aks6d1c1p2 - Mathbox for metakunt

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Theorem aks6d1c1p2 42127

Description: 𝑃 and linear factors are introspective. (Contributed by metakunt, 25-Apr-2025.)

**Hypotheses**
Ref	Expression
aks6d1c1p2.1	⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}
aks6d1c1p2.2	⊢ 𝑆 = (Poly₁‘𝐾)
aks6d1c1p2.3	⊢ 𝐵 = (Base‘𝑆)
aks6d1c1p2.4	⊢ 𝑋 = (var₁‘𝐾)
aks6d1c1p2.5	⊢ 𝑊 = (mulGrp‘𝑆)
aks6d1c1p2.6	⊢ 𝑉 = (mulGrp‘𝐾)
aks6d1c1p2.7	⊢ ↑ = (._g‘𝑉)
aks6d1c1p2.8	⊢ 𝐶 = (algSc‘𝑆)
aks6d1c1p2.9	⊢ 𝐷 = (._g‘𝑊)
aks6d1c1p2.10	⊢ 𝑃 = (chr‘𝐾)
aks6d1c1p2.11	⊢ 𝑂 = (eval₁‘𝐾)
aks6d1c1p2.12	⊢ + = (+_g‘𝑆)
aks6d1c1p2.13	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c1p2.14	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c1p2.15	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c1p2.16	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c1p2.17	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c1p2.18	⊢ 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))
aks6d1c1p2.19	⊢ (𝜑 → 𝐴 ∈ ℤ)

**Assertion**
Ref	Expression
aks6d1c1p2	⊢ (𝜑 → 𝑃 ∼ 𝐹)

Distinct variable groups: ↑ ,𝑒,𝑓 𝐵,𝑒,𝑓 𝑒,𝐹,𝑓,𝑦 𝑒,𝑂,𝑓 𝑃,𝑒,𝑓,𝑦 𝑅,𝑒,𝑓 𝑒,𝑉,𝑓 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑒,𝑓) 𝐴(𝑦,𝑒,𝑓) 𝐵(𝑦) 𝐶(𝑦,𝑒,𝑓) 𝐷(𝑦,𝑒,𝑓) + (𝑦,𝑒,𝑓) ∼ (𝑦,𝑒,𝑓) 𝑅(𝑦) 𝑆(𝑦,𝑒,𝑓) ↑ (𝑦) 𝐾(𝑦,𝑒,𝑓) 𝑁(𝑦,𝑒,𝑓) 𝑂(𝑦) 𝑉(𝑦) 𝑊(𝑦,𝑒,𝑓) 𝑋(𝑦,𝑒,𝑓)

