primrootspoweq0 - Mathbox for metakunt

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Theorem primrootspoweq0 43415

Description: The power of a 𝑅-th primitive root is zero if and only if it divides 𝑅. (Contributed by metakunt, 15-May-2025.)

**Hypotheses**
Ref	Expression
primrootspoweq0.1	⊢ (𝜑 → 𝑅 ∈ CMnd)
primrootspoweq0.2	⊢ (𝜑 → 𝐾 ∈ ℕ)
primrootspoweq0.3	⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))
primrootspoweq0.4	⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+_g‘𝑅)𝑎) = (0_g‘𝑅)}
primrootspoweq0.5	⊢ (𝜑 → 𝑁 ∈ ℤ)

**Assertion**
Ref	Expression
primrootspoweq0	⊢ (𝜑 → ((𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) ↔ 𝐾 ∥ 𝑁))

Distinct variable group: 𝑅,𝑎,𝑖 Allowed substitution hints: 𝜑(𝑖, 𝑎) 𝑈(𝑖, 𝑎) 𝐾(𝑖, 𝑎) 𝑀(𝑖, 𝑎) 𝑁(𝑖, 𝑎)

Proof of Theorem primrootspoweq0 Dummy variables 𝑙 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 781	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑁 = ((𝑥 · 𝐾) + 𝑦))
2	1	oveq1d 7464	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (((𝑥 · 𝐾) + 𝑦)(._g‘(𝑅 ↾_s 𝑈))𝑀))
3		primrootspoweq0.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ CMnd)
4		primrootspoweq0.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ ℕ)
5		primrootspoweq0.4	. . . . . . . . . . . . . 14 ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+_g‘𝑅)𝑎) = (0_g‘𝑅)}
6	3, 4, 5	primrootsunit 43407	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑅 PrimRoots 𝐾) = ((𝑅 ↾_s 𝑈) PrimRoots 𝐾) ∧ (𝑅 ↾_s 𝑈) ∈ Abel))
7	6	simprd 501	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑅 ↾_s 𝑈) ∈ Abel)
8	7	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑅 ↾_s 𝑈) ∈ Abel)
9	8	ablgrpd 20042	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑅 ↾_s 𝑈) ∈ Grp)
10		simp-4r 796	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑥 ∈ ℤ)
11	4	nnzd 12757	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ ℤ)
12	11	ad4antr 745	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝐾 ∈ ℤ)
13	10, 12	zmulcld 12847	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑥 · 𝐾) ∈ ℤ)
14		simpllr 788	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑦 ∈ (0...(𝐾 − 1)))
15	14	elfzelzd 13695	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑦 ∈ ℤ)
16		primrootspoweq0.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))
17	6	simpld 500	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑅 PrimRoots 𝐾) = ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
18	16, 17	eleqtrd 2866	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
19		ablcmn 20043	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ↾_s 𝑈) ∈ Abel → (𝑅 ↾_s 𝑈) ∈ CMnd)
20	7, 19	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑅 ↾_s 𝑈) ∈ CMnd)
21	4	nnnn0d 12705	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
22		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (._g‘(𝑅 ↾_s 𝑈)) = (._g‘(𝑅 ↾_s 𝑈))
23	20, 21, 22	isprimroot 43402	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀 ∈ ((𝑅 ↾_s 𝑈) PrimRoots 𝐾) ↔ (𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)) ∧ (𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) → 𝐾 ∥ 𝑙))))
24	23	biimpd 232	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ∈ ((𝑅 ↾_s 𝑈) PrimRoots 𝐾) → (𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)) ∧ (𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) → 𝐾 ∥ 𝑙))))
25	18, 24	mpd 16	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)) ∧ (𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) → 𝐾 ∥ 𝑙)))
26	25	simp1d 1160	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))
27	26	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))
28	13, 15, 27	3jca 1146	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → ((𝑥 · 𝐾) ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈))))
29		eqid 2763	. . . . . . . . . . 11 ⊢ (Base‘(𝑅 ↾_s 𝑈)) = (Base‘(𝑅 ↾_s 𝑈))
30		eqid 2763	. . . . . . . . . . 11 ⊢ (+_g‘(𝑅 ↾_s 𝑈)) = (+_g‘(𝑅 ↾_s 𝑈))
31	29, 22, 30	mulgdir 19357	. . . . . . . . . 10 ⊢ (((𝑅 ↾_s 𝑈) ∈ Grp ∧ ((𝑥 · 𝐾) ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))) → (((𝑥 · 𝐾) + 𝑦)(._g‘(𝑅 ↾_s 𝑈))𝑀) = (((𝑥 · 𝐾)(._g‘(𝑅 ↾_s 𝑈))𝑀)(+_g‘(𝑅 ↾_s 𝑈))(𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀)))
32	9, 28, 31	syl2anc 596	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (((𝑥 · 𝐾) + 𝑦)(._g‘(𝑅 ↾_s 𝑈))𝑀) = (((𝑥 · 𝐾)(._g‘(𝑅 ↾_s 𝑈))𝑀)(+_g‘(𝑅 ↾_s 𝑈))(𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀)))
33	10, 12, 27	3jca 1146	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑥 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈))))
