Theorem aks6d1c1p6 42571

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 7369	. . . 4 ⊢ (ℎ = 0 → (ℎ𝐷𝐹) = (0𝐷𝐹))
3	2	breq2d 5098	. . 3 ⊢ (ℎ = 0 → (𝐸 ∼ (ℎ𝐷𝐹) ↔ 𝐸 ∼ (0𝐷𝐹)))
4		oveq1 7369	. . . 4 ⊢ (ℎ = 𝑖 → (ℎ𝐷𝐹) = (𝑖𝐷𝐹))
5	4	breq2d 5098	. . 3 ⊢ (ℎ = 𝑖 → (𝐸 ∼ (ℎ𝐷𝐹) ↔ 𝐸 ∼ (𝑖𝐷𝐹)))
6		oveq1 7369	. . . 4 ⊢ (ℎ = (𝑖 + 1) → (ℎ𝐷𝐹) = ((𝑖 + 1)𝐷𝐹))
7	6	breq2d 5098	. . 3 ⊢ (ℎ = (𝑖 + 1) → (𝐸 ∼ (ℎ𝐷𝐹) ↔ 𝐸 ∼ ((𝑖 + 1)𝐷𝐹)))
8		oveq1 7369	. . . 4 ⊢ (ℎ = 𝐿 → (ℎ𝐷𝐹) = (𝐿𝐷𝐹))
9	8	breq2d 5098	. . 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 42565	. . . . . . . . . . . . . . 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 2847	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑆))
16		aks6d1c1.5	. . . . . . . . . . . . . 14 ⊢ 𝑊 = (mulGrp‘𝑆)
17		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑆) = (Base‘𝑆)
18	16, 17	mgpbas 20121	. . . . . . . . . . . . 13 ⊢ (Base‘𝑆) = (Base‘𝑊)
19	15, 18	eleqtrdi 2847	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑊))
20		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Base‘𝑊) = (Base‘𝑊)
21		eqid 2737	. . . . . . . . . . . . 13 ⊢ (0g‘𝑊) = (0g‘𝑊)
22		aks6d1c1.9	. . . . . . . . . . . . 13 ⊢ 𝐷 = (.g‘𝑊)
23	20, 21, 22	mulg0 19045	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Base‘𝑊) → (0𝐷𝐹) = (0g‘𝑊))
24	19, 23	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (0𝐷𝐹) = (0g‘𝑊))
25		eqid 2737	. . . . . . . . . . . . 13 ⊢ (1r‘𝑆) = (1r‘𝑆)
26	16, 25	ringidval 20159	. . . . . . . . . . . 12 ⊢ (1r‘𝑆) = (0g‘𝑊)
27	26	eqcomi 2746	. . . . . . . . . . 11 ⊢ (0g‘𝑊) = (1r‘𝑆)
28	24, 27	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝜑 → (0𝐷𝐹) = (1r‘𝑆))
29	28	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (0𝐷𝐹) = (1r‘𝑆))
30	29	fveq2d 6840	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘(0𝐷𝐹)) = (𝑂‘(1r‘𝑆)))
31	30	fveq1d 6838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(0𝐷𝐹))‘𝑦) = ((𝑂‘(1r‘𝑆))‘𝑦))
32	31	oveq2d 7378	. . . . . 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 20714	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ CRing)
35		crngring 20221	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (1r‘𝐾) = (1r‘𝐾)
40	37, 38, 39, 25	ply1scl1 22271	. . . . . . . . . . . . 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 2743	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (1r‘𝑆) = (𝐶‘(1r‘𝐾)))
44	43	fveq2d 6840	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘(1r‘𝑆)) = (𝑂‘(𝐶‘(1r‘𝐾))))
45	44	fveq1d 6838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(1r‘𝑆))‘𝑦) = ((𝑂‘(𝐶‘(1r‘𝐾)))‘𝑦))
46	45	oveq2d 7378	. . . . . . 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 2737	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
49	34	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ CRing)
50	48, 39	ringidcl 20241	. . . . . . . . . . . . 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 20217	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
55	34, 54	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑉 ∈ CMnd)
56		aks6d1c1.15	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ∈ ℕ)
57	56	nnnn0d 12493	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
58		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (.g‘𝑉) = (.g‘𝑉)
