gpg5nbgrvtx03starlem2 - Mathbox for Alexander van der Vekens

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Theorem gpg5nbgrvtx03starlem2 48560

Description: Lemma 2 for gpg5nbgrvtx03star 48571. (Contributed by AV, 6-Sep-2025.)

**Hypotheses**
Ref	Expression
gpg5nbgrvtx03starlem1.j	⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
gpg5nbgrvtx03starlem1.g	⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
gpg5nbgrvtx03starlem1.v	⊢ 𝑉 = (Vtx‘𝐺)
gpg5nbgrvtx03starlem1.e	⊢ 𝐸 = (Edg‘𝐺)

**Assertion**
Ref	Expression
gpg5nbgrvtx03starlem2	⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ∉ 𝐸)

Proof of Theorem gpg5nbgrvtx03starlem2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		m1modnep2mod 47821	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑋 ∈ ℤ) → ((𝑋 − 1) mod 𝑁) ≠ ((𝑋 + 2) mod 𝑁))
2	1	3adant2 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((𝑋 − 1) mod 𝑁) ≠ ((𝑋 + 2) mod 𝑁))
3		zcn 12523	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ ℤ → 𝑋 ∈ ℂ)
4	3	3ad2ant3 1136	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → 𝑋 ∈ ℂ)
5		add1p1 12422	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ ℂ → ((𝑋 + 1) + 1) = (𝑋 + 2))
6	4, 5	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((𝑋 + 1) + 1) = (𝑋 + 2))
7	6	oveq1d 7376	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → (((𝑋 + 1) + 1) mod 𝑁) = ((𝑋 + 2) mod 𝑁))
8	2, 7	neeqtrrd 3007	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((𝑋 − 1) mod 𝑁) ≠ (((𝑋 + 1) + 1) mod 𝑁))
9		zre 12522	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ ℤ → 𝑋 ∈ ℝ)
10	9	3ad2ant3 1136	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → 𝑋 ∈ ℝ)
11		1red 11139	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → 1 ∈ ℝ)
12	10, 11	readdcld 11168	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → (𝑋 + 1) ∈ ℝ)
13		eluz4nn 12834	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝑁 ∈ ℕ)
14	13	nnrpd 12978	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝑁 ∈ ℝ⁺)
15	14	3ad2ant1 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → 𝑁 ∈ ℝ⁺)
16		modaddmod 13865	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 + 1) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → ((((𝑋 + 1) mod 𝑁) + 1) mod 𝑁) = (((𝑋 + 1) + 1) mod 𝑁))
17	12, 11, 15, 16	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((((𝑋 + 1) mod 𝑁) + 1) mod 𝑁) = (((𝑋 + 1) + 1) mod 𝑁))
18	8, 17	neeqtrrd 3007	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((𝑋 − 1) mod 𝑁) ≠ ((((𝑋 + 1) mod 𝑁) + 1) mod 𝑁))
19	18	ad2antrl 729	. . . . . . . . . . . 12 ⊢ ((((𝑋 + 1) mod 𝑁) = 𝑥 ∧ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁))) → ((𝑋 − 1) mod 𝑁) ≠ ((((𝑋 + 1) mod 𝑁) + 1) mod 𝑁))
20		oveq1 7368	. . . . . . . . . . . . . 14 ⊢ (((𝑋 + 1) mod 𝑁) = 𝑥 → (((𝑋 + 1) mod 𝑁) + 1) = (𝑥 + 1))
21	20	oveq1d 7376	. . . . . . . . . . . . 13 ⊢ (((𝑋 + 1) mod 𝑁) = 𝑥 → ((((𝑋 + 1) mod 𝑁) + 1) mod 𝑁) = ((𝑥 + 1) mod 𝑁))
22	21	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑋 + 1) mod 𝑁) = 𝑥 ∧ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁))) → ((((𝑋 + 1) mod 𝑁) + 1) mod 𝑁) = ((𝑥 + 1) mod 𝑁))
23	19, 22	neeqtrd 3002	. . . . . . . . . . 11 ⊢ ((((𝑋 + 1) mod 𝑁) = 𝑥 ∧ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁))) → ((𝑋 − 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁))
24	23	olcd 875	. . . . . . . . . 10 ⊢ ((((𝑋 + 1) mod 𝑁) = 𝑥 ∧ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁))) → (((𝑋 + 1) mod 𝑁) ≠ 𝑥 ∨ ((𝑋 − 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁)))
25	24	ex 412	. . . . . . . . 9 ⊢ (((𝑋 + 1) mod 𝑁) = 𝑥 → (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → (((𝑋 + 1) mod 𝑁) ≠ 𝑥 ∨ ((𝑋 − 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁))))
26		orc 868	. . . . . . . . . 10 ⊢ (((𝑋 + 1) mod 𝑁) ≠ 𝑥 → (((𝑋 + 1) mod 𝑁) ≠ 𝑥 ∨ ((𝑋 − 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁)))
27	26	a1d 25	. . . . . . . . 9 ⊢ (((𝑋 + 1) mod 𝑁) ≠ 𝑥 → (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → (((𝑋 + 1) mod 𝑁) ≠ 𝑥 ∨ ((𝑋 − 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁))))
28	25, 27	pm2.61ine 3016	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → (((𝑋 + 1) mod 𝑁) ≠ 𝑥 ∨ ((𝑋 − 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁)))
