aks6d1c1p2 - Mathbox for metakunt

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Theorem aks6d1c1p2 42566

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 20712	. . . . . . . . . . . . . 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 20217	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
7	4, 6	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ CMnd)
8		aks6d1c1p2.15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ ℕ)
9	8	nnnn0d 12493	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
10		aks6d1c1p2.7	. . . . . . . . . . 11 ⊢ ↑ = (._g‘𝑉)
11	7, 9, 10	isprimroot 42550	. . . . . . . . . 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 1143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
15		eqid 2737	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
16	5, 15	mgpbas 20121	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝑉)
17	16	eqcomi 2746	. . . . . . . . . 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 2823	. . . . . . 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 6840	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑂‘𝐹) = (𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))))
26	25	fveq1d 6838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
27	26	oveq2d 7378	. . . . . . . . . 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 20221	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
34		aks6d1c1p2.4	. . . . . . . . . . . . . . . 16 ⊢ 𝑋 = (var₁‘𝐾)
35	34, 29, 30	vr1cl 22195	. . . . . . . . . . . . . . 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 22316	. . . . . . . . . . . . . . 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 20222	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐾 ∈ Ring)
42		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
43	42	zrhrhm 21505	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
44		rhmghm 20458	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
45		zringbas 21447	. . . . . . . . . . . . . . . . . . 19 ⊢ ℤ = (Base‘ℤ_ring)
46	45, 15	ghmf 19190	. . . . . . . . . . . . . . . . . 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 7033	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾))
50		aks6d1c1p2.8	. . . . . . . . . . . . . . . . 17 ⊢ 𝐶 = (algSc‘𝑆)
51	29, 50, 15, 30	ply1sclcl 22265	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾)) → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
52	40, 49, 51	syl2anc 585	. . . . . . . . . . . . . . 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 22314	. . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
60	28, 29, 15, 30, 31, 32, 39, 57, 58, 59	evl1addd 22320	. . . . . . . . . . . 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 7378	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)) = (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
63	27, 62	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
64	25	fveq1d 6838	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑃 ↑ 𝑦)))
65		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑉) = (Base‘𝑉)
66	5	ringmgp 20215	. . . . . . . . . . . . . . . . 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 16638	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
70	68, 69	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℕ)
71	70	nnnn0d 12493	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
72	71	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑃 ∈ ℕ₀)
73	32, 16	eleqtrdi 2847	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝑉))
74	65, 10, 67, 72, 73	mulgnn0cld 19066	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ 𝑦) ∈ (Base‘𝑉))
75	74, 17	eleqtrdi 2847	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ 𝑦) ∈ (Base‘𝐾))
76	28, 34, 15, 29, 30, 31, 75	evl1vard 22316	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝑃 ↑ 𝑦)) = (𝑃 ↑ 𝑦)))
77	28, 29, 15, 50, 30, 31, 54, 75	evl1scad 22314	. . . . . . . . . . . . . . . 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 22320	. . . . . . . . . . . . 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 2772	. . . . . . . . . . 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 6839	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘𝑊) = (._g‘(mulGrp‘𝑆))
86	83, 85	eqtri 2760	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 = (._g‘(mulGrp‘𝑆))
87	5	fveq2i 6839	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘𝑉) = (._g‘(mulGrp‘𝐾))
88	10, 87	eqtri 2760	. . . . . . . . . . . . . . . . 17 ⊢ ↑ = (._g‘(mulGrp‘𝐾))
89	28, 29, 15, 30, 31, 32, 37, 86, 88, 72	evl1expd 22324	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑃𝐷𝑋) ∈ 𝐵 ∧ ((𝑂‘(𝑃𝐷𝑋))‘𝑦) = (𝑃 ↑ 𝑦)))
90		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
91	28, 29, 15, 30, 31, 32, 89, 57, 90, 59	evl1addd 22320	. . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
94	92, 93	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
95	94	eqcomd 2743	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
96		eqid 2737	. . . . . . . . . . . . . . . . 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 22291	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))) = ((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))
99	98	fveq2d 6840	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑂‘(𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))) = (𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))))
100	99	fveq1d 6838	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑂‘(𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))))‘𝑦) = ((𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
101	100	eqcomd 2743	. . . . . . . . . . . . 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 22320	. . . . . . . . . . . . . 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 22324	. . . . . . . . . . . . 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 2776	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑃 ↑ 𝑦)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)) = (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
107	82, 106	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)) = (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
108	107	eqcomd 2743	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ (𝑦(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)))
109	63, 108	eqtrd 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑃 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝑃 ↑ 𝑦)))
110	109	adantlr 716	. . . . . . 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 3130	. 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 22176	. . . . . . . . 9 ⊢ (𝐾 ∈ CRing → 𝑆 ∈ CRing)
118	4, 117	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ CRing)
119		crngring 20221	. . . . . . . 8 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
120	118, 119	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring)
121	120	ringgrpd 20218	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Grp)
122	41, 35	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
123	121, 122, 52	3jca 1129	. . . . 5 ⊢ (𝜑 → (𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵))
124	30, 58	grpcl 18912	. . . . 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 2822	. . . 4 ⊢ (𝜑 → (𝐹 ∈ 𝐵 ↔ (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵))
128	125, 127	mpbird 257	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
129	116, 128, 70	aks6d1c1p1 42564	. 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 1542 ∈ wcel 2114 ∀wral 3052 class class class wbr 5086 {copab 5148 ⟶wf 6490 ‘cfv 6494 (class class class)co 7362 1c1 11034 ℕcn 12169 ℕ₀cn0 12432 ℤcz 12519 ∥ cdvds 16216 gcd cgcd 16458 ℙcprime 16635 Basecbs 17174 +_gcplusg 17215 0_gc0g 17397 Mndcmnd 18697 Grpcgrp 18904 ._gcmg 19038 GrpHom cghm 19182 CMndccmn 19750 mulGrpcmgp 20116 Ringcrg 20209 CRingccrg 20210 RingHom crh 20444 DivRingcdr 20701 Fieldcfield 20702 ℤ_ringczring 21440 ℤRHomczrh 21493 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 ax-pre-sup 11111 ax-addf 11112 ax-mulf 11113

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-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-oadd 8404 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-inf 9351 df-oi 9420 df-dju 9820 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-div 11803 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-xnn0 12506 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-fac 14231 df-bc 14260 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-dvds 16217 df-gcd 16459 df-prm 16636 df-phi 16731 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-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 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-od 19498 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-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-rhm 20447 df-subrng 20518 df-subrg 20542 df-drng 20703 df-field 20704 df-lmod 20852 df-lss 20922 df-lsp 20962 df-cnfld 21349 df-zring 21441 df-zrh 21497 df-chr 21499 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-vr1 22158 df-ply1 22159 df-coe1 22160 df-evl1 22295 df-primroots 42549

This theorem is referenced by: aks6d1c1 42573

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