Theorem isprimroot 42089

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 42088	. . . . . 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 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) → 𝑟 = 𝑅)
4	3	fveq2d 6918	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) → (Base‘𝑟) = (Base‘𝑅))
5		simplrl 777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → 𝑟 = 𝑅)
6	5	fveq2d 6918	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (.g‘𝑟) = (.g‘𝑅))
7		simplrr 778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → 𝑘 = 𝐾)
8		eqidd 2738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → 𝑥 = 𝑥)
9	6, 7, 8	oveq123d 7459	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (𝑘(.g‘𝑟)𝑥) = (𝐾(.g‘𝑅)𝑥))
10	5	fveq2d 6918	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (0g‘𝑟) = (0g‘𝑅))
11	9, 10	eqeq12d 2753	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → ((𝑘(.g‘𝑟)𝑥) = (0g‘𝑟) ↔ (𝐾(.g‘𝑅)𝑥) = (0g‘𝑅)))
12	3	fveq2d 6918	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) → (.g‘𝑟) = (.g‘𝑅))
13	12	oveqdr 7466	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (𝑙(.g‘𝑟)𝑥) = (𝑙(.g‘𝑅)𝑥))
14	13, 10	eqeq12d 2753	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → ((𝑙(.g‘𝑟)𝑥) = (0g‘𝑟) ↔ (𝑙(.g‘𝑅)𝑥) = (0g‘𝑅)))
15	7	breq1d 5161	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (𝑘 ∥ 𝑙 ↔ 𝐾 ∥ 𝑙))
16	14, 15	imbi12d 344	. . . . . . . . 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 3178	. . . . . . . 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 632	. . . . . . 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 3443	. . . . . 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 3920	. . . . 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 2737	. . . . . . 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 6929	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
25	23, 24	rabexd 5349	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙))} ∈ V)
26		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 = (Base‘𝑅)) → 𝑏 = (Base‘𝑅))
27	26	rabeqdv 3452	. . . . . . . 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 3949	. . . . . . 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 2826	. . . . . 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 257	. . . . 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 7592	. . . 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 2777	. . 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 2827	. 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 7446	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐾(.g‘𝑅)𝑥) = (𝐾(.g‘𝑅)𝑀))
35	34	eqeq1d 2739	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝐾(.g‘𝑅)𝑀) = (0g‘𝑅)))
36		oveq2 7446	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑙(.g‘𝑅)𝑥) = (𝑙(.g‘𝑅)𝑀))
37	36	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝑙(.g‘𝑅)𝑀) = (0g‘𝑅)))
38	37	imbi1d 341	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (((𝑙(.g‘𝑅)𝑥) = (0g‘𝑅) → 𝐾 ∥ 𝑙) ↔ ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙)))
39	38	ralbidv 3178	. . . . . 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 632	. . . . 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 3698	. . . 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 1095	. . . . . 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 224	. . . . 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 262	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ (Base‘𝑅) ↔ 𝑀 ∈ (Base‘𝑅)))
47		isprimroot.3	. . . . . . . . 9 ⊢ ↑ = (.g‘𝑅)
48	47	eqcomi 2746	. . . . . . . 8 ⊢ (.g‘𝑅) = ↑
49	48	a1i 11	. . . . . . 7 ⊢ (𝜑 → (.g‘𝑅) = ↑ )
50	49	oveqd 7455	. . . . . 6 ⊢ (𝜑 → (𝐾(.g‘𝑅)𝑀) = (𝐾 ↑ 𝑀))
51	50	eqeq1d 2739	. . . . 5 ⊢ (𝜑 → ((𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ↔ (𝐾 ↑ 𝑀) = (0g‘𝑅)))
52	49	oveqd 7455	. . . . . . . 8 ⊢ (𝜑 → (𝑙(.g‘𝑅)𝑀) = (𝑙 ↑ 𝑀))
53	52	eqeq1d 2739	. . . . . . 7 ⊢ (𝜑 → ((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) ↔ (𝑙 ↑ 𝑀) = (0g‘𝑅)))
54	53	imbi1d 341	. . . . . 6 ⊢ (𝜑 → (((𝑙(.g‘𝑅)𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙) ↔ ((𝑙 ↑ 𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙)))
55	54	ralbidv 3178	. . . . 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 1437	. . . 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 279	. . 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 279	. 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 279	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1539   ∈ wcel 2108  ∀wral 3061  {crab 3436  Vcvv 3481  ⦋csb 3911   class class class wbr 5151  ‘cfv 6569  (class class class)co 7438   ∈ cmpo 7440  ℕ₀cn0 12533   ∥ cdvds 16296  Basecbs 17254  0_gc0g 17495  ._gcmg 19107  CMndccmn 19822   PrimRoots cprimroots 42087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-iota 6522  df-fun 6571  df-fv 6577  df-ov 7441  df-oprab 7442  df-mpo 7443  df-primroots 42088

This theorem is referenced by:  isprimroot2  42090  primrootsunit1  42093  primrootscoprmpow  42095  primrootscoprbij  42098  primrootlekpowne0  42101  primrootspoweq0  42102  aks6d1c1p2  42105  aks6d1c1p3  42106  aks6d1c1p4  42107  aks6d1c1p5  42108  aks6d1c1p7  42109  aks6d1c1p6  42110  aks6d1c1p8  42111  aks6d1c2lem3  42122  aks6d1c2lem4  42123  aks6d1c6lem2  42167  aks6d1c6lem3  42168  aks6d1c6lem4  42169  aks6d1c6isolem1  42170  aks6d1c6isolem2  42171  aks6d1c6lem5  42173  aks5lem2  42183  aks5lem3a  42185