aks6d1c1p5 - Mathbox for metakunt

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Theorem aks6d1c1p5 42612

Description: The product of exponents is introspective. (Contributed by metakunt, 26-Apr-2025.)

**Hypotheses**
Ref	Expression
aks6d1c1p5.1	⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}
aks6d1c1p5.2	⊢ 𝑆 = (Poly₁‘𝐾)
aks6d1c1p5.3	⊢ 𝐵 = (Base‘𝑆)
aks6d1c1p5.4	⊢ 𝑋 = (var₁‘𝐾)
aks6d1c1p5.5	⊢ 𝑊 = (mulGrp‘𝑆)
aks6d1c1p5.6	⊢ 𝑉 = (mulGrp‘𝐾)
aks6d1c1p5.7	⊢ ↑ = (._g‘𝑉)
aks6d1c1p5.8	⊢ 𝐶 = (algSc‘𝑆)
aks6d1c1p5.10	⊢ 𝑃 = (chr‘𝐾)
aks6d1c1p5.11	⊢ 𝑂 = (eval₁‘𝐾)
aks6d1c1p5.12	⊢ + = (+_g‘𝑆)
aks6d1c1p5.13	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c1p5.14	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c1p5.15	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c1p5.16	⊢ (𝜑 → (𝐸 gcd 𝑅) = 1)
aks6d1c1p5.17	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c1p5.18	⊢ (𝜑 → 𝐷 ∼ 𝐹)
aks6d1c1p5.19	⊢ (𝜑 → 𝐸 ∼ 𝐹)

**Assertion**
Ref	Expression
aks6d1c1p5	⊢ (𝜑 → (𝐷 · 𝐸) ∼ 𝐹)

Distinct variable groups: ↑ ,𝑒,𝑓,𝑦 𝐵,𝑒,𝑓 𝐷,𝑒,𝑓,𝑦 𝑒,𝐸,𝑓,𝑦 𝑒,𝐹,𝑓,𝑦 𝑒,𝑂,𝑓,𝑦 𝑅,𝑒,𝑓,𝑦 𝑒,𝑉,𝑓,𝑦 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑒,𝑓) 𝐵(𝑦) 𝐶(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) 𝑊(𝑦,𝑒,𝑓) 𝑋(𝑦,𝑒,𝑓)

