aks6d1c1p4 - Mathbox for metakunt

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Theorem aks6d1c1p4 42068

Description: The product of polynomials is introspective. (Contributed by metakunt, 25-Apr-2025.)

**Hypotheses**
Ref	Expression
aks6d1c1p4.1	⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}
aks6d1c1p4.2	⊢ 𝑆 = (Poly₁‘𝐾)
aks6d1c1p4.3	⊢ 𝐵 = (Base‘𝑆)
aks6d1c1p4.4	⊢ 𝑋 = (var₁‘𝐾)
aks6d1c1p4.5	⊢ 𝑊 = (mulGrp‘𝑆)
aks6d1c1p4.6	⊢ 𝑉 = (mulGrp‘𝐾)
aks6d1c1p4.7	⊢ ↑ = (._g‘𝑉)
aks6d1c1p4.8	⊢ 𝐶 = (algSc‘𝑆)
aks6d1c1p4.9	⊢ 𝐷 = (._g‘𝑊)
aks6d1c1p4.10	⊢ 𝑃 = (chr‘𝐾)
aks6d1c1p4.11	⊢ 𝑂 = (eval₁‘𝐾)
aks6d1c1p4.12	⊢ + = (+_g‘𝑆)
aks6d1c1p4.13	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c1p4.14	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c1p4.15	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c1p4.16	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c1p4.17	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c1p4.18	⊢ (𝜑 → 𝐸 ∼ 𝐹)
aks6d1c1p4.19	⊢ (𝜑 → 𝐸 ∼ 𝐺)

**Assertion**
Ref	Expression
aks6d1c1p4	⊢ (𝜑 → 𝐸 ∼ (𝐹(+_g‘𝑊)𝐺))

Distinct variable groups: ↑ ,𝑒,𝑓,𝑦 𝐵,𝑒,𝑓 𝑒,𝐸,𝑓,𝑦 𝑒,𝐹,𝑓,𝑦 𝑒,𝐺,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) 𝐷(𝑦,𝑒,𝑓) 𝑃(𝑦,𝑒,𝑓) + (𝑦,𝑒,𝑓) ∼ (𝑦,𝑒,𝑓) 𝑆(𝑦,𝑒,𝑓) 𝐾(𝑦,𝑒,𝑓) 𝑁(𝑦,𝑒,𝑓) 𝑋(𝑦,𝑒,𝑓)

