aks6d1c1p2 - Mathbox for metakunt

	Mathbox for metakunt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > aks6d1c1p2		Structured version Visualization version GIF version

Theorem aks6d1c1p2 42431

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 20677	. . . . . . . . . . . . . 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 20180	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
7	4, 6	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ CMnd)
8		aks6d1c1p2.15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ ℕ)
9	8	nnnn0d 12466	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
10		aks6d1c1p2.7	. . . . . . . . . . 11 ⊢ ↑ = (._g‘𝑉)
11	7, 9, 10	isprimroot 42415	. . . . . . . . . 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 20084	. . . . . . . . . . 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 6839	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑂‘𝐹) = (𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))))
26	25	fveq1d 6837	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
27	26	oveq2d 7376	. . . . . . . . . 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 20184	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
34		aks6d1c1p2.4	. . . . . . . . . . . . . . . 16 ⊢ 𝑋 = (var₁‘𝐾)
35	34, 29, 30	vr1cl 22162	. . . . . . . . . . . . . . 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 22285	. . . . . . . . . . . . . . 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 20185	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐾 ∈ Ring)
42		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
43	42	zrhrhm 21470	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
44		rhmghm 20423	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
45		zringbas 21412	. . . . . . . . . . . . . . . . . . 19 ⊢ ℤ = (Base‘ℤ_ring)
46	45, 15	ghmf 19153	. . . . . . . . . . . . . . . . . 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 7032	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾))
50		aks6d1c1p2.8	. . . . . . . . . . . . . . . . 17 ⊢ 𝐶 = (algSc‘𝑆)
51	29, 50, 15, 30	ply1sclcl 22232	. . . . . . . . . . . . . . . 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 22283	. . . . . . . . . . . . . . 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 22289	. . . . . . . . . . . 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 7376	. . . . . . . . . 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 6837	. . . . . . . . . . . 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 20178	. . . . . . . . . . . . . . . . 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 16605	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
70	68, 69	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℕ)
71	70	nnnn0d 12466	. . . . . . . . . . . . . . . . 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 19029	. . . . . . . . . . . . . . 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 22285	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝑃 ↑ 𝑦)) = (𝑃 ↑ 𝑦)))
77	28, 29, 15, 50, 30, 31, 54, 75	evl1scad 22283	. . . . . . . . . . . . . . . 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 22289	. . . . . . . . . . . . 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 6838	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘𝑊) = (._g‘(mulGrp‘𝑆))
86	83, 85	eqtri 2760	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 = (._g‘(mulGrp‘𝑆))
87	5	fveq2i 6838	. . . . . . . . . . . . . . . . . 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 22293	. . . . . . . . . . . . . . . 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 22289	. . . . . . . . . . . . . . 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 22260	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))) = ((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))
99	98	fveq2d 6839	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑂‘(𝑃𝐷(𝑋(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴))))) = (𝑂‘((𝑃𝐷𝑋)(+_g‘𝑆)(𝐶‘((ℤRHom‘𝐾)‘𝐴)))))
100	99	fveq1d 6837	. . . . . . . . . . . . . 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 22289	. . . . . . . . . . . . . 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 22293	. . . . . . . . . . . . 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 3129	. 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 22143	. . . . . . . . 9 ⊢ (𝐾 ∈ CRing → 𝑆 ∈ CRing)
118	4, 117	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ CRing)
119		crngring 20184	. . . . . . . 8 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
120	118, 119	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring)
121	120	ringgrpd 20181	. . . . . 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 18875	. . . . 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 42429	. 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 5099 {copab 5161 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 1c1 11031 ℕcn 12149 ℕ₀cn0 12405 ℤcz 12492 ∥ cdvds 16183 gcd cgcd 16425 ℙcprime 16602 Basecbs 17140 +_gcplusg 17181 0_gc0g 17363 Mndcmnd 18663 Grpcgrp 18867 ._gcmg 19001 GrpHom cghm 19145 CMndccmn 19713 mulGrpcmgp 20079 Ringcrg 20172 CRingccrg 20173 RingHom crh 20409 DivRingcdr 20666 Fieldcfield 20667 ℤ_ringczring 21405 ℤRHomczrh 21458 chrcchr 21460 algSccascl 21811 var₁cv1 22120 Poly₁cpl1 22121 eval₁ce1 22262 PrimRoots cprimroots 42413

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 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109 ax-mulf 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 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9817 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-xnn0 12479 df-z 12493 df-dec 12612 df-uz 12756 df-rp 12910 df-fz 13428 df-fzo 13575 df-fl 13716 df-mod 13794 df-seq 13929 df-exp 13989 df-fac 14201 df-bc 14230 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-dvds 16184 df-gcd 16426 df-prm 16603 df-phi 16697 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-0g 17365 df-gsum 17366 df-prds 17371 df-pws 17373 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18712 df-submnd 18713 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-ghm 19146 df-cntz 19250 df-od 19461 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-srg 20126 df-ring 20174 df-cring 20175 df-oppr 20277 df-dvdsr 20297 df-unit 20298 df-invr 20328 df-dvr 20341 df-rhm 20412 df-subrng 20483 df-subrg 20507 df-drng 20668 df-field 20669 df-lmod 20817 df-lss 20887 df-lsp 20927 df-cnfld 21314 df-zring 21406 df-zrh 21462 df-chr 21464 df-assa 21812 df-asp 21813 df-ascl 21814 df-psr 21869 df-mvr 21870 df-mpl 21871 df-opsr 21873 df-evls 22033 df-evl 22034 df-psr1 22124 df-vr1 22125 df-ply1 22126 df-coe1 22127 df-evl1 22264 df-primroots 42414

This theorem is referenced by: aks6d1c1 42438

W3C validator