29		c0ex 11132	. . . . . . . . . . . 12 ⊢ 0 ∈ V
30		ovex 7394	. . . . . . . . . . . 12 ⊢ ((𝑋 + 1) mod 𝑁) ∈ V
31	29, 30	opthne 5431	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ↔ (0 ≠ 0 ∨ ((𝑋 + 1) mod 𝑁) ≠ 𝑥))
32		neirr 2942	. . . . . . . . . . . 12 ⊢ ¬ 0 ≠ 0
33	32	biorfi 939	. . . . . . . . . . 11 ⊢ (((𝑋 + 1) mod 𝑁) ≠ 𝑥 ↔ (0 ≠ 0 ∨ ((𝑋 + 1) mod 𝑁) ≠ 𝑥))
34	31, 33	bitr4i 278	. . . . . . . . . 10 ⊢ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ↔ ((𝑋 + 1) mod 𝑁) ≠ 𝑥)
35	34	a1i 11	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ↔ ((𝑋 + 1) mod 𝑁) ≠ 𝑥))
36		ovex 7394	. . . . . . . . . . . 12 ⊢ ((𝑋 − 1) mod 𝑁) ∈ V
37	29, 36	opthne 5431	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ↔ (0 ≠ 0 ∨ ((𝑋 − 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁)))
38	32	biorfi 939	. . . . . . . . . . 11 ⊢ (((𝑋 − 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁) ↔ (0 ≠ 0 ∨ ((𝑋 − 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁)))
39	37, 38	bitr4i 278	. . . . . . . . . 10 ⊢ (⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ↔ ((𝑋 − 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁))
40	39	a1i 11	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → (⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ↔ ((𝑋 − 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁)))
41	35, 40	orbi12d 919	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → ((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩) ↔ (((𝑋 + 1) mod 𝑁) ≠ 𝑥 ∨ ((𝑋 − 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁))))
42	28, 41	mpbird 257	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩))
43		uzuzle24 12829	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝑁 ∈ (ℤ_≥‘2))
44	43	anim1i 616	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑋 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘2) ∧ 𝑋 ∈ ℤ))
45	44	3adant2 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘2) ∧ 𝑋 ∈ ℤ))
46		zp1modne 47815	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑋 ∈ ℤ) → ((𝑋 + 1) mod 𝑁) ≠ (𝑋 mod 𝑁))
47	45, 46	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((𝑋 + 1) mod 𝑁) ≠ (𝑋 mod 𝑁))
48		npcan1 11569	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ ℂ → ((𝑋 − 1) + 1) = 𝑋)
49	4, 48	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((𝑋 − 1) + 1) = 𝑋)
50	49	oveq1d 7376	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → (((𝑋 − 1) + 1) mod 𝑁) = (𝑋 mod 𝑁))
51	47, 50	neeqtrrd 3007	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((𝑋 + 1) mod 𝑁) ≠ (((𝑋 − 1) + 1) mod 𝑁))
52	10, 11	resubcld 11572	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → (𝑋 − 1) ∈ ℝ)
53		modaddmod 13865	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 − 1) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → ((((𝑋 − 1) mod 𝑁) + 1) mod 𝑁) = (((𝑋 − 1) + 1) mod 𝑁))
54	52, 11, 15, 53	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((((𝑋 − 1) mod 𝑁) + 1) mod 𝑁) = (((𝑋 − 1) + 1) mod 𝑁))
55	51, 54	neeqtrrd 3007	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((𝑋 + 1) mod 𝑁) ≠ ((((𝑋 − 1) mod 𝑁) + 1) mod 𝑁))
56	55	ad2antrl 729	. . . . . . . . . . . 12 ⊢ ((((𝑋 − 1) mod 𝑁) = 𝑥 ∧ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁))) → ((𝑋 + 1) mod 𝑁) ≠ ((((𝑋 − 1) mod 𝑁) + 1) mod 𝑁))
57		oveq1 7368	. . . . . . . . . . . . . 14 ⊢ (((𝑋 − 1) mod 𝑁) = 𝑥 → (((𝑋 − 1) mod 𝑁) + 1) = (𝑥 + 1))
58	57	oveq1d 7376	. . . . . . . . . . . . 13 ⊢ (((𝑋 − 1) mod 𝑁) = 𝑥 → ((((𝑋 − 1) mod 𝑁) + 1) mod 𝑁) = ((𝑥 + 1) mod 𝑁))
59	58	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑋 − 1) mod 𝑁) = 𝑥 ∧ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁))) → ((((𝑋 − 1) mod 𝑁) + 1) mod 𝑁) = ((𝑥 + 1) mod 𝑁))
60	56, 59	neeqtrd 3002	. . . . . . . . . . 11 ⊢ ((((𝑋 − 1) mod 𝑁) = 𝑥 ∧ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁))) → ((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁))
61	60	orcd 874	. . . . . . . . . 10 ⊢ ((((𝑋 − 1) mod 𝑁) = 𝑥 ∧ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁))) → (((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁) ∨ ((𝑋 − 1) mod 𝑁) ≠ 𝑥))