Proof of Theorem aks6d1c1p4 Dummy variables 𝑧 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aks6d1c1p4.11	. . . . . . 7 ⊢ 𝑂 = (eval₁‘𝐾)
2		aks6d1c1p4.2	. . . . . . 7 ⊢ 𝑆 = (Poly₁‘𝐾)
3		eqid 2740	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		aks6d1c1p4.3	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
5		aks6d1c1p4.13	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Field)
6	5	fldcrngd 20764	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ CRing)
7	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ CRing)
8		aks6d1c1p4.6	. . . . . . . . 9 ⊢ 𝑉 = (mulGrp‘𝐾)
9	8, 3	mgpbas 20167	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝑉)
10		aks6d1c1p4.7	. . . . . . . 8 ⊢ ↑ = (._g‘𝑉)
11	8	crngmgp 20268	. . . . . . . . . . 11 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
12	6, 11	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ CMnd)
13	12	cmnmndd 19846	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ Mnd)
14	13	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑉 ∈ Mnd)
15		aks6d1c1p4.1	. . . . . . . . . . . 12 ⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}
16		aks6d1c1p4.18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∼ 𝐹)
17	15, 16	aks6d1c1p1rcl 42065	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
18	17	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℕ)
19	18	nnnn0d 12613	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
20	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐸 ∈ ℕ₀)
21		aks6d1c1p4.15	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℕ)
22	21	nnnn0d 12613	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
23		eqid 2740	. . . . . . . . . . . . 13 ⊢ (._g‘𝑉) = (._g‘𝑉)
24	12, 22, 23	isprimroot 42050	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) ↔ (𝑦 ∈ (Base‘𝑉) ∧ (𝑅(._g‘𝑉)𝑦) = (0_g‘𝑉) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙(._g‘𝑉)𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑙))))
25	24	biimpd 229	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅(._g‘𝑉)𝑦) = (0_g‘𝑉) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙(._g‘𝑉)𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑙))))
26	25	imp 406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅(._g‘𝑉)𝑦) = (0_g‘𝑉) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙(._g‘𝑉)𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑙)))
27	26	simp1d 1142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
28	27, 9	eleqtrrdi 2855	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
29	9, 10, 14, 20, 28	mulgnn0cld 19135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ 𝑦) ∈ (Base‘𝐾))
30	17	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
31	30	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐹 ∈ 𝐵)
32		eqidd 2741	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)))
33	31, 32	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐹 ∈ 𝐵 ∧ ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))))
34		aks6d1c1p4.19	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∼ 𝐺)
35	15, 34	aks6d1c1p1rcl 42065	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐺 ∈ 𝐵))
36	35	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
37	36	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐺 ∈ 𝐵)
38		eqidd 2741	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐺)‘(𝐸 ↑ 𝑦)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑦)))
39	37, 38	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐺 ∈ 𝐵 ∧ ((𝑂‘𝐺)‘(𝐸 ↑ 𝑦)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑦))))
40		aks6d1c1p4.5	. . . . . . . . 9 ⊢ 𝑊 = (mulGrp‘𝑆)
41		eqid 2740	. . . . . . . . 9 ⊢ (._r‘𝑆) = (._r‘𝑆)
42	40, 41	mgpplusg 20165	. . . . . . . 8 ⊢ (._r‘𝑆) = (+_g‘𝑊)
43	42	eqcomi 2749	. . . . . . 7 ⊢ (+_g‘𝑊) = (._r‘𝑆)
44		eqid 2740	. . . . . . 7 ⊢ (._r‘𝐾) = (._r‘𝐾)
45	1, 2, 3, 4, 7, 29, 33, 39, 43, 44	evl1muld 22368	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐹(+_g‘𝑊)𝐺) ∈ 𝐵 ∧ ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘(𝐸 ↑ 𝑦)) = (((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))(._r‘𝐾)((𝑂‘𝐺)‘(𝐸 ↑ 𝑦)))))
46	45	simprd 495	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘(𝐸 ↑ 𝑦)) = (((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))(._r‘𝐾)((𝑂‘𝐺)‘(𝐸 ↑ 𝑦))))
47	12	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑉 ∈ CMnd)
48	1, 2, 3, 4, 7, 28, 31	fveval1fvcl 22358	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘𝑦) ∈ (Base‘𝐾))
49	8	eqcomi 2749	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝐾) = 𝑉
50	49	fveq2i 6923	. . . . . . . . . . . . . 14 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘𝑉)
51	9, 50	eqtr4i 2771	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
52	51	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (Base‘𝐾) = (Base‘(mulGrp‘𝐾)))
