Theorem isprimroot 42081

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 42080	. . . . . 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 6862	. . . . . 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 6862	. . . . . . . . . 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 2730	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → 𝑥 = 𝑥)
9	6, 7, 8	oveq123d 7408	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (𝑘(.g‘𝑟)𝑥) = (𝐾(.g‘𝑅)𝑥))
10	5	fveq2d 6862	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (0g‘𝑟) = (0g‘𝑅))
11	9, 10	eqeq12d 2745	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → ((𝑘(.g‘𝑟)𝑥) = (0g‘𝑟) ↔ (𝐾(.g‘𝑅)𝑥) = (0g‘𝑅)))
12	3	fveq2d 6862	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) → (.g‘𝑟) = (.g‘𝑅))
13	12	oveqdr 7415	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (𝑙(.g‘𝑟)𝑥) = (𝑙(.g‘𝑅)𝑥))
14	13, 10	eqeq12d 2745	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → ((𝑙(.g‘𝑟)𝑥) = (0g‘𝑟) ↔ (𝑙(.g‘𝑅)𝑥) = (0g‘𝑅)))
15	7	breq1d 5117	. . . . . . . . . 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 3156	. . . . . . . 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 3412	. . . . . 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 3871	. . . . 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 2729	. . . . . . 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 6873	. . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 = (Base‘𝑅)) → 𝑏 = (Base‘𝑅))
27	26	rabeqdv 3421	. . . . . . . 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 3898	. . . . . . 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 2813	. . . . . 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 7541	. . . 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 2764	. . 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 2814	. 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 7395	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐾(.g‘𝑅)𝑥) = (𝐾(.g‘𝑅)𝑀))
35	34	eqeq1d 2731	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝐾(.g‘𝑅)𝑀) = (0g‘𝑅)))
36		oveq2 7395	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑙(.g‘𝑅)𝑥) = (𝑙(.g‘𝑅)𝑀))
37	36	eqeq1d 2731	. . . . . . . 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 3156	. . . . . 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 3659	. . . 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 2738	. . . . . . . 8 ⊢ (.g‘𝑅) = ↑
49	48	a1i 11	. . . . . . 7 ⊢ (𝜑 → (.g‘𝑅) = ↑ )
50	49	oveqd 7404	. . . . . 6 ⊢ (𝜑 → (𝐾(.g‘𝑅)𝑀) = (𝐾 ↑ 𝑀))
51	50	eqeq1d 2731	. . . . 5 ⊢ (𝜑 → ((𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ↔ (𝐾 ↑ 𝑀) = (0g‘𝑅)))
52	49	oveqd 7404	. . . . . . . 8 ⊢ (𝜑 → (𝑙(.g‘𝑅)𝑀) = (𝑙 ↑ 𝑀))
53	52	eqeq1d 2731	. . . . . . 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 3156	. . . . 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 3044  {crab 3405  Vcvv 3447  ⦋csb 3862   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  ℕ₀cn0 12442   ∥ cdvds 16222  Basecbs 17179  0_gc0g 17402  ._gcmg 18999  CMndccmn 19710   PrimRoots cprimroots 42079

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-primroots 42080

This theorem is referenced by:  isprimroot2  42082  primrootsunit1  42085  primrootscoprmpow  42087  primrootscoprbij  42090  primrootlekpowne0  42093  primrootspoweq0  42094  aks6d1c1p2  42097  aks6d1c1p3  42098  aks6d1c1p4  42099  aks6d1c1p5  42100  aks6d1c1p7  42101  aks6d1c1p6  42102  aks6d1c1p8  42103  aks6d1c2lem3  42114  aks6d1c2lem4  42115  aks6d1c6lem2  42159  aks6d1c6lem3  42160  aks6d1c6lem4  42161  aks6d1c6isolem1  42162  aks6d1c6isolem2  42163  aks6d1c6lem5  42165  aks5lem2  42175  aks5lem3a  42177