59	55, 57, 58	isprimroot 42550	. . . . . . . . . . . . . . 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 1143	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
63	53, 48	mgpbas 20121	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘𝑉)
64	63	eqcomi 2746	. . . . . . . . . . . 12 ⊢ (Base‘𝑉) = (Base‘𝐾)
65	62, 64	eleqtrdi 2847	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
66	47, 37, 48, 38, 14, 49, 52, 65	evl1scad 22314	. . . . . . . . . 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 7378	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(𝐶‘(1r‘𝐾)))‘𝑦)) = (𝐸 ↑ (1r‘𝐾)))
69	55	cmnmndd 19774	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ Mnd)
70	12	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℕ)
71	70	nnnn0d 12493	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℕ0)
72		eqid 2737	. . . . . . . . . . 11 ⊢ (Base‘𝑉) = (Base‘𝑉)
73		aks6d1c1.7	. . . . . . . . . . 11 ⊢ ↑ = (.g‘𝑉)
74	53, 39	ringidval 20159	. . . . . . . . . . 11 ⊢ (1r‘𝐾) = (0g‘𝑉)
75	72, 73, 74	mulgnn0z 19072	. . . . . . . . . 10 ⊢ ((𝑉 ∈ Mnd ∧ 𝐸 ∈ ℕ0) → (𝐸 ↑ (1r‘𝐾)) = (1r‘𝐾))
76	69, 71, 75	syl2anc 585	. . . . . . . . 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 19061	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ Mnd ∧ 𝐸 ∈ ℕ0 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝐸 ↑ 𝑦) ∈ (Base‘𝐾))
81	78, 79, 65, 80	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ 𝑦) ∈ (Base‘𝐾))
82	47, 37, 48, 38, 14, 49, 52, 81	evl1scad 22314	. . . . . . . . . 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 2743	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (1r‘𝐾) = ((𝑂‘(𝐶‘(1r‘𝐾)))‘(𝐸 ↑ 𝑦)))
85	68, 77, 84	3eqtrd 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(𝐶‘(1r‘𝐾)))‘𝑦)) = ((𝑂‘(𝐶‘(1r‘𝐾)))‘(𝐸 ↑ 𝑦)))
86	42	fveq2d 6840	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘(𝐶‘(1r‘𝐾))) = (𝑂‘(1r‘𝑆)))
87	86	fveq1d 6838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝐶‘(1r‘𝐾)))‘(𝐸 ↑ 𝑦)) = ((𝑂‘(1r‘𝑆))‘(𝐸 ↑ 𝑦)))
88	46, 85, 87	3eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(1r‘𝑆))‘𝑦)) = ((𝑂‘(1r‘𝑆))‘(𝐸 ↑ 𝑦)))
89	29	eqcomd 2743	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (1r‘𝑆) = (0𝐷𝐹))
90	89	fveq2d 6840	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘(1r‘𝑆)) = (𝑂‘(0𝐷𝐹)))
91	90	fveq1d 6838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(1r‘𝑆))‘(𝐸 ↑ 𝑦)) = ((𝑂‘(0𝐷𝐹))‘(𝐸 ↑ 𝑦)))
92	32, 88, 91	3eqtrd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(0𝐷𝐹))‘𝑦)) = ((𝑂‘(0𝐷𝐹))‘(𝐸 ↑ 𝑦)))
93	92	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘(0𝐷𝐹))‘𝑦)) = ((𝑂‘(0𝐷𝐹))‘(𝐸 ↑ 𝑦)))
94	37	ply1ring 22225	. . . . . . . 8 ⊢ (𝐾 ∈ Ring → 𝑆 ∈ Ring)
95	36, 94	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring)
96	17, 25	ringidcl 20241	. . . . . . 7 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ (Base‘𝑆))
97	95, 96	syl 17	. . . . . 6 ⊢ (𝜑 → (1r‘𝑆) ∈ (Base‘𝑆))
98	28	eqcomd 2743	. . . . . . 7 ⊢ (𝜑 → (1r‘𝑆) = (0𝐷𝐹))
99	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝑆))
100	99	eqcomd 2743	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑆) = 𝐵)
101	98, 100	eleq12d 2831	. . . . . 6 ⊢ (𝜑 → ((1r‘𝑆) ∈ (Base‘𝑆) ↔ (0𝐷𝐹) ∈ 𝐵))
102	97, 101	mpbid 232	. . . . 5 ⊢ (𝜑 → (0𝐷𝐹) ∈ 𝐵)
