Theorem isprimroot 42674

Description: The value of a primitive root. (Contributed by metakunt, 25-Apr-2025.)

Hypotheses

Ref	Expression
isprimroot.1	⊢ (𝜑 → 𝑅 ∈ CMnd)
isprimroot.2	⊢ (𝜑 → 𝐾 ∈ ℕ0)
isprimroot.3	⊢ ↑ = (.g‘𝑅)

Assertion

Ref	Expression
isprimroot	⊢ (𝜑 → (𝑀 ∈ (𝑅 PrimRoots 𝐾) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾 ↑ 𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 ↑ 𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙))))

Distinct variable groups:   𝐾,𝑙   𝑀,𝑙   𝑅,𝑙   𝜑,𝑙

Allowed substitution hint:   ↑ (𝑙)

Proof of Theorem isprimroot

Dummy variables 𝑏 𝑘 𝑟 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-primroots 42673	. . . . . 6 ⊢ PrimRoots = (𝑟 ∈ CMnd, 𝑘 ∈ ℕ0 ↦ ⦋(Base‘𝑟) / 𝑏⦌{𝑥 ∈ 𝑏 ∣ ((𝑘(.g‘𝑟)𝑥) = (0g‘𝑟) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑟)𝑥) = (0g‘𝑟) → 𝑘 ∥ 𝑙))})
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → PrimRoots = (𝑟 ∈ CMnd, 𝑘 ∈ ℕ0 ↦ ⦋(Base‘𝑟) / 𝑏⦌{𝑥 ∈ 𝑏 ∣ ((𝑘(.g‘𝑟)𝑥) = (0g‘𝑟) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑟)𝑥) = (0g‘𝑟) → 𝑘 ∥ 𝑙))}))
3		simprl 780	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) → 𝑟 = 𝑅)
4	3	fveq2d 6867	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) → (Base‘𝑟) = (Base‘𝑅))
5		simplrl 786	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → 𝑟 = 𝑅)
6	5	fveq2d 6867	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (.g‘𝑟) = (.g‘𝑅))
7		simplrr 787	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → 𝑘 = 𝐾)
8		eqidd 2762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → 𝑥 = 𝑥)
9	6, 7, 8	oveq123d 7413	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (𝑘(.g‘𝑟)𝑥) = (𝐾(.g‘𝑅)𝑥))
10	5	fveq2d 6867	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (0g‘𝑟) = (0g‘𝑅))
11	9, 10	eqeq12d 2777	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → ((𝑘(.g‘𝑟)𝑥) = (0g‘𝑟) ↔ (𝐾(.g‘𝑅)𝑥) = (0g‘𝑅)))
12	3	fveq2d 6867	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) → (.g‘𝑟) = (.g‘𝑅))
13	12	oveqdr 7420	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (𝑙(.g‘𝑟)𝑥) = (𝑙(.g‘𝑅)𝑥))
14	13, 10	eqeq12d 2777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → ((𝑙(.g‘𝑟)𝑥) = (0g‘𝑟) ↔ (𝑙(.g‘𝑅)𝑥) = (0g‘𝑅)))
15	7	breq1d 5109	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (𝑘 ∥ 𝑙 ↔ 𝐾 ∥ 𝑙))
16	14, 15	imbi12d 346	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (((𝑙(.g‘𝑟)𝑥) = (0g‘𝑟) → 𝑘 ∥ 𝑙) ↔ ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙)))
17	16	ralbidv 3184	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑟)𝑥) = (0g‘𝑟) → 𝑘 ∥ 𝑙) ↔ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙)))
18	11, 17	anbi12d 641	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (((𝑘(.g‘𝑟)𝑥) = (0g‘𝑟) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑟)𝑥) = (0g‘𝑟) → 𝑘 ∥ 𝑙)) ↔ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))))
19	18	rabbidva 3419	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) → {𝑥 ∈ 𝑏 ∣ ((𝑘(.g‘𝑟)𝑥) = (0g‘𝑟) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑟)𝑥) = (0g‘𝑟) → 𝑘 ∥ 𝑙))} = {𝑥 ∈ 𝑏 ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))})
20	4, 19	csbeq12dv 3861	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) → ⦋(Base‘𝑟) / 𝑏⦌{𝑥 ∈ 𝑏 ∣ ((𝑘(.g‘𝑟)𝑥) = (0g‘𝑟) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑟)𝑥) = (0g‘𝑟) → 𝑘 ∥ 𝑙))} = ⦋(Base‘𝑅) / 𝑏⦌{𝑥 ∈ 𝑏 ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))})
21		isprimroot.1	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CMnd)
22		isprimroot.2	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
23		eqid 2761	. . . . . . 7 ⊢ {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))} = {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))}
24		fvexd 6878	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
25	23, 24	rabexd 5295	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))} ∈ V)
26		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 = (Base‘𝑅)) → 𝑏 = (Base‘𝑅))
27	26	rabeqdv 3428	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 = (Base‘𝑅)) → {𝑥 ∈ 𝑏 ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))} = {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))})
28	24, 27	csbied 3888	. . . . . . 7 ⊢ (𝜑 → ⦋(Base‘𝑅) / 𝑏⦌{𝑥 ∈ 𝑏 ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))} = {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))})
29	28	eleq1d 2846	. . . . . 6 ⊢ (𝜑 → (⦋(Base‘𝑅) / 𝑏⦌{𝑥 ∈ 𝑏 ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))} ∈ V ↔ {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))} ∈ V))
30	25, 29	mpbird 259	. . . . 5 ⊢ (𝜑 → ⦋(Base‘𝑅) / 𝑏⦌{𝑥 ∈ 𝑏 ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))} ∈ V)
31	2, 20, 21, 22, 30	ovmpod 7544	. . . 4 ⊢ (𝜑 → (𝑅 PrimRoots 𝐾) = ⦋(Base‘𝑅) / 𝑏⦌{𝑥 ∈ 𝑏 ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))})