Proof of Theorem aks6d1c1p2 Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aks6d1c1p2.13	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ Field)
2		isfld 20705	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
3	1, 2	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
4	3	simprd 495	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ CRing)
5		aks6d1c1p2.6	. . . . . . . . . . . . 13 ⊢ 𝑉 = (mulGrp‘𝐾)
6	5	crngmgp 20206	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
7	4, 6	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ CMnd)
8		aks6d1c1p2.15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ ℕ)
9	8	nnnn0d 12567	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
10		aks6d1c1p2.7	. . . . . . . . . . 11 ⊢ ↑ = (._g‘𝑉)
11	7, 9, 10	isprimroot 42111	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) ↔ (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0_g‘𝑉) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙 ↑ 𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑙))))
12	11	biimpd 229	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0_g‘𝑉) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙 ↑ 𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑙))))
13	12	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0_g‘𝑉) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙 ↑ 𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑙)))
14	13	simp1d 1142	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
15		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
16	5, 15	mgpbas 20110	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝑉)
17	16	eqcomi 2745	. . . . . . . . . 10 ⊢ (Base‘𝑉) = (Base‘𝐾)
18	17	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑉) = (Base‘𝐾))
19	18	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (Base‘𝑉) = (Base‘𝐾))
20	19	eleq2d 2821	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦 ∈ (Base‘𝑉) ↔ 𝑦 ∈ (Base‘𝐾)))
21	14, 20	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
22	21	ex 412	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) → 𝑦 ∈ (Base‘𝐾)))
23		aks6d1c1p2.18	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))
24	23	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))
25	24	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑂‘𝐹) = (𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))))
26	25	fveq1d 6883	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
27	26	oveq2d 7426	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑃 ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)))
28		aks6d1c1p2.11	. . . . . . . . . . . . 13 ⊢ 𝑂 = (eval₁‘𝐾)
29		aks6d1c1p2.2	. . . . . . . . . . . . 13 ⊢ 𝑆 = (Poly₁‘𝐾)
30		aks6d1c1p2.3	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑆)
31	4	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → 𝐾 ∈ CRing)
32		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝐾))
33		crngring 20210	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
34		aks6d1c1p2.4	. . . . . . . . . . . . . . . 16 ⊢ 𝑋 = (var₁‘𝐾)
35	34, 29, 30	vr1cl 22158	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ 𝐵)
36	31, 33, 35	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑋 ∈ 𝐵)
37	28, 34, 15, 29, 30, 31, 32	evl1vard 22280	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘𝑦) = 𝑦))
38	37	simprd 495	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘𝑋)‘𝑦) = 𝑦)
39	36, 38	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘𝑦) = 𝑦))
40	4, 33	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ Ring)
41	4	crngringd 20211	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐾 ∈ Ring)
42		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
43	42	zrhrhm 21477	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
44		rhmghm 20449	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
45		zringbas 21419	. . . . . . . . . . . . . . . . . . 19 ⊢ ℤ = (Base‘ℤ_ring)
46	45, 15	ghmf 19208	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
47	41, 43, 44, 46	4syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
48		aks6d1c1p2.19	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℤ)
49	47, 48	ffvelcdmd 7080	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾))
50		aks6d1c1p2.8	. . . . . . . . . . . . . . . . 17 ⊢ 𝐶 = (algSc‘𝑆)
51	29, 50, 15, 30	ply1sclcl 22228	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾)) → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
52	40, 49, 51	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
53	52	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
54	49	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾))
55	28, 29, 15, 50, 30, 31, 54, 32	evl1scad 22278	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘𝑦) = ((ℤRHom‘𝐾)‘𝐴)))
56	55	simprd 495	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘𝑦) = ((ℤRHom‘𝐾)‘𝐴))
57	53, 56	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘𝑦) = ((ℤRHom‘𝐾)‘𝐴)))
58		aks6d1c1p2.12	. . . . . . . . . . . . 13 ⊢ + = (+_g‘𝑆)
59		eqid 2736	. . . . . . . . . . . . 13 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
60	28, 29, 15, 30, 31, 32, 39, 57, 58, 59	evl1addd 22284	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵 ∧ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
61	60	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
62	61	oveq2d 7426	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)) = (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
63	27, 62	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
64	25	fveq1d 6883	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑃 ↑ 𝑦)))
65		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑉) = (Base‘𝑉)
66	5	ringmgp 20204	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Ring → 𝑉 ∈ Mnd)
67	31, 33, 66	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑉 ∈ Mnd)
68		aks6d1c1p2.14	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ ℙ)
69		prmnn 16698	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
70	68, 69	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℕ)
71	70	nnnn0d 12567	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
72	71	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑃 ∈ ℕ₀)
73	32, 16	eleqtrdi 2845	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝑉))
74	65, 10, 67, 72, 73	mulgnn0cld 19083	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ 𝑦) ∈ (Base‘𝑉))
75	74, 17	eleqtrdi 2845	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ 𝑦) ∈ (Base‘𝐾))
76	28, 34, 15, 29, 30, 31, 75	evl1vard 22280	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝑃 ↑ 𝑦)) = (𝑃 ↑ 𝑦)))
77	28, 29, 15, 50, 30, 31, 54, 75	evl1scad 22278	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑃 ↑ 𝑦)) = ((ℤRHom‘𝐾)‘𝐴)))
78	77	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑃 ↑ 𝑦)) = ((ℤRHom‘𝐾)‘𝐴))
79	53, 78	jca 511	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑃 ↑ 𝑦)) = ((ℤRHom‘𝐾)‘𝐴)))
80	28, 29, 15, 30, 31, 75, 76, 79, 58, 59	evl1addd 22284	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵 ∧ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑃 ↑ 𝑦)) = ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
81	80	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑃 ↑ 𝑦)) = ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