53	52	eleq2d 2830	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑂‘𝐹)‘𝑦) ∈ (Base‘𝐾) ↔ ((𝑂‘𝐹)‘𝑦) ∈ (Base‘(mulGrp‘𝐾))))
54	48, 53	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘𝑦) ∈ (Base‘(mulGrp‘𝐾)))
55	1, 2, 3, 4, 7, 28, 37	fveval1fvcl 22358	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐺)‘𝑦) ∈ (Base‘𝐾))
56	52	eleq2d 2830	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑂‘𝐺)‘𝑦) ∈ (Base‘𝐾) ↔ ((𝑂‘𝐺)‘𝑦) ∈ (Base‘(mulGrp‘𝐾))))
57	55, 56	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐺)‘𝑦) ∈ (Base‘(mulGrp‘𝐾)))
58	20, 54, 57	3jca 1128	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ∈ ℕ₀ ∧ ((𝑂‘𝐹)‘𝑦) ∈ (Base‘(mulGrp‘𝐾)) ∧ ((𝑂‘𝐺)‘𝑦) ∈ (Base‘(mulGrp‘𝐾))))
59	49	fveq2i 6923	. . . . . . . . . 10 ⊢ (+_g‘(mulGrp‘𝐾)) = (+_g‘𝑉)
60	50, 10, 59	mulgnn0di 19867	. . . . . . . . 9 ⊢ ((𝑉 ∈ CMnd ∧ (𝐸 ∈ ℕ₀ ∧ ((𝑂‘𝐹)‘𝑦) ∈ (Base‘(mulGrp‘𝐾)) ∧ ((𝑂‘𝐺)‘𝑦) ∈ (Base‘(mulGrp‘𝐾)))) → (𝐸 ↑ (((𝑂‘𝐹)‘𝑦)(+_g‘(mulGrp‘𝐾))((𝑂‘𝐺)‘𝑦))) = ((𝐸 ↑ ((𝑂‘𝐹)‘𝑦))(+_g‘(mulGrp‘𝐾))(𝐸 ↑ ((𝑂‘𝐺)‘𝑦))))
61	47, 58, 60	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ (((𝑂‘𝐹)‘𝑦)(+_g‘(mulGrp‘𝐾))((𝑂‘𝐺)‘𝑦))) = ((𝐸 ↑ ((𝑂‘𝐹)‘𝑦))(+_g‘(mulGrp‘𝐾))(𝐸 ↑ ((𝑂‘𝐺)‘𝑦))))
62	8, 44	mgpplusg 20165	. . . . . . . . . . . 12 ⊢ (._r‘𝐾) = (+_g‘𝑉)
63	8	fveq2i 6923	. . . . . . . . . . . 12 ⊢ (+_g‘𝑉) = (+_g‘(mulGrp‘𝐾))
64	62, 63	eqtri 2768	. . . . . . . . . . 11 ⊢ (._r‘𝐾) = (+_g‘(mulGrp‘𝐾))
65	64	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (._r‘𝐾) = (+_g‘(mulGrp‘𝐾)))
66	65	eqcomd 2746	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (+_g‘(mulGrp‘𝐾)) = (._r‘𝐾))
67		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘𝑧) = ((𝑂‘𝐹)‘𝑦))
68	67	oveq2d 7464	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)))
69		oveq2 7456	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝐸 ↑ 𝑧) = (𝐸 ↑ 𝑦))
70	69	fveq2d 6924	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)))
71	68, 70	eqeq12d 2756	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → ((𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)) ↔ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))))
72	15, 30, 18	aks6d1c1p1 42064	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))))
73	16, 72	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)))
74		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘𝐹)‘𝑧))
75	74	oveq2d 7464	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = (𝐸 ↑ ((𝑂‘𝐹)‘𝑧)))
76		oveq2 7456	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝐸 ↑ 𝑦) = (𝐸 ↑ 𝑧))
77	76	fveq2d 6924	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)))
78	75, 77	eqeq12d 2756	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → ((𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)) ↔ (𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧))))
79	78	cbvralvw 3243	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)) ↔ ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)))
80	73, 79	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)))
81	80	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)))
82		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (𝑉 PrimRoots 𝑅))
83	71, 81, 82	rspcdva 3636	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)))
84		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐺)‘𝑧) = ((𝑂‘𝐺)‘𝑦))
85	84	oveq2d 7464	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝐸 ↑ ((𝑂‘𝐺)‘𝑧)) = (𝐸 ↑ ((𝑂‘𝐺)‘𝑦)))
86	69	fveq2d 6924	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐺)‘(𝐸 ↑ 𝑧)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑦)))
87	85, 86	eqeq12d 2756	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → ((𝐸 ↑ ((𝑂‘𝐺)‘𝑧)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑧)) ↔ (𝐸 ↑ ((𝑂‘𝐺)‘𝑦)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑦))))
88	15, 36, 18	aks6d1c1p1 42064	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ∼ 𝐺 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐺)‘𝑦)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑦))))
89	34, 88	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐺)‘𝑦)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑦)))
90		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → ((𝑂‘𝐺)‘𝑦) = ((𝑂‘𝐺)‘𝑧))
91	90	oveq2d 7464	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝐸 ↑ ((𝑂‘𝐺)‘𝑦)) = (𝐸 ↑ ((𝑂‘𝐺)‘𝑧)))
92	76	fveq2d 6924	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → ((𝑂‘𝐺)‘(𝐸 ↑ 𝑦)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑧)))
93	91, 92	eqeq12d 2756	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → ((𝐸 ↑ ((𝑂‘𝐺)‘𝑦)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑦)) ↔ (𝐸 ↑ ((𝑂‘𝐺)‘𝑧)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑧))))
94	93	cbvralvw 3243	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐺)‘𝑦)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑦)) ↔ ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐺)‘𝑧)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑧)))