34	29, 22	mulgass 19362	. . . . . . . . . . . . 13 ⊢ (((𝑅 ↾_s 𝑈) ∈ Grp ∧ (𝑥 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))) → ((𝑥 · 𝐾)(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑥(._g‘(𝑅 ↾_s 𝑈))(𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀)))
35	9, 33, 34	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → ((𝑥 · 𝐾)(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑥(._g‘(𝑅 ↾_s 𝑈))(𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀)))
36	25	simp2d 1161	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)))
37	36	ad4antr 745	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)))
38	37	oveq2d 7465	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑥(._g‘(𝑅 ↾_s 𝑈))(𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀)) = (𝑥(._g‘(𝑅 ↾_s 𝑈))(0_g‘(𝑅 ↾_s 𝑈))))
39		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (0_g‘(𝑅 ↾_s 𝑈)) = (0_g‘(𝑅 ↾_s 𝑈))
40	29, 22, 39	mulgz 19353	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ↾_s 𝑈) ∈ Grp ∧ 𝑥 ∈ ℤ) → (𝑥(._g‘(𝑅 ↾_s 𝑈))(0_g‘(𝑅 ↾_s 𝑈))) = (0_g‘(𝑅 ↾_s 𝑈)))
41	9, 10, 40	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑥(._g‘(𝑅 ↾_s 𝑈))(0_g‘(𝑅 ↾_s 𝑈))) = (0_g‘(𝑅 ↾_s 𝑈)))
42	38, 41	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑥(._g‘(𝑅 ↾_s 𝑈))(𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀)) = (0_g‘(𝑅 ↾_s 𝑈)))
43	35, 42	eqtrd 2798	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → ((𝑥 · 𝐾)(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)))
44	43	oveq1d 7464	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (((𝑥 · 𝐾)(._g‘(𝑅 ↾_s 𝑈))𝑀)(+_g‘(𝑅 ↾_s 𝑈))(𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀)) = ((0_g‘(𝑅 ↾_s 𝑈))(+_g‘(𝑅 ↾_s 𝑈))(𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀)))
45	29, 22, 9, 15, 27	mulgcld 19347	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ (Base‘(𝑅 ↾_s 𝑈)))
46	29, 30, 39, 9, 45	grplidd 19221	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → ((0_g‘(𝑅 ↾_s 𝑈))(+_g‘(𝑅 ↾_s 𝑈))(𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀)) = (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀))
47	44, 46	eqtrd 2798	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (((𝑥 · 𝐾)(._g‘(𝑅 ↾_s 𝑈))𝑀)(+_g‘(𝑅 ↾_s 𝑈))(𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀)) = (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀))
48	32, 47	eqtrd 2798	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (((𝑥 · 𝐾) + 𝑦)(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀))
49	2, 48	eqtrd 2798	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀))
50	8, 19	syl 18	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑅 ↾_s 𝑈) ∈ CMnd)
51	4	ad4antr 745	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝐾 ∈ ℕ)
52	16	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑀 ∈ (𝑅 PrimRoots 𝐾))
53	17	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑅 PrimRoots 𝐾) = ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
54	52, 53	eleqtrd 2866	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑀 ∈ ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
55		1cnd 11339	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ ℂ)
56	55	addlidd 11548	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0 + 1) = 1)
57	4	nnge1d 12424	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ≤ 𝐾)
58	56, 57	eqbrtrd 5131	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 + 1) ≤ 𝐾)
59		0red 11348	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℝ)
60		1red 11346	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
61	4	nnred 12388	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ ℝ)
62		leaddsub 11830	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐾 ∈ ℝ) → ((0 + 1) ≤ 𝐾 ↔ 0 ≤ (𝐾 − 1)))
63	59, 60, 61, 62	syl3anc 1399	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((0 + 1) ≤ 𝐾 ↔ 0 ≤ (𝐾 − 1)))
64	58, 63	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (𝐾 − 1))
65		0zd 12743	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℤ)
66		1zzd 12765	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℤ)
67	11, 66	zsubcld 12846	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾 − 1) ∈ ℤ)
68		eluz 13017	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℤ ∧ (𝐾 − 1) ∈ ℤ) → ((𝐾 − 1) ∈ (ℤ_≥‘0) ↔ 0 ≤ (𝐾 − 1)))
69	65, 67, 68	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐾 − 1) ∈ (ℤ_≥‘0) ↔ 0 ≤ (𝐾 − 1)))
70	64, 69	mpbird 260	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾 − 1) ∈ (ℤ_≥‘0))
71		elfzp12 13775	. . . . . . . . . . . . 13 ⊢ ((𝐾 − 1) ∈ (ℤ_≥‘0) → (𝑦 ∈ (0...(𝐾 − 1)) ↔ (𝑦 = 0 ∨ 𝑦 ∈ ((0 + 1)...(𝐾 − 1)))))
