Theorem aks6d1c1p6 42965

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 7423	. . . 4 ⊢ (ℎ = 0 → (ℎ𝐷𝐹) = (0𝐷𝐹))
3	2	breq2d 5119	. . 3 ⊢ (ℎ = 0 → (𝐸 ∼ (ℎ𝐷𝐹) ↔ 𝐸 ∼ (0𝐷𝐹)))
4		oveq1 7423	. . . 4 ⊢ (ℎ = 𝑖 → (ℎ𝐷𝐹) = (𝑖𝐷𝐹))
5	4	breq2d 5119	. . 3 ⊢ (ℎ = 𝑖 → (𝐸 ∼ (ℎ𝐷𝐹) ↔ 𝐸 ∼ (𝑖𝐷𝐹)))
6		oveq1 7423	. . . 4 ⊢ (ℎ = (𝑖 + 1) → (ℎ𝐷𝐹) = ((𝑖 + 1)𝐷𝐹))
7	6	breq2d 5119	. . 3 ⊢ (ℎ = (𝑖 + 1) → (𝐸 ∼ (ℎ𝐷𝐹) ↔ 𝐸 ∼ ((𝑖 + 1)𝐷𝐹)))
8		oveq1 7423	. . . 4 ⊢ (ℎ = 𝐿 → (ℎ𝐷𝐹) = (𝐿𝐷𝐹))
9	8	breq2d 5119	. . 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 42959	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
13	12	simprd 501	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
14		aks6d1c1.3	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝑆)
15	13, 14	eleqtrdi 2872	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑆))
16		aks6d1c1.5	. . . . . . . . . . . . . 14 ⊢ 𝑊 = (mulGrp‘𝑆)
17		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑆) = (Base‘𝑆)
18	16, 17	mgpbas 20277	. . . . . . . . . . . . 13 ⊢ (Base‘𝑆) = (Base‘𝑊)
19	15, 18	eleqtrdi 2872	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑊))
20		eqid 2762	. . . . . . . . . . . . 13 ⊢ (Base‘𝑊) = (Base‘𝑊)
21		eqid 2762	. . . . . . . . . . . . 13 ⊢ (0g‘𝑊) = (0g‘𝑊)
22		aks6d1c1.9	. . . . . . . . . . . . 13 ⊢ 𝐷 = (.g‘𝑊)
23	20, 21, 22	mulg0 19196	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Base‘𝑊) → (0𝐷𝐹) = (0g‘𝑊))
24	19, 23	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (0𝐷𝐹) = (0g‘𝑊))
25		eqid 2762	. . . . . . . . . . . . 13 ⊢ (1r‘𝑆) = (1r‘𝑆)
26	16, 25	ringidval 20321	. . . . . . . . . . . 12 ⊢ (1r‘𝑆) = (0g‘𝑊)
27	26	eqcomi 2771	. . . . . . . . . . 11 ⊢ (0g‘𝑊) = (1r‘𝑆)
28	24, 27	eqtrdi 2813	. . . . . . . . . 10 ⊢ (𝜑 → (0𝐷𝐹) = (1r‘𝑆))
29	28	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (0𝐷𝐹) = (1r‘𝑆))
30	29	fveq2d 6886	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘(0𝐷𝐹)) = (𝑂‘(1r‘𝑆)))
31	30	fveq1d 6884	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(0𝐷𝐹))‘𝑦) = ((𝑂‘(1r‘𝑆))‘𝑦))
32	31	oveq2d 7432	. . . . . 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 20904	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ CRing)
35		crngring 20383	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
36	34, 35	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Ring)
37		aks6d1c1.2	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (Poly1‘𝐾)
38		aks6d1c1.8	. . . . . . . . . . . . . 14 ⊢ 𝐶 = (algSc‘𝑆)
39		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (1r‘𝐾) = (1r‘𝐾)
40	37, 38, 39, 25	ply1scl1 22517	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Ring → (𝐶‘(1r‘𝐾)) = (1r‘𝑆))
41	36, 40	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶‘(1r‘𝐾)) = (1r‘𝑆))
42	41	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐶‘(1r‘𝐾)) = (1r‘𝑆))
43	42	eqcomd 2768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (1r‘𝑆) = (𝐶‘(1r‘𝐾)))
44	43	fveq2d 6886	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘(1r‘𝑆)) = (𝑂‘(𝐶‘(1r‘𝐾))))
45	44	fveq1d 6884	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(1r‘𝑆))‘𝑦) = ((𝑂‘(𝐶‘(1r‘𝐾)))‘𝑦))
