primrootspoweq0 - Mathbox for metakunt

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Theorem primrootspoweq0 42906

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 7431	. . . . . . . 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 42898	. . . . . . . . . . . . 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 19879	. . . . . . . . . 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 12628	. . . . . . . . . . . . 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 12717	. . . . . . . . . . 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 13564	. . . . . . . . . . 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 2867	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
19		ablcmn 19880	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ↾_s 𝑈) ∈ Abel → (𝑅 ↾_s 𝑈) ∈ CMnd)
20	7, 19	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑅 ↾_s 𝑈) ∈ CMnd)
21	4	nnnn0d 12576	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
22		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ (._g‘(𝑅 ↾_s 𝑈)) = (._g‘(𝑅 ↾_s 𝑈))
23	20, 21, 22	isprimroot 42893	. . . . . . . . . . . . . . 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 2765	. . . . . . . . . . 11 ⊢ (Base‘(𝑅 ↾_s 𝑈)) = (Base‘(𝑅 ↾_s 𝑈))
30		eqid 2765	. . . . . . . . . . 11 ⊢ (+_g‘(𝑅 ↾_s 𝑈)) = (+_g‘(𝑅 ↾_s 𝑈))
31	29, 22, 30	mulgdir 19195	. . . . . . . . . 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 19200	. . . . . . . . . . . . 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 7432	. . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . 15 ⊢ (0_g‘(𝑅 ↾_s 𝑈)) = (0_g‘(𝑅 ↾_s 𝑈))
40	29, 22, 39	mulgz 19191	. . . . . . . . . . . . . 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 2800	. . . . . . . . . . . 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 2800	. . . . . . . . . . 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 7431	. . . . . . . . . 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 19185	. . . . . . . . . . 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 19059	. . . . . . . . . 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 2800	. . . . . . . . 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 2800	. . . . . . . 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 2800	. . . . . . 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 2867	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑀 ∈ ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
55		1cnd 11213	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ ℂ)
56	55	addlidd 11422	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0 + 1) = 1)
57	4	nnge1d 12295	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ≤ 𝐾)
58	56, 57	eqbrtrd 5135	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 + 1) ≤ 𝐾)
59		0red 11222	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℝ)
60		1red 11220	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
61	4	nnred 12259	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ ℝ)
62		leaddsub 11701	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐾 ∈ ℝ) → ((0 + 1) ≤ 𝐾 ↔ 0 ≤ (𝐾 − 1)))
63	59, 60, 61, 62	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((0 + 1) ≤ 𝐾 ↔ 0 ≤ (𝐾 − 1)))
64	58, 63	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (𝐾 − 1))
65		0zd 12614	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℤ)
66		1zzd 12636	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℤ)
67	11, 66	zsubcld 12716	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾 − 1) ∈ ℤ)
68		eluz 12887	. . . . . . . . . . . . . . 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 13643	. . . . . . . . . . . . 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 12628	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝐾 ∈ ℤ)
79		dvdsmul2 16353	. . . . . . . . . . . . . . 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 12712	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝑥 ∈ ℂ)
82	77	nncnd 12260	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝐾 ∈ ℂ)
83	81, 82	mulcld 11240	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → (𝑥 · 𝐾) ∈ ℂ)
84	83	addridd 11421	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → ((𝑥 · 𝐾) + 0) = (𝑥 · 𝐾))
85	84	eqcomd 2771	. . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 0 = 𝑦)
88	87	oveq2d 7432	. . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . 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 5139	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → ((𝑥 · 𝐾) + 𝑦) = 𝑁)
93	90, 92	breqtrd 5139	. . . . . . . . . . . 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 11213	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 1 ∈ ℂ)
98	97	addlidd 11422	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (0 + 1) = 1)
99	98	oveq1d 7431	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → ((0 + 1)...(𝐾 − 1)) = (1...(𝐾 − 1)))
100		ssidd 3961	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (1...(𝐾 − 1)) ⊆ (1...(𝐾 − 1)))
101	99, 100	eqsstrd 3972	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → ((0 + 1)...(𝐾 − 1)) ⊆ (1...(𝐾 − 1)))
102	101	sseld 3937	. . . . . . . . . 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 42905	. . . . . . 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 3027	. . . . . 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 2965	. . . . 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 16329	. . . . . . . . 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 2771	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → 𝑁 = (𝑦 · 𝐾))
120	119	oveq1d 7431	. . . . . . . . . 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 19878	. . . . . . . . . . . . 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 19200	. . . . . . . . . . . 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 7432	. . . . . . . . . . . 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 19191	. . . . . . . . . . . . 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 2800	. . . . . . . . . . 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 2800	. . . . . . . . . 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 2800	. . . . . . . . 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 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑥 · 𝐾) = 𝑁
138		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑦 · 𝐾) = 𝑁
139		oveq1 7423	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝐾) = (𝑦 · 𝐾))
140	139	eqeq1d 2767	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 · 𝐾) = 𝑁 ↔ (𝑦 · 𝐾) = 𝑁))
141	137, 138, 140	cbvrexw 3310	. . . . . . . . . 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 3175	. . . . . . . 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 42904	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (0...(𝐾 − 1))𝑁 = ((𝑥 · 𝐾) + 𝑦))
151	149, 150	r19.29vva 3227	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 2146 ∀wral 3081 ∃wrex 3091 {crab 3418 class class class wbr 5111 ‘cfv 6540 (class class class)co 7416 ℝcr 11110 0cc0 11111 1c1 11112 + caddc 11114 · cmul 11116 ≤ cle 11255 − cmin 11452 ℕcn 12244 ℕ₀cn0 12515 ℤcz 12602 ℤ_≥cuz 12873 ...cfz 13546 ∥ cdvds 16327 Basecbs 17286 ↾_s cress 17307 +_gcplusg 17327 0_gc0g 17509 Grpcgrp 19023 ._gcmg 19156 CMndccmn 19873 Abelcabl 19874 PrimRoots cprimroots 42891

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189

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 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9405 df-inf 9406 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-n0 12516 df-z 12603 df-uz 12874 df-rp 13028 df-ico 13389 df-fz 13547 df-fzo 13695 df-fl 13838 df-mod 13916 df-seq 14051 df-dvds 16328 df-sets 17241 df-slot 17259 df-ndx 17271 df-base 17287 df-ress 17308 df-plusg 17340 df-0g 17511 df-mgm 18715 df-sgrp 18798 df-mnd 18814 df-submnd 18865 df-grp 19026 df-minusg 19027 df-mulg 19157 df-cmn 19875 df-abl 19876 df-primroots 42892

This theorem is used by: aks6d1c6isolem3 42976

W3C validator