aks6d1c1p2 - Mathbox for metakunt

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Theorem aks6d1c1p2 42066

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 20762	. . . . . . . . . . . . . 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 20268	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
7	4, 6	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ CMnd)
8		aks6d1c1p2.15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ ℕ)
9	8	nnnn0d 12613	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
10		aks6d1c1p2.7	. . . . . . . . . . 11 ⊢ ↑ = (._g‘𝑉)
11	7, 9, 10	isprimroot 42050	. . . . . . . . . 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 2740	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
16	5, 15	mgpbas 20167	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝑉)
17	16	eqcomi 2749	. . . . . . . . . 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 2830	. . . . . . 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 6924	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑂‘𝐹) = (𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))))
26	25	fveq1d 6922	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
27	26	oveq2d 7464	. . . . . . . . . 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 20272	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
34		aks6d1c1p2.4	. . . . . . . . . . . . . . . 16 ⊢ 𝑋 = (var₁‘𝐾)
35	34, 29, 30	vr1cl 22240	. . . . . . . . . . . . . . 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 22362	. . . . . . . . . . . . . . 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 20273	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐾 ∈ Ring)
42		eqid 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
43	42	zrhrhm 21545	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
44		rhmghm 20510	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
45		zringbas 21487	. . . . . . . . . . . . . . . . . . 19 ⊢ ℤ = (Base‘ℤ_ring)
46	45, 15	ghmf 19260	. . . . . . . . . . . . . . . . . 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 7119	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾))
50		aks6d1c1p2.8	. . . . . . . . . . . . . . . . 17 ⊢ 𝐶 = (algSc‘𝑆)
51	29, 50, 15, 30	ply1sclcl 22310	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾)) → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
52	40, 49, 51	syl2anc 583	. . . . . . . . . . . . . . 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 22360	. . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . 13 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
60	28, 29, 15, 30, 31, 32, 39, 57, 58, 59	evl1addd 22366	. . . . . . . . . . . 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 7464	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)) = (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
63	27, 62	eqtrd 2780	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
64	25	fveq1d 6922	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑃 ↑ 𝑦)))
65		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑉) = (Base‘𝑉)
66	5	ringmgp 20266	. . . . . . . . . . . . . . . . 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 16721	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
70	68, 69	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℕ)
71	70	nnnn0d 12613	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
72	71	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑃 ∈ ℕ₀)
73	32, 16	eleqtrdi 2854	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝑉))
74	65, 10, 67, 72, 73	mulgnn0cld 19135	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ 𝑦) ∈ (Base‘𝑉))
75	74, 17	eleqtrdi 2854	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ 𝑦) ∈ (Base‘𝐾))
76	28, 34, 15, 29, 30, 31, 75	evl1vard 22362	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝑃 ↑ 𝑦)) = (𝑃 ↑ 𝑦)))
77	28, 29, 15, 50, 30, 31, 54, 75	evl1scad 22360	. . . . . . . . . . . . . . . 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 22366	. . . . . . . . . . . . 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 2780	. . . . . . . . . . 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 6923	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘𝑊) = (._g‘(mulGrp‘𝑆))
86	83, 85	eqtri 2768	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 = (._g‘(mulGrp‘𝑆))
87	5	fveq2i 6923	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘𝑉) = (._g‘(mulGrp‘𝐾))
88	10, 87	eqtri 2768	. . . . . . . . . . . . . . . . 17 ⊢ ↑ = (._g‘(mulGrp‘𝐾))
89	28, 29, 15, 30, 31, 32, 37, 86, 88, 72	evl1expd 22370	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑃𝐷𝑋) ∈ 𝐵 ∧ ((𝑂‘(𝑃𝐷𝑋))‘𝑦) = (𝑃 ↑ 𝑦)))
90		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
91	28, 29, 15, 30, 31, 32, 89, 57, 90, 59	evl1addd 22366	. . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
94	92, 93	eqtrd 2780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
95	94	eqcomd 2746	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
96		eqid 2740	. . . . . . . . . . . . . . . . 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 22337	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))) = ((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))
99	98	fveq2d 6924	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑂‘(𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))) = (𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))))
100	99	fveq1d 6922	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑂‘(𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))))‘𝑦) = ((𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
101	100	eqcomd 2746	. . . . . . . . . . . . 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 22366	. . . . . . . . . . . . . 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 22370	. . . . . . . . . . . . 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 2784	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
107	82, 106	eqtrd 2780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)) = (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
108	107	eqcomd 2746	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)))
109	63, 108	eqtrd 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)))
110	109	adantlr 714	. . . . . . 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 3152	. 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 22221	. . . . . . . . 9 ⊢ (𝐾 ∈ CRing → 𝑆 ∈ CRing)
118	4, 117	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ CRing)
119		crngring 20272	. . . . . . . 8 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
120	118, 119	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring)
121	120	ringgrpd 20269	. . . . . 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 18981	. . . . 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 2829	. . . 4 ⊢ (𝜑 → (𝐹 ∈ 𝐵 ↔ (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵))
128	125, 127	mpbird 257	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
129	116, 128, 70	aks6d1c1p1 42064	. 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 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 class class class wbr 5166 {copab 5228 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 1c1 11185 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 ∥ cdvds 16302 gcd cgcd 16540 ℙcprime 16718 Basecbs 17258 +_gcplusg 17311 0_gc0g 17499 Mndcmnd 18772 Grpcgrp 18973 ._gcmg 19107 GrpHom cghm 19252 CMndccmn 19822 mulGrpcmgp 20161 Ringcrg 20260 CRingccrg 20261 RingHom crh 20495 DivRingcdr 20751 Fieldcfield 20752 ℤ_ringczring 21480 ℤRHomczrh 21533 chrcchr 21535 algSccascl 21895 var₁cv1 22198 Poly₁cpl1 22199 eval₁ce1 22339 PrimRoots cprimroots 42048

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-gcd 16541 df-prm 16719 df-phi 16813 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-ghm 19253 df-cntz 19357 df-od 19570 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-srg 20214 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-drng 20753 df-field 20754 df-lmod 20882 df-lss 20953 df-lsp 20993 df-cnfld 21388 df-zring 21481 df-zrh 21537 df-chr 21539 df-assa 21896 df-asp 21897 df-ascl 21898 df-psr 21952 df-mvr 21953 df-mpl 21954 df-opsr 21956 df-evls 22121 df-evl 22122 df-psr1 22202 df-vr1 22203 df-ply1 22204 df-coe1 22205 df-evl1 22341 df-primroots 42049

This theorem is referenced by: aks6d1c1 42073

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