46	45	oveq2d 7432	. . . . . . 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 2762	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
49	34	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ CRing)
50	48, 39	ringidcl 20405	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Ring → (1r‘𝐾) ∈ (Base‘𝐾))
51	36, 50	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (1r‘𝐾) ∈ (Base‘𝐾))
52	51	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (1r‘𝐾) ∈ (Base‘𝐾))
53		aks6d1c1.6	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑉 = (mulGrp‘𝐾)
54	53	crngmgp 20379	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
55	34, 54	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑉 ∈ CMnd)
56		aks6d1c1.15	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ∈ ℕ)
57	56	nnnn0d 12590	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
58		eqid 2762	. . . . . . . . . . . . . . . 16 ⊢ (.g‘𝑉) = (.g‘𝑉)
59	55, 57, 58	isprimroot 42944	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 1160	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
63	53, 48	mgpbas 20277	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘𝑉)
64	63	eqcomi 2771	. . . . . . . . . . . 12 ⊢ (Base‘𝑉) = (Base‘𝐾)
65	62, 64	eleqtrdi 2872	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
66	47, 37, 48, 38, 14, 49, 52, 65	evl1scad 22559	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘(1r‘𝐾)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘(1r‘𝐾)))‘𝑦) = (1r‘𝐾)))
67	66	simprd 501	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝐶‘(1r‘𝐾)))‘𝑦) = (1r‘𝐾))
68	67	oveq2d 7432	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(𝐶‘(1r‘𝐾)))‘𝑦)) = (𝐸 ↑ (1r‘𝐾)))
69	55	cmnmndd 19930	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ Mnd)
70	12	simpld 500	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℕ)
71	70	nnnn0d 12590	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℕ0)
72		eqid 2762	. . . . . . . . . . 11 ⊢ (Base‘𝑉) = (Base‘𝑉)
73		aks6d1c1.7	. . . . . . . . . . 11 ⊢ ↑ = (.g‘𝑉)
74	53, 39	ringidval 20321	. . . . . . . . . . 11 ⊢ (1r‘𝐾) = (0g‘𝑉)
75	72, 73, 74	mulgnn0z 19223	. . . . . . . . . 10 ⊢ ((𝑉 ∈ Mnd ∧ 𝐸 ∈ ℕ0) → (𝐸 ↑ (1r‘𝐾)) = (1r‘𝐾))
76	69, 71, 75	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↑ (1r‘𝐾)) = (1r‘𝐾))
77	76	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ (1r‘𝐾)) = (1r‘𝐾))
78	69	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑉 ∈ Mnd)
79	71	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐸 ∈ ℕ0)
80	63, 73	mulgnn0cl 19212	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ Mnd ∧ 𝐸 ∈ ℕ0 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝐸 ↑ 𝑦) ∈ (Base‘𝐾))
81	78, 79, 65, 80	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ 𝑦) ∈ (Base‘𝐾))
82	47, 37, 48, 38, 14, 49, 52, 81	evl1scad 22559	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘(1r‘𝐾)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘(1r‘𝐾)))‘(𝐸 ↑ 𝑦)) = (1r‘𝐾)))
83	82	simprd 501	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝐶‘(1r‘𝐾)))‘(𝐸 ↑ 𝑦)) = (1r‘𝐾))