Proof of Theorem aks6d1c1p5 Dummy variables 𝑖 𝑙 𝑧 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aks6d1c1p5.13	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Field)
2	1	fldcrngd 20718	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ CRing)
3		aks6d1c1p5.6	. . . . . . . . 9 ⊢ 𝑉 = (mulGrp‘𝐾)
4	3	crngmgp 20217	. . . . . . . 8 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ CMnd)
6	5	cmnmndd 19774	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ Mnd)
7	6	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑉 ∈ Mnd)
8		aks6d1c1p5.1	. . . . . . . . . 10 ⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}
9		aks6d1c1p5.18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∼ 𝐹)
10	8, 9	aks6d1c1p1rcl 42608	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
11	10	simpld 496	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℕ)
12	11	nnnn0d 12493	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℕ₀)
13	12	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐷 ∈ ℕ₀)
14		aks6d1c1p5.19	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∼ 𝐹)
15	8, 14	aks6d1c1p1rcl 42608	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
16	15	simpld 496	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℕ)
17	16	nnnn0d 12493	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
18	17	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐸 ∈ ℕ₀)
19		aks6d1c1p5.11	. . . . . . . 8 ⊢ 𝑂 = (eval₁‘𝐾)
20		aks6d1c1p5.2	. . . . . . . 8 ⊢ 𝑆 = (Poly₁‘𝐾)
21		eqid 2741	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
22		aks6d1c1p5.3	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑆)
23	2	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ CRing)
24		aks6d1c1p5.15	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℕ)
25	24	nnnn0d 12493	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
26		aks6d1c1p5.7	. . . . . . . . . . . . 13 ⊢ ↑ = (._g‘𝑉)
27	5, 25, 26	isprimroot 42593	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) ↔ (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0_g‘𝑉) ∧ ∀𝑞 ∈ ℕ₀ ((𝑞 ↑ 𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑞))))
28	27	biimpd 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0_g‘𝑉) ∧ ∀𝑞 ∈ ℕ₀ ((𝑞 ↑ 𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑞))))
29	28	imp 408	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0_g‘𝑉) ∧ ∀𝑞 ∈ ℕ₀ ((𝑞 ↑ 𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑞)))
30	29	simp1d 1149	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
31	3, 21	mgpbas 20121	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝑉)
32	31	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝑉))
33	32	eqcomd 2747	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑉) = (Base‘𝐾))
34	33	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (Base‘𝑉) = (Base‘𝐾))
35	30, 34	eleqtrd 2843	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
36	10	simprd 497	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
37	36	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐹 ∈ 𝐵)
38	19, 20, 21, 22, 23, 35, 37	fveval1fvcl 22323	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘𝑦) ∈ (Base‘𝐾))
39	34	eleq2d 2827	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑂‘𝐹)‘𝑦) ∈ (Base‘𝑉) ↔ ((𝑂‘𝐹)‘𝑦) ∈ (Base‘𝐾)))
40	38, 39	mpbird 259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘𝑦) ∈ (Base‘𝑉))
41	13, 18, 40	3jca 1135	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ∈ ℕ₀ ∧ 𝐸 ∈ ℕ₀ ∧ ((𝑂‘𝐹)‘𝑦) ∈ (Base‘𝑉)))
42		eqid 2741	. . . . . 6 ⊢ (Base‘𝑉) = (Base‘𝑉)
43	42, 26	mulgnn0ass 19081	. . . . 5 ⊢ ((𝑉 ∈ Mnd ∧ (𝐷 ∈ ℕ₀ ∧ 𝐸 ∈ ℕ₀ ∧ ((𝑂‘𝐹)‘𝑦) ∈ (Base‘𝑉))) → ((𝐷 · 𝐸) ↑ ((𝑂‘𝐹)‘𝑦)) = (𝐷 ↑ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦))))
44	7, 41, 43	syl2anc 591	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐷 · 𝐸) ↑ ((𝑂‘𝐹)‘𝑦)) = (𝐷 ↑ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦))))
45		eqidd 2742	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙)) = (𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙)))
46		simpr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑙 = 𝑦) → 𝑙 = 𝑦)
47	46	oveq2d 7376	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑙 = 𝑦) → (𝐸 ↑ 𝑙) = (𝐸 ↑ 𝑦))
48		simpr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (𝑉 PrimRoots 𝑅))
49	42, 26, 7, 18, 30	mulgnn0cld 19066	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ 𝑦) ∈ (Base‘𝑉))
50	45, 47, 48, 49	fvmptd 6947	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦) = (𝐸 ↑ 𝑦))
51	50	fveq2d 6835	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)))
52	51	oveq2d 7376	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦))) = (𝐷 ↑ ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))))
53	52	eqcomd 2747	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ↑ ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))) = (𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦))))
54		2fveq3 6836	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑦 → ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖)) = ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦)))