62	61	ex 412	. . . . . . . . 9 ⊢ (((𝑋 − 1) mod 𝑁) = 𝑥 → (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → (((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁) ∨ ((𝑋 − 1) mod 𝑁) ≠ 𝑥)))
63		olc 869	. . . . . . . . . 10 ⊢ (((𝑋 − 1) mod 𝑁) ≠ 𝑥 → (((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁) ∨ ((𝑋 − 1) mod 𝑁) ≠ 𝑥))
64	63	a1d 25	. . . . . . . . 9 ⊢ (((𝑋 − 1) mod 𝑁) ≠ 𝑥 → (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → (((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁) ∨ ((𝑋 − 1) mod 𝑁) ≠ 𝑥)))
65	62, 64	pm2.61ine 3016	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → (((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁) ∨ ((𝑋 − 1) mod 𝑁) ≠ 𝑥))
66	29, 30	opthne 5431	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ↔ (0 ≠ 0 ∨ ((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁)))
67	32	biorfi 939	. . . . . . . . . . 11 ⊢ (((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁) ↔ (0 ≠ 0 ∨ ((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁)))
68	66, 67	bitr4i 278	. . . . . . . . . 10 ⊢ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ↔ ((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁))
69	68	a1i 11	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ↔ ((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁)))
70	29, 36	opthne 5431	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ↔ (0 ≠ 0 ∨ ((𝑋 − 1) mod 𝑁) ≠ 𝑥))
71	32	biorfi 939	. . . . . . . . . . 11 ⊢ (((𝑋 − 1) mod 𝑁) ≠ 𝑥 ↔ (0 ≠ 0 ∨ ((𝑋 − 1) mod 𝑁) ≠ 𝑥))
72	70, 71	bitr4i 278	. . . . . . . . . 10 ⊢ (⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ↔ ((𝑋 − 1) mod 𝑁) ≠ 𝑥)
73	72	a1i 11	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → (⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ↔ ((𝑋 − 1) mod 𝑁) ≠ 𝑥))
74	69, 73	orbi12d 919	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → ((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩) ↔ (((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁) ∨ ((𝑋 − 1) mod 𝑁) ≠ 𝑥)))
75	65, 74	mpbird 257	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩))
76		opex 5412	. . . . . . . . . 10 ⊢ ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∈ V
77		opex 5412	. . . . . . . . . 10 ⊢ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ∈ V
78	76, 77	pm3.2i 470	. . . . . . . . 9 ⊢ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∈ V ∧ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ∈ V)
79		opex 5412	. . . . . . . . . 10 ⊢ ⟨0, 𝑥⟩ ∈ V
80		opex 5412	. . . . . . . . . 10 ⊢ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ∈ V
81	79, 80	pm3.2i 470	. . . . . . . . 9 ⊢ (⟨0, 𝑥⟩ ∈ V ∧ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ∈ V)
82	78, 81	pm3.2i 470	. . . . . . . 8 ⊢ ((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∈ V ∧ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ∈ V) ∧ (⟨0, 𝑥⟩ ∈ V ∧ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ∈ V))
83		prneimg2 4799	. . . . . . . 8 ⊢ (((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∈ V ∧ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ∈ V) ∧ (⟨0, 𝑥⟩ ∈ V ∧ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ∈ V)) → ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ↔ ((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩) ∧ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩))))
84	82, 83	mp1i 13	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ↔ ((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩) ∧ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩))))
85	42, 75, 84	mpbir2and 714	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩})
86		0ne1 12246	. . . . . . . . . . . 12 ⊢ 0 ≠ 1
87	86	orci 866	. . . . . . . . . . 11 ⊢ (0 ≠ 1 ∨ ((𝑋 − 1) mod 𝑁) ≠ 𝑥)
88	29, 36	opthne 5431	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩ ↔ (0 ≠ 1 ∨ ((𝑋 − 1) mod 𝑁) ≠ 𝑥))
89	87, 88	mpbir 231	. . . . . . . . . 10 ⊢ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩
90	89	olci 867	. . . . . . . . 9 ⊢ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩)
91	86	orci 866	. . . . . . . . . . 11 ⊢ (0 ≠ 1 ∨ ((𝑋 + 1) mod 𝑁) ≠ 𝑥)
92	29, 30	opthne 5431	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩ ↔ (0 ≠ 1 ∨ ((𝑋 + 1) mod 𝑁) ≠ 𝑥))