72	70, 71	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ (0...(𝐾 − 1)) ↔ (𝑦 = 0 ∨ 𝑦 ∈ ((0 + 1)...(𝐾 − 1)))))
73	72	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑦 ∈ (0...(𝐾 − 1)) ↔ (𝑦 = 0 ∨ 𝑦 ∈ ((0 + 1)...(𝐾 − 1)))))
74	73	biimpd 232	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑦 ∈ (0...(𝐾 − 1)) → (𝑦 = 0 ∨ 𝑦 ∈ ((0 + 1)...(𝐾 − 1)))))
75	14, 74	mpd 16	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑦 = 0 ∨ 𝑦 ∈ ((0 + 1)...(𝐾 − 1))))
76		simp-5r 798	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝑥 ∈ ℤ)
77	51	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝐾 ∈ ℕ)
78	77	nnzd 12757	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝐾 ∈ ℤ)
79		dvdsmul2 16489	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝐾 ∥ (𝑥 · 𝐾))
80	76, 78, 79	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝐾 ∥ (𝑥 · 𝐾))
81	76	zcnd 12842	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝑥 ∈ ℂ)
82	77	nncnd 12389	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝐾 ∈ ℂ)
83	81, 82	mulcld 11366	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → (𝑥 · 𝐾) ∈ ℂ)
84	83	addridd 11547	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → ((𝑥 · 𝐾) + 0) = (𝑥 · 𝐾))
85	84	eqcomd 2769	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → (𝑥 · 𝐾) = ((𝑥 · 𝐾) + 0))
86		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝑦 = 0)
87	86	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 0 = 𝑦)
88	87	oveq2d 7465	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → ((𝑥 · 𝐾) + 0) = ((𝑥 · 𝐾) + 𝑦))
89	85, 88	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → (𝑥 · 𝐾) = ((𝑥 · 𝐾) + 𝑦))
90	80, 89	breqtrd 5135	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝐾 ∥ ((𝑥 · 𝐾) + 𝑦))
91	1	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝑁 = ((𝑥 · 𝐾) + 𝑦))
92	91	eqcomd 2769	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → ((𝑥 · 𝐾) + 𝑦) = 𝑁)
93	90, 92	breqtrd 5135	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝐾 ∥ 𝑁)
94		simplr 781	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → ¬ 𝐾 ∥ 𝑁)
95	93, 94	pm2.21dd 198	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝑦 ∈ (1...(𝐾 − 1)))
96	95	ex 418	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑦 = 0 → 𝑦 ∈ (1...(𝐾 − 1))))
97		1cnd 11339	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 1 ∈ ℂ)
98	97	addlidd 11548	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (0 + 1) = 1)
99	98	oveq1d 7464	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → ((0 + 1)...(𝐾 − 1)) = (1...(𝐾 − 1)))
100		ssidd 3936	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (1...(𝐾 − 1)) ⊆ (1...(𝐾 − 1)))
101	99, 100	eqsstrd 3947	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → ((0 + 1)...(𝐾 − 1)) ⊆ (1...(𝐾 − 1)))
102	101	sseld 3912	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑦 ∈ ((0 + 1)...(𝐾 − 1)) → 𝑦 ∈ (1...(𝐾 − 1))))
103	96, 102	jaod 873	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → ((𝑦 = 0 ∨ 𝑦 ∈ ((0 + 1)...(𝐾 − 1))) → 𝑦 ∈ (1...(𝐾 − 1))))
104	75, 103	mpd 16	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑦 ∈ (1...(𝐾 − 1)))
105	50, 51, 54, 104	primrootlekpowne0 43414	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀) ≠ (0_g‘(𝑅 ↾_s 𝑈)))
106	49, 105	eqnetrd 3026	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) ≠ (0_g‘(𝑅 ↾_s 𝑈)))
107	106	neneqd 2964	. . . . 5 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → ¬ (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)))
108	107	ex 418	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) → (¬ 𝐾 ∥ 𝑁 → ¬ (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈))))
109	108	con4d 116	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) → ((𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) → 𝐾 ∥ 𝑁))
110		simp-4l 795	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ 𝐾 ∥ 𝑁) → 𝜑)
111		simpr 490	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ 𝐾 ∥ 𝑁) → 𝐾 ∥ 𝑁)
112	110, 111	jca 521	. . . . 5 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ 𝐾 ∥ 𝑁) → (𝜑 ∧ 𝐾 ∥ 𝑁))
113		primrootspoweq0.5	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
114		divides 16465	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ 𝑁 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁))
115	11, 113, 114	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐾 ∥ 𝑁 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁))
116	115	biimpd 232	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∥ 𝑁 → ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁))
117	116	imp 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∥ 𝑁) → ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁)
