 16, 53, 73, 38, 22, 106, 47, 107, 108, 110, 111, 113, 115, 116, 117	aks6d1c1p4 42106	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐸 ∼ ((𝑖𝐷𝐹)(+g‘𝑊)𝐹))
119	16	ringmgp 20155	. . . . . . . 8 ⊢ (𝑆 ∈ Ring → 𝑊 ∈ Mnd)
120	95, 119	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Mnd)
121	120	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑊 ∈ Mnd)
122	121	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝑊 ∈ Mnd)
123		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝑖 ∈ ℕ0)
124	18	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑊))
125	99, 124	eqtrd 2765	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝑊))
126	125	eleq2d 2815	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∈ 𝐵 ↔ 𝐹 ∈ (Base‘𝑊)))
127	13, 126	mpbid 232	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑊))
128	127	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐹 ∈ (Base‘𝑊))
129	128	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐹 ∈ (Base‘𝑊))
130		eqid 2730	. . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊)
131	20, 22, 130	mulgnn0p1 19024	. . . . 5 ⊢ ((𝑊 ∈ Mnd ∧ 𝑖 ∈ ℕ0 ∧ 𝐹 ∈ (Base‘𝑊)) → ((𝑖 + 1)𝐷𝐹) = ((𝑖𝐷𝐹)(+g‘𝑊)𝐹))
132	122, 123, 129, 131	syl3anc 1373	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → ((𝑖 + 1)𝐷𝐹) = ((𝑖𝐷𝐹)(+g‘𝑊)𝐹))
133	118, 132	breqtrrd 5138	. . 3 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐸 ∼ ((𝑖 + 1)𝐷𝐹))
134	3, 5, 7, 9, 104, 133	nn0indd 12638	. 2 ⊢ ((𝜑 ∧ 𝐿 ∈ ℕ0) → 𝐸 ∼ (𝐿𝐷𝐹))
135	1, 134	mpdan 687	1 ⊢ (𝜑 → 𝐸 ∼ (𝐿𝐷𝐹))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045   class class class wbr 5110  {copab 5172  ‘cfv 6514  (class class class)co 7390  0cc0 11075  1c1 11076   + caddc 11078  ℕcn 12193  ℕ₀cn0 12449   ∥ cdvds 16229   gcd cgcd 16471  ℙcprime 16648  Basecbs 17186  +_gcplusg 17227  0_gc0g 17409  Mndcmnd 18668  ._gcmg 19006  CMndccmn 19717  mulGrpcmgp 20056  1_rcur 20097  Ringcrg 20149  CRingccrg 20150  Fieldcfield 20646  chrcchr 21418  algSccascl 21768  var₁cv1 22067  Poly₁cpl1 22068  eval₁ce1 22208   PrimRoots cprimroots 42086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-ofr 7657  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-pm 8805  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-sup 9400  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-fzo 13623  df-seq 13974  df-hash 14303  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-hom 17251  df-cco 17252  df-0g 17411  df-gsum 17412  df-prds 17417  df-pws 17419  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-submnd 18718  df-grp 18875  df-minusg 18876  df-sbg 18877  df-mulg 19007  df-subg 19062  df-ghm 19152  df-cntz 19256  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-srg 20103  df-ring 20151  df-cring 20152  df-rhm 20388  df-subrng 20462  df-subrg 20486  df-field 20648  df-lmod 20775  df-lss 20845  df-lsp 20885  df-assa 21769  df-asp 21770  df-ascl 21771  df-psr 21825  df-mvr 21826  df-mpl 21827  df-opsr 21829  df-evls 21988  df-evl 21989  df-psr1 22071  df-ply1 22073  df-evl1 22210  df-primroots 42087

This theorem is referenced by:  aks6d1c1  42111