gpg5nbgrvtx03starlem2 - Mathbox for Alexander van der Vekens

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Theorem gpg5nbgrvtx03starlem2 48050

Description: Lemma 2 for gpg5nbgrvtx03star 48061. (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 47343	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑋 ∈ ℤ) → ((𝑋 − 1) mod 𝑁) ≠ ((𝑋 + 2) mod 𝑁))
2	1	3adant2 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((𝑋 − 1) mod 𝑁) ≠ ((𝑋 + 2) mod 𝑁))
3		zcn 12540	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ ℤ → 𝑋 ∈ ℂ)
4	3	3ad2ant3 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → 𝑋 ∈ ℂ)
5		add1p1 12439	. . . . . . . . . . . . . . . . 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 7404	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → (((𝑋 + 1) + 1) mod 𝑁) = ((𝑋 + 2) mod 𝑁))
8	2, 7	neeqtrrd 3000	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((𝑋 − 1) mod 𝑁) ≠ (((𝑋 + 1) + 1) mod 𝑁))
9		zre 12539	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ ℤ → 𝑋 ∈ ℝ)
10	9	3ad2ant3 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → 𝑋 ∈ ℝ)
11		1red 11181	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → 1 ∈ ℝ)
12	10, 11	readdcld 11209	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → (𝑋 + 1) ∈ ℝ)
13		eluz4nn 12855	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝑁 ∈ ℕ)
14	13	nnrpd 12999	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝑁 ∈ ℝ⁺)
15	14	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → 𝑁 ∈ ℝ⁺)
16		modaddmod 13880	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((((𝑋 + 1) mod 𝑁) + 1) mod 𝑁) = (((𝑋 + 1) + 1) mod 𝑁))
18	8, 17	neeqtrrd 3000	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((𝑋 − 1) mod 𝑁) ≠ ((((𝑋 + 1) mod 𝑁) + 1) mod 𝑁))
19	18	ad2antrl 728	. . . . . . . . . . . 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 7396	. . . . . . . . . . . . . 14 ⊢ (((𝑋 + 1) mod 𝑁) = 𝑥 → (((𝑋 + 1) mod 𝑁) + 1) = (𝑥 + 1))
21	20	oveq1d 7404	. . . . . . . . . . . . 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 2995	. . . . . . . . . . 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 874	. . . . . . . . . 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 867	. . . . . . . . . 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 3009	. . . . . . . 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 11174	. . . . . . . . . . . 12 ⊢ 0 ∈ V
30		ovex 7422	. . . . . . . . . . . 12 ⊢ ((𝑋 + 1) mod 𝑁) ∈ V
31	29, 30	opthne 5444	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ↔ (0 ≠ 0 ∨ ((𝑋 + 1) mod 𝑁) ≠ 𝑥))
32		neirr 2935	. . . . . . . . . . . 12 ⊢ ¬ 0 ≠ 0
33	32	biorfi 938	. . . . . . . . . . 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 7422	. . . . . . . . . . . 12 ⊢ ((𝑋 − 1) mod 𝑁) ∈ V
37	29, 36	opthne 5444	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ↔ (0 ≠ 0 ∨ ((𝑋 − 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁)))
38	32	biorfi 938	. . . . . . . . . . 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 918	. . . . . . . 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 12850	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝑁 ∈ (ℤ_≥‘2))
44	43	anim1i 615	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑋 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘2) ∧ 𝑋 ∈ ℤ))
45	44	3adant2 1131	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘2) ∧ 𝑋 ∈ ℤ))
46		zp1modne 47337	. . . . . . . . . . . . . . . 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 11609	. . . . . . . . . . . . . . . . 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 7404	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → (((𝑋 − 1) + 1) mod 𝑁) = (𝑋 mod 𝑁))
51	47, 50	neeqtrrd 3000	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((𝑋 + 1) mod 𝑁) ≠ (((𝑋 − 1) + 1) mod 𝑁))
52	10, 11	resubcld 11612	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → (𝑋 − 1) ∈ ℝ)
53		modaddmod 13880	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((((𝑋 − 1) mod 𝑁) + 1) mod 𝑁) = (((𝑋 − 1) + 1) mod 𝑁))
55	51, 54	neeqtrrd 3000	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → ((𝑋 + 1) mod 𝑁) ≠ ((((𝑋 − 1) mod 𝑁) + 1) mod 𝑁))
56	55	ad2antrl 728	. . . . . . . . . . . 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 7396	. . . . . . . . . . . . . 14 ⊢ (((𝑋 − 1) mod 𝑁) = 𝑥 → (((𝑋 − 1) mod 𝑁) + 1) = (𝑥 + 1))