118		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → (𝑦 · 𝐾) = 𝑁)
119	118	eqcomd 2769	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → 𝑁 = (𝑦 · 𝐾))
120	119	oveq1d 7464	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = ((𝑦 · 𝐾)(._g‘(𝑅 ↾_s 𝑈))𝑀))
121	7	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → (𝑅 ↾_s 𝑈) ∈ Abel)
122		ablgrp 20041	. . . . . . . . . . . . 13 ⊢ ((𝑅 ↾_s 𝑈) ∈ Abel → (𝑅 ↾_s 𝑈) ∈ Grp)
123	121, 122	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → (𝑅 ↾_s 𝑈) ∈ Grp)
124		simplr 781	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → 𝑦 ∈ ℤ)
125	11	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → 𝐾 ∈ ℤ)
126	26	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))
127	124, 125, 126	3jca 1146	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → (𝑦 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈))))
128	29, 22	mulgass 19362	. . . . . . . . . . . 12 ⊢ (((𝑅 ↾_s 𝑈) ∈ Grp ∧ (𝑦 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))) → ((𝑦 · 𝐾)(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑦(._g‘(𝑅 ↾_s 𝑈))(𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀)))
129	123, 127, 128	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → ((𝑦 · 𝐾)(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑦(._g‘(𝑅 ↾_s 𝑈))(𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀)))
130	36	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → (𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)))
131	130	oveq2d 7465	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → (𝑦(._g‘(𝑅 ↾_s 𝑈))(𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀)) = (𝑦(._g‘(𝑅 ↾_s 𝑈))(0_g‘(𝑅 ↾_s 𝑈))))
132	29, 22, 39	mulgz 19353	. . . . . . . . . . . . 13 ⊢ (((𝑅 ↾_s 𝑈) ∈ Grp ∧ 𝑦 ∈ ℤ) → (𝑦(._g‘(𝑅 ↾_s 𝑈))(0_g‘(𝑅 ↾_s 𝑈))) = (0_g‘(𝑅 ↾_s 𝑈)))
133	123, 124, 132	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → (𝑦(._g‘(𝑅 ↾_s 𝑈))(0_g‘(𝑅 ↾_s 𝑈))) = (0_g‘(𝑅 ↾_s 𝑈)))
134	131, 133	eqtrd 2798	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → (𝑦(._g‘(𝑅 ↾_s 𝑈))(𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀)) = (0_g‘(𝑅 ↾_s 𝑈)))
135	129, 134	eqtrd 2798	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → ((𝑦 · 𝐾)(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)))
136	120, 135	eqtrd 2798	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)))
137		nfv 1949	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑥 · 𝐾) = 𝑁
138		nfv 1949	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑦 · 𝐾) = 𝑁
139		oveq1 7456	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝐾) = (𝑦 · 𝐾))
140	139	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 · 𝐾) = 𝑁 ↔ (𝑦 · 𝐾) = 𝑁))
141	137, 138, 140	cbvrexw 3309	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁 ↔ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝑁)
142	141	bilani 510	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) → ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝑁)
143	136, 142	r19.29a 3174	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)))
144	143	ex 418	. . . . . . 7 ⊢ (𝜑 → (∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁 → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈))))
145	144	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∥ 𝑁) → (∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁 → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈))))
146	117, 145	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∥ 𝑁) → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)))
147	112, 146	syl 18	. . . 4 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ 𝐾 ∥ 𝑁) → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)))
148	147	ex 418	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) → (𝐾 ∥ 𝑁 → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈))))
149	109, 148	impbid 215	. 2 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) → ((𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) ↔ 𝐾 ∥ 𝑁))
150	113, 4	remexz 43413	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (0...(𝐾 − 1))𝑁 = ((𝑥 · 𝐾) + 𝑦))
151	149, 150	r19.29vva 3226	1 ⊢ (𝜑 → ((𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) ↔ 𝐾 ∥ 𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ∃wrex 3090 {crab 3416 class class class wbr 5106 ‘cfv 6564 (class class class)co 7449 ℝcr 11236 0cc0 11237 1c1 11238 + caddc 11240 · cmul 11242 ≤ cle 11381 − cmin 11578 ℕcn 12373 ℕ₀cn0 12644 ℤcz 12731 ℤ_≥cuz 13003 ...cfz 13677 ∥ cdvds 16463 Basecbs 17428 ↾_s cress 17449 +_gcplusg 17469 0_gc0g 17651 Grpcgrp 19185 ._gcmg 19318 CMndccmn 20036 Abelcabl 20037 PrimRoots cprimroots 43400

