primrootspoweq0 - Mathbox for metakunt

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Theorem primrootspoweq0 42088

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 769	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑁 = ((𝑥 · 𝐾) + 𝑦))
2	1	oveq1d 7446	. . . . . . . 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 42080	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑅 PrimRoots 𝐾) = ((𝑅 ↾_s 𝑈) PrimRoots 𝐾) ∧ (𝑅 ↾_s 𝑈) ∈ Abel))
7	6	simprd 495	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑅 ↾_s 𝑈) ∈ Abel)
8	7	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑅 ↾_s 𝑈) ∈ Abel)
9	8	ablgrpd 19819	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑅 ↾_s 𝑈) ∈ Grp)
10		simp-4r 784	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑥 ∈ ℤ)
11	4	nnzd 12638	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ ℤ)
12	11	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝐾 ∈ ℤ)
13	10, 12	zmulcld 12726	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑥 · 𝐾) ∈ ℤ)
14		simpllr 776	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑦 ∈ (0...(𝐾 − 1)))
15	14	elfzelzd 13562	. . . . . . . . . . 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 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑅 PrimRoots 𝐾) = ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
18	16, 17	eleqtrd 2841	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
19		ablcmn 19820	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ↾_s 𝑈) ∈ Abel → (𝑅 ↾_s 𝑈) ∈ CMnd)
20	7, 19	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑅 ↾_s 𝑈) ∈ CMnd)
21	4	nnnn0d 12585	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
22		eqid 2735	. . . . . . . . . . . . . . . 16 ⊢ (._g‘(𝑅 ↾_s 𝑈)) = (._g‘(𝑅 ↾_s 𝑈))
23	20, 21, 22	isprimroot 42075	. . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . 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 15	. . . . . . . . . . . . 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 1141	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))
27	26	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))
28	13, 15, 27	3jca 1127	. . . . . . . . . 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 2735	. . . . . . . . . . 11 ⊢ (Base‘(𝑅 ↾_s 𝑈)) = (Base‘(𝑅 ↾_s 𝑈))
30		eqid 2735	. . . . . . . . . . 11 ⊢ (+_g‘(𝑅 ↾_s 𝑈)) = (+_g‘(𝑅 ↾_s 𝑈))
31	29, 22, 30	mulgdir 19137	. . . . . . . . . 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 584	. . . . . . . . 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 1127	. . . . . . . . . . . . 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 19142	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 1142	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)))
37	36	ad4antr 732	. . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . 15 ⊢ (0_g‘(𝑅 ↾_s 𝑈)) = (0_g‘(𝑅 ↾_s 𝑈))
40	29, 22, 39	mulgz 19133	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 2775	. . . . . . . . . . . 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 2775	. . . . . . . . . . 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 7446	. . . . . . . . . 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 19127	. . . . . . . . . . 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 19000	. . . . . . . . . 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 2775	. . . . . . . . 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 2775	. . . . . . . 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 2775	. . . . . . 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 17	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑅 ↾_s 𝑈) ∈ CMnd)
51	4	ad4antr 732	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝐾 ∈ ℕ)
52	16	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑀 ∈ (𝑅 PrimRoots 𝐾))
53	17	ad4antr 732	. . . . . . . . 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 2841	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑀 ∈ ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
55		1cnd 11254	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ ℂ)
56	55	addlidd 11460	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0 + 1) = 1)
57	4	nnge1d 12312	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ≤ 𝐾)
58	56, 57	eqbrtrd 5170	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 + 1) ≤ 𝐾)
59		0red 11262	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℝ)
60		1red 11260	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
61	4	nnred 12279	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ ℝ)
62		leaddsub 11737	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐾 ∈ ℝ) → ((0 + 1) ≤ 𝐾 ↔ 0 ≤ (𝐾 − 1)))
63	59, 60, 61, 62	syl3anc 1370	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((0 + 1) ≤ 𝐾 ↔ 0 ≤ (𝐾 − 1)))
64	58, 63	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (𝐾 − 1))
65		0zd 12623	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℤ)
66		1zzd 12646	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℤ)
67	11, 66	zsubcld 12725	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾 − 1) ∈ ℤ)
68		eluz 12890	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℤ ∧ (𝐾 − 1) ∈ ℤ) → ((𝐾 − 1) ∈ (ℤ_≥‘0) ↔ 0 ≤ (𝐾 − 1)))
69	65, 67, 68	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐾 − 1) ∈ (ℤ_≥‘0) ↔ 0 ≤ (𝐾 − 1)))
70	64, 69	mpbird 257	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾 − 1) ∈ (ℤ_≥‘0))
71		elfzp12 13640	. . . . . . . . . . . . 13 ⊢ ((𝐾 − 1) ∈ (ℤ_≥‘0) → (𝑦 ∈ (0...(𝐾 − 1)) ↔ (𝑦 = 0 ∨ 𝑦 ∈ ((0 + 1)...(𝐾 − 1)))))
72	70, 71	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ (0...(𝐾 − 1)) ↔ (𝑦 = 0 ∨ 𝑦 ∈ ((0 + 1)...(𝐾 − 1)))))
73	72	ad4antr 732	. . . . . . . . . . 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 229	. . . . . . . . . 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 15	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑦 = 0 ∨ 𝑦 ∈ ((0 + 1)...(𝐾 − 1))))
76		simp-5r 786	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝑥 ∈ ℤ)
77	51	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝐾 ∈ ℕ)
78	77	nnzd 12638	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝐾 ∈ ℤ)
79		dvdsmul2 16313	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝐾 ∥ (𝑥 · 𝐾))
80	76, 78, 79	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝐾 ∥ (𝑥 · 𝐾))
81	76	zcnd 12721	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝑥 ∈ ℂ)
82	77	nncnd 12280	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝐾 ∈ ℂ)
83	81, 82	mulcld 11279	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → (𝑥 · 𝐾) ∈ ℂ)
84	83	addridd 11459	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → ((𝑥 · 𝐾) + 0) = (𝑥 · 𝐾))
85	84	eqcomd 2741	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → (𝑥 · 𝐾) = ((𝑥 · 𝐾) + 0))