84	83	eqcomd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (1r‘𝐾) = ((𝑂‘(𝐶‘(1r‘𝐾)))‘(𝐸 ↑ 𝑦)))
85	68, 77, 84	3eqtrd 2801	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(𝐶‘(1r‘𝐾)))‘𝑦)) = ((𝑂‘(𝐶‘(1r‘𝐾)))‘(𝐸 ↑ 𝑦)))
86	42	fveq2d 6886	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘(𝐶‘(1r‘𝐾))) = (𝑂‘(1r‘𝑆)))
87	86	fveq1d 6884	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝐶‘(1r‘𝐾)))‘(𝐸 ↑ 𝑦)) = ((𝑂‘(1r‘𝑆))‘(𝐸 ↑ 𝑦)))
88	46, 85, 87	3eqtrd 2801	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(1r‘𝑆))‘𝑦)) = ((𝑂‘(1r‘𝑆))‘(𝐸 ↑ 𝑦)))
89	29	eqcomd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (1r‘𝑆) = (0𝐷𝐹))
90	89	fveq2d 6886	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂‘(1r‘𝑆)) = (𝑂‘(0𝐷𝐹)))
91	90	fveq1d 6884	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(1r‘𝑆))‘(𝐸 ↑ 𝑦)) = ((𝑂‘(0𝐷𝐹))‘(𝐸 ↑ 𝑦)))
92	32, 88, 91	3eqtrd 2801	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(0𝐷𝐹))‘𝑦)) = ((𝑂‘(0𝐷𝐹))‘(𝐸 ↑ 𝑦)))
93	92	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘(0𝐷𝐹))‘𝑦)) = ((𝑂‘(0𝐷𝐹))‘(𝐸 ↑ 𝑦)))
94	37	ply1ring 22471	. . . . . . . 8 ⊢ (𝐾 ∈ Ring → 𝑆 ∈ Ring)
95	36, 94	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring)
96	17, 25	ringidcl 20405	. . . . . . 7 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ (Base‘𝑆))
97	95, 96	syl 18	. . . . . 6 ⊢ (𝜑 → (1r‘𝑆) ∈ (Base‘𝑆))
98	28	eqcomd 2768	. . . . . . 7 ⊢ (𝜑 → (1r‘𝑆) = (0𝐷𝐹))
99	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝑆))
100	99	eqcomd 2768	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑆) = 𝐵)
101	98, 100	eleq12d 2856	. . . . . 6 ⊢ (𝜑 → ((1r‘𝑆) ∈ (Base‘𝑆) ↔ (0𝐷𝐹) ∈ 𝐵))
102	97, 101	mpbid 235	. . . . 5 ⊢ (𝜑 → (0𝐷𝐹) ∈ 𝐵)
103	10, 102, 70	aks6d1c1p1 42958	. . . 4 ⊢ (𝜑 → (𝐸 ∼ (0𝐷𝐹) ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘(0𝐷𝐹))‘𝑦)) = ((𝑂‘(0𝐷𝐹))‘(𝐸 ↑ 𝑦))))
104	93, 103	mpbird 260	. . 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 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐾 ∈ Field)
109		aks6d1c1.14	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ)
110	109	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝑃 ∈ ℙ)
111	56	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝑅 ∈ ℕ)
112		aks6d1c1.18	. . . . . 6 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
113	112	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → (𝑁 gcd 𝑅) = 1)
114		aks6d1c1.17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
115	114	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝑃 ∥ 𝑁)
116		simpr 490	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐸 ∼ (𝑖𝐷𝐹))
117	11	ad2antrr 739	. . . . 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 42962	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐸 ∼ ((𝑖𝐷𝐹)(+g‘𝑊)𝐹))
119	16	ringmgp 20377	. . . . . . . 8 ⊢ (𝑆 ∈ Ring → 𝑊 ∈ Mnd)
120	95, 119	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Mnd)
121	120	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑊 ∈ Mnd)
122	121	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝑊 ∈ Mnd)
123		simplr 781	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝑖 ∈ ℕ0)
