aks6d1c1p5 - Mathbox for metakunt

	Mathbox for metakunt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > aks6d1c1p5		Structured version Visualization version GIF version

Theorem aks6d1c1p5 42215

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 20657	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ CRing)
3		aks6d1c1p5.6	. . . . . . . . 9 ⊢ 𝑉 = (mulGrp‘𝐾)
4	3	crngmgp 20159	. . . . . . . 8 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ CMnd)
6	5	cmnmndd 19716	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ Mnd)
7	6	adantr 480	. . . . 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 42211	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
11	10	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℕ)
12	11	nnnn0d 12442	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℕ₀)
13	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐷 ∈ ℕ₀)
14		aks6d1c1p5.19	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∼ 𝐹)
15	8, 14	aks6d1c1p1rcl 42211	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
16	15	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℕ)
17	16	nnnn0d 12442	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
18	17	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐸 ∈ ℕ₀)
19		aks6d1c1p5.11	. . . . . . . 8 ⊢ 𝑂 = (eval₁‘𝐾)
20		aks6d1c1p5.2	. . . . . . . 8 ⊢ 𝑆 = (Poly₁‘𝐾)
21		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
22		aks6d1c1p5.3	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑆)
23	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ CRing)
24		aks6d1c1p5.15	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℕ)
25	24	nnnn0d 12442	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
26		aks6d1c1p5.7	. . . . . . . . . . . . 13 ⊢ ↑ = (._g‘𝑉)
27	5, 25, 26	isprimroot 42196	. . . . . . . . . . . 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 229	. . . . . . . . . . 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 406	. . . . . . . . . 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 1142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
31	3, 21	mgpbas 20063	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝑉)
32	31	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝑉))
33	32	eqcomd 2737	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑉) = (Base‘𝐾))
34	33	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (Base‘𝑉) = (Base‘𝐾))
35	30, 34	eleqtrd 2833	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
36	10	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
37	36	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐹 ∈ 𝐵)
38	19, 20, 21, 22, 23, 35, 37	fveval1fvcl 22248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘𝑦) ∈ (Base‘𝐾))
39	34	eleq2d 2817	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑂‘𝐹)‘𝑦) ∈ (Base‘𝑉) ↔ ((𝑂‘𝐹)‘𝑦) ∈ (Base‘𝐾)))
40	38, 39	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘𝑦) ∈ (Base‘𝑉))
41	13, 18, 40	3jca 1128	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ∈ ℕ₀ ∧ 𝐸 ∈ ℕ₀ ∧ ((𝑂‘𝐹)‘𝑦) ∈ (Base‘𝑉)))
42		eqid 2731	. . . . . 6 ⊢ (Base‘𝑉) = (Base‘𝑉)
43	42, 26	mulgnn0ass 19023	. . . . 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 584	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐷 · 𝐸) ↑ ((𝑂‘𝐹)‘𝑦)) = (𝐷 ↑ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦))))
45		eqidd 2732	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙)) = (𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙)))
46		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑙 = 𝑦) → 𝑙 = 𝑦)
47	46	oveq2d 7362	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑙 = 𝑦) → (𝐸 ↑ 𝑙) = (𝐸 ↑ 𝑦))
48		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (𝑉 PrimRoots 𝑅))
49	42, 26, 7, 18, 30	mulgnn0cld 19008	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ 𝑦) ∈ (Base‘𝑉))
50	45, 47, 48, 49	fvmptd 6936	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦) = (𝐸 ↑ 𝑦))
51	50	fveq2d 6826	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)))
52	51	oveq2d 7362	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦))) = (𝐷 ↑ ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))))
53	52	eqcomd 2737	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ↑ ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))) = (𝐷 ↑ ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦))))
54		2fveq3 6827	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑦 → ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖)) = ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦)))
55	54	oveq2d 7362	. . . . . . . . . 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 6822	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑦 → ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖) = ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦))
57	56	oveq2d 7362	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑦 → (𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖)) = (𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦)))
58	57	fveq2d 6826	. . . . . . . . . 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 2747	. . . . . . . . 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 42210	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐷 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐷 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐷 ↑ 𝑦))))
61	60	biimpd 229	. . . . . . . . . . . 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 7359	. . . . . . . . . . . . . . . 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 5184	. . . . . . . . . . . . . 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 42206	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙)):(𝑉 PrimRoots 𝑅)–1-1-onto→(𝑉 PrimRoots 𝑅))