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7781 ax-cnex 11293 ax-resscn 11294 ax-1cn 11295 ax-icn 11296 ax-addcl 11297 ax-addrcl 11298 ax-mulcl 11299 ax-mulrcl 11300 ax-mulcom 11301 ax-addass 11302 ax-mulass 11303 ax-distr 11304 ax-i2m1 11305 ax-1ne0 11306 ax-1rid 11307 ax-rnegex 11308 ax-rrecex 11309 ax-cnre 11310 ax-pre-lttri 11311 ax-pre-lttrn 11312 ax-pre-ltadd 11313 ax-pre-mulgt0 11314 ax-pre-sup 11315

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6328 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6518 df-fun 6566 df-fn 6567 df-f 6568 df-f1 6569 df-fo 6570 df-f1o 6571 df-fv 6572 df-riota 7406 df-ov 7452 df-oprab 7453 df-mpo 7454 df-om 7910 df-1st 8033 df-2nd 8034 df-frecs 8335 df-wrecs 8366 df-recs 8415 df-rdg 8454 df-er 8753 df-en 9010 df-dom 9011 df-sdom 9012 df-sup 9469 df-inf 9470 df-pnf 11382 df-mnf 11383 df-xr 11384 df-ltxr 11385 df-le 11386 df-sub 11580 df-neg 11581 df-div 12012 df-nn 12374 df-2 12443 df-n0 12645 df-z 12732 df-uz 13004 df-rp 13159 df-ico 13520 df-fz 13678 df-fzo 13827 df-fl 13970 df-mod 14048 df-seq 14183 df-dvds 16464 df-sets 17383 df-slot 17401 df-ndx 17413 df-base 17429 df-ress 17450 df-plusg 17482 df-0g 17653 df-mgm 18857 df-sgrp 18949 df-mnd 18965 df-submnd 19020 df-grp 19188 df-minusg 19189 df-mulg 19319 df-cmn 20038 df-abl 20039 df-primroots 43401

This theorem is used by: aks6d1c6isolem3 43485

W3C validator