93	91, 92	mpbir 231	. . . . . . . . . 10 ⊢ ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩
94	93	orci 866	. . . . . . . . 9 ⊢ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩)
95	90, 94	pm3.2i 470	. . . . . . . 8 ⊢ ((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩) ∧ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩))
96	95	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → ((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩) ∧ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩)))
97		opex 5412	. . . . . . . . . 10 ⊢ ⟨1, 𝑥⟩ ∈ V
98	79, 97	pm3.2i 470	. . . . . . . . 9 ⊢ (⟨0, 𝑥⟩ ∈ V ∧ ⟨1, 𝑥⟩ ∈ V)
99	78, 98	pm3.2i 470	. . . . . . . 8 ⊢ ((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∈ V ∧ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ∈ V) ∧ (⟨0, 𝑥⟩ ∈ V ∧ ⟨1, 𝑥⟩ ∈ V))
100		prneimg2 4799	. . . . . . . 8 ⊢ (((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∈ V ∧ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ∈ V) ∧ (⟨0, 𝑥⟩ ∈ V ∧ ⟨1, 𝑥⟩ ∈ V)) → ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ ((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩) ∧ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩))))
101	99, 100	mp1i 13	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ ((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩) ∧ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩))))
102	96, 101	mpbird 257	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩})
103		opex 5412	. . . . . . . . 9 ⊢ ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩ ∈ V
104	97, 103	pm3.2i 470	. . . . . . . 8 ⊢ (⟨1, 𝑥⟩ ∈ V ∧ ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩ ∈ V)
105	78, 104	pm3.2i 470	. . . . . . 7 ⊢ ((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∈ V ∧ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ∈ V) ∧ (⟨1, 𝑥⟩ ∈ V ∧ ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩ ∈ V))
106	86	orci 866	. . . . . . . . . . 11 ⊢ (0 ≠ 1 ∨ ((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 𝐾) mod 𝑁))
107	29, 30	opthne 5431	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩ ↔ (0 ≠ 1 ∨ ((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 𝐾) mod 𝑁)))
108	106, 107	mpbir 231	. . . . . . . . . 10 ⊢ ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩
109	93, 108	pm3.2i 470	. . . . . . . . 9 ⊢ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩ ∧ ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩)
110	109	a1i 11	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩ ∧ ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩))
111	110	orcd 874	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → ((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩ ∧ ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩) ∨ (⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩ ∧ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩)))
112		prneimg 4798	. . . . . . 7 ⊢ (((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∈ V ∧ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ∈ V) ∧ (⟨1, 𝑥⟩ ∈ V ∧ ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩ ∈ V)) → (((⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩ ∧ ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩) ∨ (⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩ ∧ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩)) → {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}))
113	105, 111, 112	mpsyl 68	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})
114	85, 102, 113	3jca 1129	. . . . 5 ⊢ (((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) ∧ 𝑥 ∈ (0..^𝑁)) → ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∧ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}))
115	114	ralrimiva 3130	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ∀𝑥 ∈ (0..^𝑁)({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∧ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}))
116		ralnex 3064	. . . . 5 ⊢ (∀𝑥 ∈ (0..^𝑁) ¬ ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}) ↔ ¬ ∃𝑥 ∈ (0..^𝑁)({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}))