124	18	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑊))
125	99, 124	eqtrd 2797	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝑊))
126	125	eleq2d 2848	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∈ 𝐵 ↔ 𝐹 ∈ (Base‘𝑊)))
127	13, 126	mpbid 235	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑊))
128	127	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐹 ∈ (Base‘𝑊))
129	128	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐹 ∈ (Base‘𝑊))
130		eqid 2762	. . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊)
131	20, 22, 130	mulgnn0p1 19207	. . . . 5 ⊢ ((𝑊 ∈ Mnd ∧ 𝑖 ∈ ℕ0 ∧ 𝐹 ∈ (Base‘𝑊)) → ((𝑖 + 1)𝐷𝐹) = ((𝑖𝐷𝐹)(+g‘𝑊)𝐹))
132	122, 123, 129, 131	syl3anc 1398	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → ((𝑖 + 1)𝐷𝐹) = ((𝑖𝐷𝐹)(+g‘𝑊)𝐹))
133	118, 132	breqtrrd 5137	. . 3 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝐸 ∼ (𝑖𝐷𝐹)) → 𝐸 ∼ ((𝑖 + 1)𝐷𝐹))
134	3, 5, 7, 9, 104, 133	nn0indd 12719	. 2 ⊢ ((𝜑 ∧ 𝐿 ∈ ℕ0) → 𝐸 ∼ (𝐿𝐷𝐹))
135	1, 134	mpdan 700	1 ⊢ (𝜑 → 𝐸 ∼ (𝐿𝐷𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078   class class class wbr 5107  {copab 5171  ‘cfv 6537  (class class class)co 7416  0cc0 11125  1c1 11126   + caddc 11128  ℕcn 12258  ℕ₀cn0 12529   ∥ cdvds 16344   gcd cgcd 16586  ℙcprime 16763  Basecbs 17303  +_gcplusg 17344  0_gc0g 17526  Mndcmnd 18836  ._gcmg 19189  CMndccmn 19906  mulGrpcmgp 20272  1_rcur 20319  Ringcrg 20371  CRingccrg 20372  Fieldcfield 20890  chrcchr 21713  algSccascl 22066  var₁cv1 22400  Poly₁cpl1 22401  eval₁ce1 22538   PrimRoots cprimroots 42942

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7681  df-ofr 7682  df-om 7866  df-1st 7989  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-er 8699  df-map 8831  df-pm 8832  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-sup 9415  df-oi 9485  df-card 9947  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-2 12328  df-3 12329  df-4 12330  df-5 12331  df-6 12332  df-7 12333  df-8 12334  df-9 12335  df-n0 12530  df-z 12617  df-dec 12738  df-uz 12889  df-fz 13562  df-fzo 13710  df-seq 14066  df-hash 14395  df-struct 17241  df-sets 17258  df-slot 17276  df-ndx 17288  df-base 17304  df-ress 17325  df-plusg 17357  df-mulr 17358  df-sca 17360  df-vsca 17361  df-ip 17362  df-tset 17363  df-ple 17364  df-ds 17366  df-hom 17368  df-cco 17369  df-0g 17528  df-gsum 17529  df-prds 17534  df-pws 17536  df-mre 17672  df-mrc 17673  df-acs 17675  df-mgm 18732  df-sgrp 18821  df-mnd 18837  df-mhm 18890  df-submnd 18891  df-grp 19059  df-minusg 19060  df-sbg 19061  df-mulg 19190  df-subg 19245  df-ghm 19340  df-cntz 19443  df-cmn 19908  df-abl 19909  df-mgp 20273  df-rng 20287  df-ur 20320  df-srg 20325  df-ring 20373  df-cring 20374  df-rhm 20612  df-subrng 20707  df-subrg 20731  df-field 20892  df-lmod 21045  df-lss 21115  df-lsp 21155  df-assa 22067  df-asp 22068  df-ascl 22069  df-psr 22123  df-mvr 22124  df-mpl 22125  df-opsr 22127  df-evls 22289  df-evl 22290  df-psr1 22404  df-ply1 22406  df-evl1 22540  df-primroots 42943

This theorem is used by:  aks6d1c1  42967