32	31, 28	eqtrd 2796	. . 3 ⊢ (𝜑 → (𝑅 PrimRoots 𝐾) = {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))})
33	32	eleq2d 2847	. 2 ⊢ (𝜑 → (𝑀 ∈ (𝑅 PrimRoots 𝐾) ↔ 𝑀 ∈ {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))}))
34		oveq2 7400	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐾(.g‘𝑅)𝑥) = (𝐾(.g‘𝑅)𝑀))
35	34	eqeq1d 2763	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝐾(.g‘𝑅)𝑀) = (0g‘𝑅)))
36		oveq2 7400	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑙(.g‘𝑅)𝑥) = (𝑙(.g‘𝑅)𝑀))
37	36	eqeq1d 2763	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝑙(.g‘𝑅)𝑀) = (0g‘𝑅)))
38	37	imbi1d 343	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙) ↔ ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙)))
39	38	ralbidv 3184	. . . . . 6 ⊢ (𝑥 = 𝑀 → (∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙) ↔ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙)))
40	35, 39	anbi12d 641	. . . . 5 ⊢ (𝑥 = 𝑀 → (((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙)) ↔ ((𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙))))
41	40	elrab 3650	. . . 4 ⊢ (𝑀 ∈ {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))} ↔ (𝑀 ∈ (Base‘𝑅) ∧ ((𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙))))
42	41	a1i 11	. . 3 ⊢ (𝜑 → (𝑀 ∈ {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))} ↔ (𝑀 ∈ (Base‘𝑅) ∧ ((𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙)))))
43		3anass 1105	. . . . . 6 ⊢ ((𝑀 ∈ (Base‘𝑅) ∧ (𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙)) ↔ (𝑀 ∈ (Base‘𝑅) ∧ ((𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙))))
44	43	bicomi 226	. . . . 5 ⊢ ((𝑀 ∈ (Base‘𝑅) ∧ ((𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙))) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙)))
45	44	a1i 11	. . . 4 ⊢ (𝜑 → ((𝑀 ∈ (Base‘𝑅) ∧ ((𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙))) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙))))
46		biidd 264	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ (Base‘𝑅) ↔ 𝑀 ∈ (Base‘𝑅)))
47		isprimroot.3	. . . . . . . . 9 ⊢ ↑ = (.g‘𝑅)
48	47	eqcomi 2770	. . . . . . . 8 ⊢ (.g‘𝑅) = ↑
49	48	a1i 11	. . . . . . 7 ⊢ (𝜑 → (.g‘𝑅) = ↑ )
50	49	oveqd 7409	. . . . . 6 ⊢ (𝜑 → (𝐾(.g‘𝑅)𝑀) = (𝐾 ↑ 𝑀))
51	50	eqeq1d 2763	. . . . 5 ⊢ (𝜑 → ((𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ↔ (𝐾 ↑ 𝑀) = (0g‘𝑅)))
52	49	oveqd 7409	. . . . . . . 8 ⊢ (𝜑 → (𝑙(.g‘𝑅)𝑀) = (𝑙 ↑ 𝑀))
53	52	eqeq1d 2763	. . . . . . 7 ⊢ (𝜑 → ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) ↔ (𝑙 ↑ 𝑀) = (0g‘𝑅)))
54	53	imbi1d 343	. . . . . 6 ⊢ (𝜑 → (((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙) ↔ ((𝑙 ↑ 𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙)))
55	54	ralbidv 3184	. . . . 5 ⊢ (𝜑 → (∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙) ↔ ∀𝑙 ∈ ℕ0 ((𝑙 ↑ 𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙)))
56	46, 51, 55	3anbi123d 1456	. . . 4 ⊢ (𝜑 → ((𝑀 ∈ (Base‘𝑅) ∧ (𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙)) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾 ↑ 𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 ↑ 𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙))))
57	45, 56	bitrd 281	. . 3 ⊢ (𝜑 → ((𝑀 ∈ (Base‘𝑅) ∧ ((𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙))) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾 ↑ 𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 ↑ 𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙))))
58	42, 57	bitrd 281	. 2 ⊢ (𝜑 → (𝑀 ∈ {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))} ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾 ↑ 𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 ↑ 𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙))))
59	33, 58	bitrd 281	1 ⊢ (𝜑 → (𝑀 ∈ (𝑅 PrimRoots 𝐾) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾 ↑ 𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 ↑ 𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413  Vcvv 3453  ⦋csb 3852   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  ℕ₀cn0 12478   ∥ cdvds 16269  Basecbs 17228  0_gc0g 17451  ._gcmg 19092  CMndccmn 19803   PrimRoots cprimroots 42672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-primroots 42673

This theorem is referenced by:  isprimroot2  42675  primrootsunit1  42678  primrootscoprmpow  42680  primrootscoprbij  42683  primrootlekpowne0  42686  primrootspoweq0  42687  aks6d1c1p2  42690  aks6d1c1p3  42691  aks6d1c1p4  42692  aks6d1c1p5  42693  aks6d1c1p7  42694  aks6d1c1p6  42695  aks6d1c1p8  42696  aks6d1c2lem3  42707  aks6d1c2lem4  42708  aks6d1c6lem2  42752  aks6d1c6lem3  42753  aks6d1c6lem4  42754  aks6d1c6isolem1  42755  aks6d1c6isolem2  42756  aks6d1c6lem5  42758  aks5lem2  42768  aks5lem3a  42770