117		3ioran 1106	. . . . . . 7 ⊢ (¬ ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}) ↔ (¬ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∧ ¬ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ ¬ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}))
118		df-ne 2934	. . . . . . . 8 ⊢ ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ↔ ¬ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩})
119		df-ne 2934	. . . . . . . 8 ⊢ ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ ¬ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩})
120		df-ne 2934	. . . . . . . 8 ⊢ ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩} ↔ ¬ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})
121	118, 119, 120	3anbi123i 1156	. . . . . . 7 ⊢ (({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∧ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}) ↔ (¬ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∧ ¬ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ ¬ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}))
122	117, 121	bitr4i 278	. . . . . 6 ⊢ (¬ ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}) ↔ ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∧ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}))
123	122	ralbii 3084	. . . . 5 ⊢ (∀𝑥 ∈ (0..^𝑁) ¬ ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}) ↔ ∀𝑥 ∈ (0..^𝑁)({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∧ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}))
124	116, 123	bitr3i 277	. . . 4 ⊢ (¬ ∃𝑥 ∈ (0..^𝑁)({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}) ↔ ∀𝑥 ∈ (0..^𝑁)({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∧ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}))
125	115, 124	sylibr 234	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ¬ ∃𝑥 ∈ (0..^𝑁)({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}))
126		uzuzle34 12830	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝑁 ∈ (ℤ_≥‘3))
127		eqid 2737	. . . . . 6 ⊢ (0..^𝑁) = (0..^𝑁)
128		gpg5nbgrvtx03starlem1.j	. . . . . 6 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
129		gpg5nbgrvtx03starlem1.g	. . . . . 6 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
130		gpg5nbgrvtx03starlem1.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐺)
131	127, 128, 129, 130	gpgedgel 48541	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) → ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ∈ 𝐸 ↔ ∃𝑥 ∈ (0..^𝑁)({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})))
132	126, 131	sylan 581	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽) → ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ∈ 𝐸 ↔ ∃𝑥 ∈ (0..^𝑁)({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})))
133	132	3adant3 1133	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ∈ 𝐸 ↔ ∃𝑥 ∈ (0..^𝑁)({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})))
134	125, 133	mtbird 325	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ¬ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ∈ 𝐸)
135		df-nel 3038	. 2 ⊢ ({⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ∉ 𝐸 ↔ ¬ {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ∈ 𝐸)
136	134, 135	sylibr 234	1 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ∉ 𝐸)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∨ w3o 1086 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∉ wnel 3037 ∀wral 3052 ∃wrex 3062 Vcvv 3430 {cpr 4570 ⟨cop 4574 ‘cfv 6493 (class class class)co 7361 ℂcc 11030 ℝcr 11031 0cc0 11032 1c1 11033 + caddc 11035 − cmin 11371 / cdiv 11801 2c2 12230 3c3 12231 4c4 12232 ℤcz 12518 ℤ_≥cuz 12782 ℝ⁺crp 12936 ..^cfzo 13602 ⌈cceil 13744 mod cmo 13822 Vtxcvtx 29082 Edgcedg 29133 gPetersenGr cgpg 48531

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-oadd 8403 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-dju 9819 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-xnn0 12505 df-z 12519 df-dec 12639 df-uz 12783 df-rp 12937 df-fz 13456 df-fzo 13603 df-fl 13745 df-mod 13823 df-hash 14287 df-dvds 16216 df-struct 17111 df-slot 17146 df-ndx 17158 df-base 17174 df-edgf 29075 df-iedg 29085 df-edg 29134 df-gpg 48532

This theorem is referenced by: gpg5nbgrvtx03star 48571

W3C validator