55	54	oveq2d 7376	. . . . . . . . . 10 ⊢ (𝑖 = 𝑦 → (𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))) = (𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦))))
56		fveq2 6831	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑦 → ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖) = ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦))
57	56	oveq2d 7376	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑦 → (𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖)) = (𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦)))
58	57	fveq2d 6835	. . . . . . . . . 10 ⊢ (𝑖 = 𝑦 → ((𝑂‘𝐹)‘(𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))) = ((𝑂‘𝐹)‘(𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦))))
59	55, 58	eqeq12d 2757	. . . . . . . . 9 ⊢ (𝑖 = 𝑦 → ((𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))) = ((𝑂‘𝐹)‘(𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))) ↔ (𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦))) = ((𝑂‘𝐹)‘(𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦)))))
60	8, 36, 11	aks6d1c1p1 42607	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐷 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐷 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐷 ↑ 𝑦))))
61	60	biimpd 231	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷 ∼ 𝐹 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐷 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐷 ↑ 𝑦))))
62	9, 61	mpd 15	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐷 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐷 ↑ 𝑦)))
63	26	oveqi 7373	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ↑ 𝑙) = (𝐸(._g‘𝑉)𝑙)
64	63	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (𝑉 PrimRoots 𝑅) → (𝐸 ↑ 𝑙) = (𝐸(._g‘𝑉)𝑙))
65	64	mpteq2ia 5170	. . . . . . . . . . . . . 14 ⊢ (𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙)) = (𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸(._g‘𝑉)𝑙))
66		aks6d1c1p5.16	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 gcd 𝑅) = 1)
67	65, 5, 24, 16, 66	primrootscoprbij2 42603	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙)):(𝑉 PrimRoots 𝑅)–1-1-onto→(𝑉 PrimRoots 𝑅))
68		f1ofo 6778	. . . . . . . . . . . . 13 ⊢ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙)):(𝑉 PrimRoots 𝑅)–1-1-onto→(𝑉 PrimRoots 𝑅) → (𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙)):(𝑉 PrimRoots 𝑅)–onto→(𝑉 PrimRoots 𝑅))
69	67, 68	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙)):(𝑉 PrimRoots 𝑅)–onto→(𝑉 PrimRoots 𝑅))
70		fveq2 6831	. . . . . . . . . . . . . . 15 ⊢ (((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖) = 𝑦 → ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖)) = ((𝑂‘𝐹)‘𝑦))
71	70	oveq2d 7376	. . . . . . . . . . . . . 14 ⊢ (((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖) = 𝑦 → (𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))) = (𝐷 ↑ ((𝑂‘𝐹)‘𝑦)))
72		oveq2 7368	. . . . . . . . . . . . . . 15 ⊢ (((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖) = 𝑦 → (𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖)) = (𝐷 ↑ 𝑦))
73	72	fveq2d 6835	. . . . . . . . . . . . . 14 ⊢ (((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖) = 𝑦 → ((𝑂‘𝐹)‘(𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))) = ((𝑂‘𝐹)‘(𝐷 ↑ 𝑦)))
74	71, 73	eqeq12d 2757	. . . . . . . . . . . . 13 ⊢ (((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖) = 𝑦 → ((𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))) = ((𝑂‘𝐹)‘(𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))) ↔ (𝐷 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐷 ↑ 𝑦))))
75	74	cbvfo 7237	. . . . . . . . . . . 12 ⊢ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙)):(𝑉 PrimRoots 𝑅)–onto→(𝑉 PrimRoots 𝑅) → (∀𝑖 ∈ (𝑉 PrimRoots 𝑅)(𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))) = ((𝑂‘𝐹)‘(𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))) ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐷 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐷 ↑ 𝑦))))
76	69, 75	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑖 ∈ (𝑉 PrimRoots 𝑅)(𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))) = ((𝑂‘𝐹)‘(𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))) ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐷 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐷 ↑ 𝑦))))
77	62, 76	mpbird 259	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑖 ∈ (𝑉 PrimRoots 𝑅)(𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))) = ((𝑂‘𝐹)‘(𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))))
78	77	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ∀𝑖 ∈ (𝑉 PrimRoots 𝑅)(𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))) = ((𝑂‘𝐹)‘(𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖))))
79	59, 78, 48	rspcdva 3563	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦))) = ((𝑂‘𝐹)‘(𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦))))
80	50	oveq2d 7376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦)) = (𝐷 ↑ (𝐸 ↑ 𝑦)))