103	10, 102, 70	aks6d1c1p1 42564	. . . 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 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐾 ∈ Field)
109		aks6d1c1.14	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ)
110	109	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝑃 ∈ ℙ)
111	56	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝑅 ∈ ℕ)
112		aks6d1c1.18	. . . . . 6 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
113	112	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → (𝑁 gcd 𝑅) = 1)
114		aks6d1c1.17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
115	114	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝑃 ∥ 𝑁)
116		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐸 ∼ (𝑖𝐷𝐹))
117	11	ad2antrr 727	. . . . 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 42568	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐸 ∼ ((𝑖𝐷𝐹)(+g‘𝑊)𝐹))
119	16	ringmgp 20215	. . . . . . . 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 769	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝑖 ∈ ℕ0)
124	18	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑊))
125	99, 124	eqtrd 2772	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝑊))
126	125	eleq2d 2823	. . . . . . . 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 2737	. . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊)
131	20, 22, 130	mulgnn0p1 19056	. . . . 5 ⊢ ((𝑊 ∈ Mnd ∧ 𝑖 ∈ ℕ0 ∧ 𝐹 ∈ (Base‘𝑊)) → ((𝑖 + 1)𝐷𝐹) = ((𝑖𝐷𝐹)(+g‘𝑊)𝐹))
132	122, 123, 129, 131	syl3anc 1374	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → ((𝑖 + 1)𝐷𝐹) = ((𝑖𝐷𝐹)(+g‘𝑊)𝐹))
133	118, 132	breqtrrd 5114	. . 3 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐸 ∼ ((𝑖 + 1)𝐷𝐹))
134	3, 5, 7, 9, 104, 133	nn0indd 12621	. 2 ⊢ ((𝜑 ∧ 𝐿 ∈ ℕ0) → 𝐸 ∼ (𝐿𝐷𝐹))
135	1, 134	mpdan 688	1 ⊢ (𝜑 → 𝐸 ∼ (𝐿𝐷𝐹))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5086  {copab 5148  ‘cfv 6494  (class class class)co 7362  0cc0 11033  1c1 11034   + caddc 11036  ℕcn 12169  ℕ₀cn0 12432   ∥ cdvds 16216   gcd cgcd 16458  ℙcprime 16635  Basecbs 17174  +_gcplusg 17215  0_gc0g 17397  Mndcmnd 18697  ._gcmg 19038  CMndccmn 19750  mulGrpcmgp 20116  1_rcur 20157  Ringcrg 20209  CRingccrg 20210  Fieldcfield 20702  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

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-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  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-oi 9420  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-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-z 12520  df-dec 12640  df-uz 12784  df-fz 13457  df-fzo 13604  df-seq 13959  df-hash 14288  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-sca 17231  df-vsca 17232  df-ip 17233  df-tset 17234  df-ple 17235  df-ds 17237  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-cmn 19752  df-abl 19753  df-mgp 20117  df-rng 20129  df-ur 20158  df-srg 20163  df-ring 20211  df-cring 20212  df-rhm 20447  df-subrng 20518  df-subrg 20542  df-field 20704  df-lmod 20852  df-lss 20922  df-lsp 20962  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-ply1 22159  df-evl1 22295  df-primroots 42549

This theorem is referenced by:  aks6d1c1  42573