68		f1ofo 6770	. . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . 15 ⊢ (((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖) = 𝑦 → ((𝑂‘𝐹)‘((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖)) = ((𝑂‘𝐹)‘𝑦))
71	70	oveq2d 7362	. . . . . . . . . . . . . 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 7354	. . . . . . . . . . . . . . 15 ⊢ (((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖) = 𝑦 → (𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑖)) = (𝐷 ↑ 𝑦))
73	72	fveq2d 6826	. . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . 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 7223	. . . . . . . . . . . 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 257	. . . . . . . . . 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 480	. . . . . . . . 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 3573	. . . . . . . 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 7362	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ↑ ((𝑙 ∈ (𝑉 PrimRoots 𝑅) ↦ (𝐸 ↑ 𝑙))‘𝑦)) = (𝐷 ↑ (𝐸 ↑ 𝑦)))
81	80	fveq2d 6826	. . . . . . . 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 2766	. . . . . . 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 2767	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝐷 ↑ (𝐸 ↑ 𝑦))) = (𝐷 ↑ ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))))
84		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘𝑧) = ((𝑂‘𝐹)‘𝑦))
85	84	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)))
86		oveq2 7354	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝐸 ↑ 𝑧) = (𝐸 ↑ 𝑦))
87	86	fveq2d 6826	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)))
88	85, 87	eqeq12d 2747	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)) ↔ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))))
89	8, 36, 16	aks6d1c1p1 42210	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))))
90	89	biimpd 229	. . . . . . . . . . . 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 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧))
93		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))
94	92, 93, 88	cbvralw 3274	. . . . . . . . . . 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 234	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)))
96	95	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑧)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑧)))
97	88, 96, 48	rspcdva 3573	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)))
98	97	eqcomd 2737	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦)) = (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)))
99	98	oveq2d 7362	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ↑ ((𝑂‘𝐹)‘(𝐸 ↑ 𝑦))) = (𝐷 ↑ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦))))
100	83, 99	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝐷 ↑ (𝐸 ↑ 𝑦))) = (𝐷 ↑ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦))))
101	100	eqcomd 2737	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ↑ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦))) = ((𝑂‘𝐹)‘(𝐷 ↑ (𝐸 ↑ 𝑦))))
102	13, 18, 30	3jca 1128	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ∈ ℕ₀ ∧ 𝐸 ∈ ℕ₀ ∧ 𝑦 ∈ (Base‘𝑉)))
103	42, 26	mulgnn0ass 19023	. . . . . . 7 ⊢ ((𝑉 ∈ Mnd ∧ (𝐷 ∈ ℕ₀ ∧ 𝐸 ∈ ℕ₀ ∧ 𝑦 ∈ (Base‘𝑉))) → ((𝐷 · 𝐸) ↑ 𝑦) = (𝐷 ↑ (𝐸 ↑ 𝑦)))
104	103	eqcomd 2737	. . . . . 6 ⊢ ((𝑉 ∈ Mnd ∧ (𝐷 ∈ ℕ₀ ∧ 𝐸 ∈ ℕ₀ ∧ 𝑦 ∈ (Base‘𝑉))) → (𝐷 ↑ (𝐸 ↑ 𝑦)) = ((𝐷 · 𝐸) ↑ 𝑦))
105	7, 102, 104	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐷 ↑ (𝐸 ↑ 𝑦)) = ((𝐷 · 𝐸) ↑ 𝑦))
106	105	fveq2d 6826	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝐹)‘(𝐷 ↑ (𝐸 ↑ 𝑦))) = ((𝑂‘𝐹)‘((𝐷 · 𝐸) ↑ 𝑦)))
107	44, 101, 106	3eqtrd 2770	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐷 · 𝐸) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝐷 · 𝐸) ↑ 𝑦)))
108	107	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)((𝐷 · 𝐸) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝐷 · 𝐸) ↑ 𝑦)))
109	11, 16	nnmulcld 12178	. . 3 ⊢ (𝜑 → (𝐷 · 𝐸) ∈ ℕ)
110	8, 36, 109	aks6d1c1p1 42210	. 2 ⊢ (𝜑 → ((𝐷 · 𝐸) ∼ 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)((𝐷 · 𝐸) ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘((𝐷 · 𝐸) ↑ 𝑦))))
111	108, 110	mpbird 257	1 ⊢ (𝜑 → (𝐷 · 𝐸) ∼ 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∀wral 3047 class class class wbr 5089 {copab 5151 ↦ cmpt 5170 –onto→wfo 6479 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 1c1 11007 · cmul 11011 ℕcn 12125 ℕ₀cn0 12381 ∥ cdvds 16163 gcd cgcd 16405 ℙcprime 16582 Basecbs 17120 +_gcplusg 17161 0_gc0g 17343 Mndcmnd 18642 ._gcmg 18980 CMndccmn 19692 mulGrpcmgp 20058 CRingccrg 20152 Fieldcfield 20645 chrcchr 21438 algSccascl 21789 var₁cv1 22088 Poly₁cpl1 22089 eval₁ce1 22229 PrimRoots cprimroots 42194

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-rp 12891 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-dvds 16164 df-gcd 16406 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19125 df-cntz 19229 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-srg 20105 df-ring 20153 df-cring 20154 df-rhm 20390 df-subrng 20461 df-subrg 20485 df-field 20647 df-lmod 20795 df-lss 20865 df-lsp 20905 df-assa 21790 df-asp 21791 df-ascl 21792 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-evls 22009 df-evl 22010 df-psr1 22092 df-ply1 22094 df-evl1 22231 df-primroots 42195

This theorem is referenced by: aks6d1c1p8 42218 aks6d1c1 42219

W3C validator