81	80	fveq2d 6835	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦))) = ((𝑂‘𝐹)‘(𝐷 ↑ (𝐸 ↑ 𝑦))))
82	79, 81	eqtrd 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦))) = ((𝑂‘𝐹)‘(𝐷 ↑ (𝐸 ↑ 𝑦))))
83	53, 82	eqtr2d 2777	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝐷 ↑ (𝐸 ↑ 𝑦))) = (𝐷 ↑ ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))))
84		fveq2 6831	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘𝑧) = ((𝑂‘𝐹)‘𝑦))
85	84	oveq2d 7376	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)))
86		oveq2 7368	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝐸 ↑ 𝑧) = (𝐸 ↑ 𝑦))
87	86	fveq2d 6835	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)))
88	85, 87	eqeq12d 2757	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)) ↔ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))))
89	8, 36, 16	aks6d1c1p1 42607	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))))
90	89	biimpd 231	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 ∼ 𝐹 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))))
91	14, 90	mpd 15	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)))
92		nfv 1922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧))
93		nfv 1922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))
94	92, 93, 88	cbvralw 3283	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)) ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)))
95	91, 94	sylibr 236	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)))
96	95	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)))
97	88, 96, 48	rspcdva 3563	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)))
98	97	eqcomd 2747	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)) = (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)))
99	98	oveq2d 7376	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ↑ ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))) = (𝐷 ↑ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦))))
100	83, 99	eqtrd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝐷 ↑ (𝐸 ↑ 𝑦))) = (𝐷 ↑ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦))))
101	100	eqcomd 2747	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ↑ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦))) = ((𝑂‘𝐹)‘(𝐷 ↑ (𝐸 ↑ 𝑦))))
102	13, 18, 30	3jca 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ∈ ℕ₀ ∧ 𝐸 ∈ ℕ₀ ∧ 𝑦 ∈ (Base‘𝑉)))
103	42, 26	mulgnn0ass 19081	. . . . . . 7 ⊢ ((𝑉 ∈ Mnd ∧ (𝐷 ∈ ℕ₀ ∧ 𝐸 ∈ ℕ₀ ∧ 𝑦 ∈ (Base‘𝑉))) → ((𝐷 · 𝐸) ↑ 𝑦) = (𝐷 ↑ (𝐸 ↑ 𝑦)))
104	103	eqcomd 2747	. . . . . 6 ⊢ ((𝑉 ∈ Mnd ∧ (𝐷 ∈ ℕ₀ ∧ 𝐸 ∈ ℕ₀ ∧ 𝑦 ∈ (Base‘𝑉))) → (𝐷 ↑ (𝐸 ↑ 𝑦)) = ((𝐷 · 𝐸) ↑ 𝑦))
105	7, 102, 104	syl2anc 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ↑ (𝐸 ↑ 𝑦)) = ((𝐷 · 𝐸) ↑ 𝑦))
106	105	fveq2d 6835	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝐷 ↑ (𝐸 ↑ 𝑦))) = ((𝑂‘𝐹)‘((𝐷 · 𝐸) ↑ 𝑦)))
107	44, 101, 106	3eqtrd 2780	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐷 · 𝐸) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝐷 · 𝐸) ↑ 𝑦)))
108	107	ralrimiva 3133	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)((𝐷 · 𝐸) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝐷 · 𝐸) ↑ 𝑦)))
109	11, 16	nnmulcld 12225	. . 3 ⊢ (𝜑 → (𝐷 · 𝐸) ∈ ℕ)
110	8, 36, 109	aks6d1c1p1 42607	. 2 ⊢ (𝜑 → ((𝐷 · 𝐸) ∼ 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)((𝐷 · 𝐸) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝐷 · 𝐸) ↑ 𝑦))))
111	108, 110	mpbird 259	1 ⊢ (𝜑 → (𝐷 · 𝐸) ∼ 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ∀wral 3055 class class class wbr 5075 {copab 5137 ↦ cmpt 5156 –onto→wfo 6487 –1-1-onto→wf1o 6488 ‘cfv 6489 (class class class)co 7360 1c1 11034 · cmul 11038 ℕcn 12169 ℕ₀cn0 12432 ∥ cdvds 16216 gcd cgcd 16458 ℙcprime 16635 Basecbs 17174 +_gcplusg 17215 0_gc0g 17397 Mndcmnd 18697 ._gcmg 19038 CMndccmn 19750 mulGrpcmgp 20116 CRingccrg 20210 Fieldcfield 20706 chrcchr 21480 algSccascl 21831 var₁cv1 22165 Poly₁cpl1 22166 eval₁ce1 22304 PrimRoots cprimroots 42591

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 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

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 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-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 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-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-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-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-dvds 16217 df-gcd 16459 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-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 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-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-srg 20163 df-ring 20211 df-cring 20212 df-rhm 20447 df-subrng 20522 df-subrg 20546 df-field 20708 df-lmod 20856 df-lss 20926 df-lsp 20966 df-assa 21832 df-asp 21833 df-ascl 21834 df-psr 21888 df-mvr 21889 df-mpl 21890 df-opsr 21892 df-evls 22054 df-evl 22055 df-psr1 22169 df-ply1 22171 df-evl1 22306 df-primroots 42592

This theorem is referenced by: aks6d1c1p8 42615 aks6d1c1 42616

W3C validator