86		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝑦 = 0)
87	86	eqcomd 2741	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 0 = 𝑦)
88	87	oveq2d 7447	. . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . 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 5174	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝐾 ∥ ((𝑥 · 𝐾) + 𝑦))
91	1	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝑁 = ((𝑥 · 𝐾) + 𝑦))
92	91	eqcomd 2741	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → ((𝑥 · 𝐾) + 𝑦) = 𝑁)
93	90, 92	breqtrd 5174	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝐾 ∥ 𝑁)
94		simplr 769	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → ¬ 𝐾 ∥ 𝑁)
95	93, 94	pm2.21dd 195	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) ∧ 𝑦 = 0) → 𝑦 ∈ (1...(𝐾 − 1)))
96	95	ex 412	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (𝑦 = 0 → 𝑦 ∈ (1...(𝐾 − 1))))
97		1cnd 11254	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 1 ∈ ℂ)
98	97	addlidd 11460	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (0 + 1) = 1)
99	98	oveq1d 7446	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → ((0 + 1)...(𝐾 − 1)) = (1...(𝐾 − 1)))
100		ssidd 4019	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → (1...(𝐾 − 1)) ⊆ (1...(𝐾 − 1)))
101	99, 100	eqsstrd 4034	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → ((0 + 1)...(𝐾 − 1)) ⊆ (1...(𝐾 − 1)))
102	101	sseld 3994	. . . . . . . . . 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 859	. . . . . . . . 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 15	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ ¬ 𝐾 ∥ 𝑁) → 𝑦 ∈ (1...(𝐾 − 1)))
105	50, 51, 54, 104	primrootlekpowne0 42087	. . . . . . 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 3006	. . . . . 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 2943	. . . . 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 412	. . . 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 115	. . 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 783	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ 𝐾 ∥ 𝑁) → 𝜑)
111		simpr 484	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...(𝐾 − 1))) ∧ 𝑁 = ((𝑥 · 𝐾) + 𝑦)) ∧ 𝐾 ∥ 𝑁) → 𝐾 ∥ 𝑁)
112	110, 111	jca 511	. . . . 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 16289	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ 𝑁 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁))
115	11, 113, 114	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐾 ∥ 𝑁 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁))
116	115	biimpd 229	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∥ 𝑁 → ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁))
117	116	imp 406	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∥ 𝑁) → ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁)
118		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → (𝑦 · 𝐾) = 𝑁)
119	118	eqcomd 2741	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → 𝑁 = (𝑦 · 𝐾))
120	119	oveq1d 7446	. . . . . . . . . 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 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → (𝑅 ↾_s 𝑈) ∈ Abel)
122		ablgrp 19818	. . . . . . . . . . . . 13 ⊢ ((𝑅 ↾_s 𝑈) ∈ Abel → (𝑅 ↾_s 𝑈) ∈ Grp)
123	121, 122	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → (𝑅 ↾_s 𝑈) ∈ Grp)
124		simplr 769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → 𝑦 ∈ ℤ)
125	11	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → 𝐾 ∈ ℤ)
126	26	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) ∧ 𝑦 ∈ ℤ) ∧ (𝑦 · 𝐾) = 𝑁) → 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))
127	124, 125, 126	3jca 1127	. . . . . . . . . . . 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 19142	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 730	. . . . . . . . . . . . 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 7447	. . . . . . . . . . . 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 19133	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 2775	. . . . . . . . . . 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 2775	. . . . . . . . . 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 2775	. . . . . . . . 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 1912	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝑥 · 𝐾) = 𝑁
138		nfv 1912	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑦 · 𝐾) = 𝑁
139		oveq1 7438	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝐾) = (𝑦 · 𝐾))
140	139	eqeq1d 2737	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 · 𝐾) = 𝑁 ↔ (𝑦 · 𝐾) = 𝑁))
141	137, 138, 140	cbvrexw 3305	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁 ↔ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝑁)
142	141	biimpi 216	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁 → ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝑁)
143	142	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) → ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝑁)
144	136, 143	r19.29a 3160	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁) → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)))
145	144	ex 412	. . . . . . 7 ⊢ (𝜑 → (∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁 → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈))))
146	145	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∥ 𝑁) → (∃𝑥 ∈ ℤ (𝑥 · 𝐾) = 𝑁 → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈))))
147	117, 146	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∥ 𝑁) → (𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)))
148	112, 147	syl 17	. . . 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)))
149	148	ex 412	. . 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))))
150	109, 149	impbid 212	. 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))
151	113, 4	remexz 42086	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (0...(𝐾 − 1))𝑁 = ((𝑥 · 𝐾) + 𝑦))
152	150, 151	r19.29vva 3214	1 ⊢ (𝜑 → ((𝑁(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) ↔ 𝐾 ∥ 𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 {crab 3433 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 ≤ cle 11294 − cmin 11490 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ...cfz 13544 ∥ cdvds 16287 Basecbs 17245 ↾_s cress 17274 +_gcplusg 17298 0_gc0g 17486 Grpcgrp 18964 ._gcmg 19098 CMndccmn 19813 Abelcabl 19814 PrimRoots cprimroots 42073

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-ico 13390 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-dvds 16288 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-mulg 19099 df-cmn 19815 df-abl 19816 df-primroots 42074

This theorem is referenced by: aks6d1c6isolem3 42158

W3C validator