82	64, 81	eqtrd 2771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)) = ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
83		aks6d1c1p2.9	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 = (._g‘𝑊)
84		aks6d1c1p2.5	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑊 = (mulGrp‘𝑆)
85	84	fveq2i 6884	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘𝑊) = (._g‘(mulGrp‘𝑆))
86	83, 85	eqtri 2759	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 = (._g‘(mulGrp‘𝑆))
87	5	fveq2i 6884	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘𝑉) = (._g‘(mulGrp‘𝐾))
88	10, 87	eqtri 2759	. . . . . . . . . . . . . . . . 17 ⊢ ↑ = (._g‘(mulGrp‘𝐾))
89	28, 29, 15, 30, 31, 32, 37, 86, 88, 72	evl1expd 22288	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑃𝐷𝑋) ∈ 𝐵 ∧ ((𝑂‘(𝑃𝐷𝑋))‘𝑦) = (𝑃 ↑ 𝑦)))
90		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
91	28, 29, 15, 30, 31, 32, 89, 57, 90, 59	evl1addd 22284	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵 ∧ ((𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
92	91	simprd 495	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
93		eqidd 2737	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
94	92, 93	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
95	94	eqcomd 2742	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
96		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶‘((ℤRHom‘𝐾)‘𝐴)) = (𝐶‘((ℤRHom‘𝐾)‘𝐴))
97		aks6d1c1p2.10	. . . . . . . . . . . . . . . . 17 ⊢ 𝑃 = (chr‘𝐾)
98	29, 34, 90, 84, 83, 50, 96, 97, 4, 68, 48	ply1fermltlchr 22255	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))) = ((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))
99	98	fveq2d 6885	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑂‘(𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))) = (𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))))
100	99	fveq1d 6883	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑂‘(𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))))‘𝑦) = ((𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
101	100	eqcomd 2742	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = ((𝑂‘(𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))))‘𝑦))
102	101	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = ((𝑂‘(𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))))‘𝑦))
103	28, 29, 15, 30, 31, 32, 37, 55, 90, 59	evl1addd 22284	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵 ∧ ((𝑂‘(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
104	28, 29, 15, 30, 31, 32, 103, 86, 88, 72	evl1expd 22288	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))) ∈ 𝐵 ∧ ((𝑂‘(𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))))‘𝑦) = (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))))
105	104	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘(𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))))‘𝑦) = (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
106	95, 102, 105	3eqtrd 2775	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
107	82, 106	eqtrd 2771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)) = (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
108	107	eqcomd 2742	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)))
109	63, 108	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)))
110	109	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)))
111	110	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦 ∈ (Base‘𝐾) → (𝑃 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦))))
112	111	ex 412	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) → (𝑦 ∈ (Base‘𝐾) → (𝑃 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)))))
113	22, 112	mpdd 43	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) → (𝑃 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦))))
114	113	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)))
115	114	ralrimiva 3133	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑃 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)))
116		aks6d1c1p2.1	. . 3 ⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}
117	29	ply1crng 22139	. . . . . . . . 9 ⊢ (𝐾 ∈ CRing → 𝑆 ∈ CRing)
118	4, 117	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ CRing)
119		crngring 20210	. . . . . . . 8 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
120	118, 119	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring)
121	120	ringgrpd 20207	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Grp)
122	41, 35	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
123	121, 122, 52	3jca 1128	. . . . 5 ⊢ (𝜑 → (𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵))
124	30, 58	grpcl 18929	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵)
125	123, 124	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵)
126	23	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))
127	126	eleq1d 2820	. . . 4 ⊢ (𝜑 → (𝐹 ∈ 𝐵 ↔ (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵))
128	125, 127	mpbird 257	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
129	116, 128, 70	aks6d1c1p1 42125	. 2 ⊢ (𝜑 → (𝑃 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑃 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦))))
130	115, 129	mpbird 257	1 ⊢ (𝜑 → 𝑃 ∼ 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3052 class class class wbr 5124 {copab 5186 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 1c1 11135 ℕcn 12245 ℕ₀cn0 12506 ℤcz 12593 ∥ cdvds 16277 gcd cgcd 16518 ℙcprime 16695 Basecbs 17233 +_gcplusg 17276 0_gc0g 17458 Mndcmnd 18717 Grpcgrp 18921 ._gcmg 19055 GrpHom cghm 19200 CMndccmn 19766 mulGrpcmgp 20105 Ringcrg 20198 CRingccrg 20199 RingHom crh 20434 DivRingcdr 20694 Fieldcfield 20695 ℤ_ringczring 21412 ℤRHomczrh 21465 chrcchr 21467 algSccascl 21817 var₁cv1 22116 Poly₁cpl1 22117 eval₁ce1 22257 PrimRoots cprimroots 42109

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213 ax-mulf 11214

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 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 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-ofr 7677 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-xnn0 12580 df-z 12594 df-dec 12714 df-uz 12858 df-rp 13014 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-dvds 16278 df-gcd 16519 df-prm 16696 df-phi 16790 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-0g 17460 df-gsum 17461 df-prds 17466 df-pws 17468 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-ghm 19201 df-cntz 19305 df-od 19514 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-srg 20152 df-ring 20200 df-cring 20201 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-dvr 20366 df-rhm 20437 df-subrng 20511 df-subrg 20535 df-drng 20696 df-field 20697 df-lmod 20824 df-lss 20894 df-lsp 20934 df-cnfld 21321 df-zring 21413 df-zrh 21469 df-chr 21471 df-assa 21818 df-asp 21819 df-ascl 21820 df-psr 21874 df-mvr 21875 df-mpl 21876 df-opsr 21878 df-evls 22037 df-evl 22038 df-psr1 22120 df-vr1 22121 df-ply1 22122 df-coe1 22123 df-evl1 22259 df-primroots 42110

This theorem is referenced by: aks6d1c1 42134

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