Theorem isprimroot 42028

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 42027	. . . . . 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 6889	. . . . . 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 6889	. . . . . . . . . 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 2735	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → 𝑥 = 𝑥)
9	6, 7, 8	oveq123d 7433	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (𝑘(.g‘𝑟)𝑥) = (𝐾(.g‘𝑅)𝑥))
10	5	fveq2d 6889	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (0g‘𝑟) = (0g‘𝑅))
11	9, 10	eqeq12d 2750	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → ((𝑘(.g‘𝑟)𝑥) = (0g‘𝑟) ↔ (𝐾(.g‘𝑅)𝑥) = (0g‘𝑅)))
12	3	fveq2d 6889	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) → (.g‘𝑟) = (.g‘𝑅))
13	12	oveqdr 7440	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → (𝑙(.g‘𝑟)𝑥) = (𝑙(.g‘𝑅)𝑥))
14	13, 10	eqeq12d 2750	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑘 = 𝐾)) ∧ 𝑥 ∈ 𝑏) → ((𝑙(.g‘𝑟)𝑥) = (0g‘𝑟) ↔ (𝑙(.g‘𝑅)𝑥) = (0g‘𝑅)))
15	7	breq1d 5133	. . . . . . . . . 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 3165	. . . . . . . 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 3426	. . . . . 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 3888	. . . . 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 2734	. . . . . . 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 6900	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
25	23, 24	rabexd 5320	. . . . . 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 3435	. . . . . . . 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 3915	. . . . . . 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 2818	. . . . . 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 7566	. . . 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 2769	. . 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 2819	. 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 7420	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐾(.g‘𝑅)𝑥) = (𝐾(.g‘𝑅)𝑀))
35	34	eqeq1d 2736	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝐾(.g‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝐾(.g‘𝑅)𝑀) = (0g‘𝑅)))
36		oveq2 7420	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑙(.g‘𝑅)𝑥) = (𝑙(.g‘𝑅)𝑀))
37	36	eqeq1d 2736	. . . . . . . 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 3165	. . . . . 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 3675	. . . 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 2743	. . . . . . . 8 ⊢ (.g‘𝑅) = ↑
49	48	a1i 11	. . . . . . 7 ⊢ (𝜑 → (.g‘𝑅) = ↑ )
50	49	oveqd 7429	. . . . . 6 ⊢ (𝜑 → (𝐾(.g‘𝑅)𝑀) = (𝐾 ↑ 𝑀))
51	50	eqeq1d 2736	. . . . 5 ⊢ (𝜑 → ((𝐾(.g‘𝑅)𝑀) = (0g‘𝑅) ↔ (𝐾 ↑ 𝑀) = (0g‘𝑅)))
52	49	oveqd 7429	. . . . . . . 8 ⊢ (𝜑 → (𝑙(.g‘𝑅)𝑀) = (𝑙 ↑ 𝑀))
53	52	eqeq1d 2736	. . . . . . 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 3165	. . . . 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 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050  {crab 3419  Vcvv 3463  ⦋csb 3879   class class class wbr 5123  ‘cfv 6540  (class class class)co 7412   ∈ cmpo 7414  ℕ₀cn0 12508   ∥ cdvds 16271  Basecbs 17228  0_gc0g 17454  ._gcmg 19053  CMndccmn 19765   PrimRoots cprimroots 42026

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 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6493  df-fun 6542  df-fv 6548  df-ov 7415  df-oprab 7416  df-mpo 7417  df-primroots 42027

This theorem is referenced by:  isprimroot2  42029  primrootsunit1  42032  primrootscoprmpow  42034  primrootscoprbij  42037  primrootlekpowne0  42040  primrootspoweq0  42041  aks6d1c1p2  42044  aks6d1c1p3  42045  aks6d1c1p4  42046  aks6d1c1p5  42047  aks6d1c1p7  42048  aks6d1c1p6  42049  aks6d1c1p8  42050  aks6d1c2lem3  42061  aks6d1c2lem4  42062  aks6d1c6lem2  42106  aks6d1c6lem3  42107  aks6d1c6lem4  42108  aks6d1c6isolem1  42109  aks6d1c6isolem2  42110  aks6d1c6lem5  42112  aks5lem2  42122  aks5lem3a  42124