95	89, 94	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐺)‘𝑧)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑧)))
96	95	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐺)‘𝑧)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑧)))
97	87, 96, 82	rspcdva 3636	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘𝐺)‘𝑦)) = ((𝑂‘𝐺)‘(𝐸 ↑ 𝑦)))
98	66, 83, 97	oveq123d 7469	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐸 ↑ ((𝑂‘𝐹)‘𝑦))(+_g‘(mulGrp‘𝐾))(𝐸 ↑ ((𝑂‘𝐺)‘𝑦))) = (((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))(._r‘𝐾)((𝑂‘𝐺)‘(𝐸 ↑ 𝑦))))
99	61, 98	eqtr2d 2781	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))(._r‘𝐾)((𝑂‘𝐺)‘(𝐸 ↑ 𝑦))) = (𝐸 ↑ (((𝑂‘𝐹)‘𝑦)(+_g‘(mulGrp‘𝐾))((𝑂‘𝐺)‘𝑦))))
100	64	eqcomi 2749	. . . . . . . . . 10 ⊢ (+_g‘(mulGrp‘𝐾)) = (._r‘𝐾)
101	100	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (+_g‘(mulGrp‘𝐾)) = (._r‘𝐾))
102	101	oveqd 7465	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑂‘𝐹)‘𝑦)(+_g‘(mulGrp‘𝐾))((𝑂‘𝐺)‘𝑦)) = (((𝑂‘𝐹)‘𝑦)(._r‘𝐾)((𝑂‘𝐺)‘𝑦)))
103	102	oveq2d 7464	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ (((𝑂‘𝐹)‘𝑦)(+_g‘(mulGrp‘𝐾))((𝑂‘𝐺)‘𝑦))) = (𝐸 ↑ (((𝑂‘𝐹)‘𝑦)(._r‘𝐾)((𝑂‘𝐺)‘𝑦))))
104	99, 103	eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))(._r‘𝐾)((𝑂‘𝐺)‘(𝐸 ↑ 𝑦))) = (𝐸 ↑ (((𝑂‘𝐹)‘𝑦)(._r‘𝐾)((𝑂‘𝐺)‘𝑦))))
105		eqidd 2741	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘𝑦) = ((𝑂‘𝐹)‘𝑦))
106	31, 105	jca 511	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐹 ∈ 𝐵 ∧ ((𝑂‘𝐹)‘𝑦) = ((𝑂‘𝐹)‘𝑦)))
107		eqidd 2741	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐺)‘𝑦) = ((𝑂‘𝐺)‘𝑦))
108	37, 107	jca 511	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐺 ∈ 𝐵 ∧ ((𝑂‘𝐺)‘𝑦) = ((𝑂‘𝐺)‘𝑦)))
109	1, 2, 3, 4, 7, 28, 106, 108, 43, 44	evl1muld 22368	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐹(+_g‘𝑊)𝐺) ∈ 𝐵 ∧ ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘𝑦) = (((𝑂‘𝐹)‘𝑦)(._r‘𝐾)((𝑂‘𝐺)‘𝑦))))
110	109	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘𝑦) = (((𝑂‘𝐹)‘𝑦)(._r‘𝐾)((𝑂‘𝐺)‘𝑦)))
111	110	eqcomd 2746	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑂‘𝐹)‘𝑦)(._r‘𝐾)((𝑂‘𝐺)‘𝑦)) = ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘𝑦))
112	111	oveq2d 7464	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ (((𝑂‘𝐹)‘𝑦)(._r‘𝐾)((𝑂‘𝐺)‘𝑦))) = (𝐸 ↑ ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘𝑦)))
113	104, 112	eqtrd 2780	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))(._r‘𝐾)((𝑂‘𝐺)‘(𝐸 ↑ 𝑦))) = (𝐸 ↑ ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘𝑦)))
114	46, 113	eqtrd 2780	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘(𝐸 ↑ 𝑦)) = (𝐸 ↑ ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘𝑦)))
115	114	eqcomd 2746	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘𝑦)) = ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘(𝐸 ↑ 𝑦)))
116	115	ralrimiva 3152	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘𝑦)) = ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘(𝐸 ↑ 𝑦)))
117	2	ply1crng 22221	. . . . . 6 ⊢ (𝐾 ∈ CRing → 𝑆 ∈ CRing)
118	6, 117	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ CRing)
119	118	crngringd 20273	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Ring)
120	4, 43, 119, 30, 36	ringcld 20286	. . 3 ⊢ (𝜑 → (𝐹(+_g‘𝑊)𝐺) ∈ 𝐵)
121	15, 120, 18	aks6d1c1p1 42064	. 2 ⊢ (𝜑 → (𝐸 ∼ (𝐹(+_g‘𝑊)𝐺) ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘𝑦)) = ((𝑂‘(𝐹(+_g‘𝑊)𝐺))‘(𝐸 ↑ 𝑦))))
122	116, 121	mpbird 257	1 ⊢ (𝜑 → 𝐸 ∼ (𝐹(+_g‘𝑊)𝐺))

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 ‘cfv 6573 (class class class)co 7448 1c1 11185 ℕcn 12293 ℕ₀cn0 12553 ∥ cdvds 16302 gcd cgcd 16540 ℙcprime 16718 Basecbs 17258 +_gcplusg 17311 ._rcmulr 17312 0_gc0g 17499 Mndcmnd 18772 ._gcmg 19107 CMndccmn 19822 mulGrpcmgp 20161 CRingccrg 20261 Fieldcfield 20752 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

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-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 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-oi 9579 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-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-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 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-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 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-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-srg 20214 df-ring 20262 df-cring 20263 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-field 20754 df-lmod 20882 df-lss 20953 df-lsp 20993 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-ply1 22204 df-evl1 22341 df-primroots 42049

This theorem is referenced by: aks6d1c1p6 42071 aks6d1c1 42073

W3C validator