Theorem isprimroot 42076

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 42075	. . . . . 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 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) → 𝑟 = 𝑅)
4	3	fveq2d 6864	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) → (Base‘𝑟) = (Base‘𝑅))
5		simplrl 776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → 𝑟 = 𝑅)
6	5	fveq2d 6864	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (.g‘𝑟) = (.g‘𝑅))
7		simplrr 777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → 𝑘 = 𝐾)
8		eqidd 2731	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → 𝑥 = 𝑥)
9	6, 7, 8	oveq123d 7410	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (𝑘(.g‘𝑟)𝑥) = (𝐾(.g‘𝑅)𝑥))
10	5	fveq2d 6864	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (0g‘𝑟) = (0g‘𝑅))
11	9, 10	eqeq12d 2746	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → ((𝑘(.g‘𝑟)𝑥) = (0g‘𝑟) ↔ (𝐾(.g‘𝑅)𝑥) = (0g‘𝑅)))
12	3	fveq2d 6864	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) → (.g‘𝑟) = (.g‘𝑅))
13	12	oveqdr 7417	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (𝑙(.g‘𝑟)𝑥) = (𝑙(.g‘𝑅)𝑥))
14	13, 10	eqeq12d 2746	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → ((𝑙(.g‘𝑟)𝑥) = (0g‘𝑟) ↔ (𝑙(.g‘𝑅)𝑥) = (0g‘𝑅)))
15	7	breq1d 5119	. . . . . . . . . 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 3157	. . . . . . . 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 3415	. . . . . 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 3873	. . . . 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 2730	. . . . . . 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 6875	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
25	23, 24	rabexd 5297	. . . . . 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 3424	. . . . . . . 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 3900	. . . . . . 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 2814	. . . . . 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 7543	. . . 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 2765	. . 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 2815	. 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 7397	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐾(.g‘𝑅)𝑥) = (𝐾(.g‘𝑅)𝑀))
35	34	eqeq1d 2732	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝐾(.g‘𝑅)𝑀) = (0g‘𝑅)))
36		oveq2 7397	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑙(.g‘𝑅)𝑥) = (𝑙(.g‘𝑅)𝑀))
37	36	eqeq1d 2732	. . . . . . . 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 3157	. . . . . 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 3661	. . . 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 1094	. . . . . 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 2739	. . . . . . . 8 ⊢ (.g‘𝑅) = ↑
49	48	a1i 11	. . . . . . 7 ⊢ (𝜑 → (.g‘𝑅) = ↑ )
50	49	oveqd 7406	. . . . . 6 ⊢ (𝜑 → (𝐾(.g‘𝑅)𝑀) = (𝐾 ↑ 𝑀))
51	50	eqeq1d 2732	. . . . 5 ⊢ (𝜑 → ((𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ↔ (𝐾 ↑ 𝑀) = (0g‘𝑅)))
52	49	oveqd 7406	. . . . . . . 8 ⊢ (𝜑 → (𝑙(.g‘𝑅)𝑀) = (𝑙 ↑ 𝑀))
53	52	eqeq1d 2732	. . . . . . 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 3157	. . . . 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 1438	. . . 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 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408  Vcvv 3450  ⦋csb 3864   class class class wbr 5109  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391  ℕ₀cn0 12448   ∥ cdvds 16228  Basecbs 17185  0_gc0g 17408  ._gcmg 19005  CMndccmn 19716   PrimRoots cprimroots 42074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6515  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-primroots 42075

This theorem is referenced by:  isprimroot2  42077  primrootsunit1  42080  primrootscoprmpow  42082  primrootscoprbij  42085  primrootlekpowne0  42088  primrootspoweq0  42089  aks6d1c1p2  42092  aks6d1c1p3  42093  aks6d1c1p4  42094  aks6d1c1p5  42095  aks6d1c1p7  42096  aks6d1c1p6  42097  aks6d1c1p8  42098  aks6d1c2lem3  42109  aks6d1c2lem4  42110  aks6d1c6lem2  42154  aks6d1c6lem3  42155  aks6d1c6lem4  42156  aks6d1c6isolem1  42157  aks6d1c6isolem2  42158  aks6d1c6lem5  42160  aks5lem2  42170  aks5lem3a  42172