58	57	oveq1d 7404	. . . . . . . . . . . . 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 2995	. . . . . . . . . . 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 873	. . . . . . . . . 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 868	. . . . . . . . . 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 3009	. . . . . . . 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 5444	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, ((𝑥 + 1) mod 𝑁)⟩ ↔ (0 ≠ 0 ∨ ((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 1) mod 𝑁)))
67	32	biorfi 938	. . . . . . . . . . 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 5444	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ↔ (0 ≠ 0 ∨ ((𝑋 − 1) mod 𝑁) ≠ 𝑥))
71	32	biorfi 938	. . . . . . . . . . 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 918	. . . . . . . 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 5426	. . . . . . . . . 10 ⊢ ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∈ V
77		opex 5426	. . . . . . . . . 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 5426	. . . . . . . . . 10 ⊢ ⟨0, 𝑥⟩ ∈ V
80		opex 5426	. . . . . . . . . 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 4821	. . . . . . . 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 713	. . . . . 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 12258	. . . . . . . . . . . 12 ⊢ 0 ≠ 1
87	86	orci 865	. . . . . . . . . . 11 ⊢ (0 ≠ 1 ∨ ((𝑋 − 1) mod 𝑁) ≠ 𝑥)
88	29, 36	opthne 5444	. . . . . . . . . . 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 866	. . . . . . . . 9 ⊢ (⟨0, ((𝑋 + 1) mod 𝑁)⟩ ≠ ⟨0, 𝑥⟩ ∨ ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑥⟩)
91	86	orci 865	. . . . . . . . . . 11 ⊢ (0 ≠ 1 ∨ ((𝑋 + 1) mod 𝑁) ≠ 𝑥)
92	29, 30	opthne 5444	. . . . . . . . . . 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 865	. . . . . . . . 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 5426	. . . . . . . . . 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 4821	. . . . . . . 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 5426	. . . . . . . . 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 865	. . . . . . . . . . 11 ⊢ (0 ≠ 1 ∨ ((𝑋 + 1) mod 𝑁) ≠ ((𝑥 + 𝐾) mod 𝑁))
107	29, 30	opthne 5444	. . . . . . . . . . 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 873	. . . . . . 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 4820	. . . . . . 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 1128	. . . . 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 3126	. . . 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 3056	. . . . 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 1105	. . . . . . 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 2927	. . . . . . . 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 2927	. . . . . . . 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 2927	. . . . . . . 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 1155	. . . . . . 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 3076	. . . . 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 12851	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝑁 ∈ (ℤ_≥‘3))
127		eqid 2730	. . . . . 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 48031	. . . . 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 580	. . . 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 1132	. . 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 3031	. 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 847 ∨ w3o 1085 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∉ wnel 3030 ∀wral 3045 ∃wrex 3054 Vcvv 3450 {cpr 4593 ⟨cop 4597 ‘cfv 6513 (class class class)co 7389 ℂcc 11072 ℝcr 11073 0cc0 11074 1c1 11075 + caddc 11077 − cmin 11411 / cdiv 11841 2c2 12242 3c3 12243 4c4 12244 ℤcz 12535 ℤ_≥cuz 12799 ℝ⁺crp 12957 ..^cfzo 13621 ⌈cceil 13759 mod cmo 13837 Vtxcvtx 28929 Edgcedg 28980 gPetersenGr cgpg 48021

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-oadd 8440 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-sup 9399 df-inf 9400 df-dju 9860 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-xnn0 12522 df-z 12536 df-dec 12656 df-uz 12800 df-rp 12958 df-fz 13475 df-fzo 13622 df-fl 13760 df-mod 13838 df-hash 14302 df-dvds 16229 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17186 df-edgf 28922 df-iedg 28932 df-edg 28981 df-gpg 48022

This theorem is referenced by: gpg5nbgrvtx03star 48061

W3C validator