gpgedg2ov - Mathbox for Alexander van der Vekens

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Theorem gpgedg2ov 48889

Description: The edges of the generalized Petersen graph GPG(N,K) between two outside vertices. (Contributed by AV, 15-Nov-2025.)

**Hypotheses**
Ref	Expression
gpgedgiov.j	⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
gpgedgiov.i	⊢ 𝐼 = (0..^𝑁)
gpgedgiov.g	⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
gpgedgiov.e	⊢ 𝐸 = (Edg‘𝐺)

**Assertion**
Ref	Expression
gpgedg2ov	⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∈ 𝐸 ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸) ↔ 𝑋 = 𝑌))

**Proof of Theorem gpgedg2ov**
Step	Hyp	Ref	Expression
1		prcom 4700	. . . . . 6 ⊢ {⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} = {⟨0, 𝑋⟩, ⟨0, ((𝑌 − 1) mod 𝑁)⟩}
2	1	eleq1i 2856	. . . . 5 ⊢ ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∈ 𝐸 ↔ {⟨0, 𝑋⟩, ⟨0, ((𝑌 − 1) mod 𝑁)⟩} ∈ 𝐸)
3		uzuzle35 12927	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘5) → 𝑁 ∈ (ℤ_≥‘3))
4	3	anim1i 627	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) → (𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽))
5	4	adantr 486	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽))
6	5	adantr 486	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 − 1) mod 𝑁)⟩} ∈ 𝐸) → (𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽))
7		c0ex 11215	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
8	7	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ 𝐼 → 0 ∈ V)
9	8	anim1i 627	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝐼) → (0 ∈ V ∧ 𝑋 ∈ 𝐼))
10	9	ancoms 464	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (0 ∈ V ∧ 𝑋 ∈ 𝐼))
11		op1stg 8004	. . . . . . . . . 10 ⊢ ((0 ∈ V ∧ 𝑋 ∈ 𝐼) → (1^st ‘⟨0, 𝑋⟩) = 0)
12	10, 11	syl 18	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (1^st ‘⟨0, 𝑋⟩) = 0)
13	12	adantl 487	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (1^st ‘⟨0, 𝑋⟩) = 0)
14	13	adantr 486	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 − 1) mod 𝑁)⟩} ∈ 𝐸) → (1^st ‘⟨0, 𝑋⟩) = 0)
15		simpr 490	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 − 1) mod 𝑁)⟩} ∈ 𝐸) → {⟨0, 𝑋⟩, ⟨0, ((𝑌 − 1) mod 𝑁)⟩} ∈ 𝐸)
16		gpgedgiov.j	. . . . . . . 8 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
17		gpgedgiov.g	. . . . . . . 8 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
18		eqid 2765	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
19		gpgedgiov.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
20	16, 17, 18, 19	gpgvtxedg0 48886	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘⟨0, 𝑋⟩) = 0 ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 − 1) mod 𝑁)⟩} ∈ 𝐸) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩))
21	6, 14, 15, 20	syl3anc 1398	. . . . . 6 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 − 1) mod 𝑁)⟩} ∈ 𝐸) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩))
22	21	ex 418	. . . . 5 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, 𝑋⟩, ⟨0, ((𝑌 − 1) mod 𝑁)⟩} ∈ 𝐸 → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩)))
23	2, 22	biimtrid 245	. . . 4 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∈ 𝐸 → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩)))
24	5	adantr 486	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸) → (𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽))
25	13	adantr 486	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸) → (1^st ‘⟨0, 𝑋⟩) = 0)
26		simpr 490	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸) → {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸)
27	16, 17, 18, 19	gpgvtxedg0 48886	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘⟨0, 𝑋⟩) = 0 ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸) → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩))
28	24, 25, 26, 27	syl3anc 1398	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸) → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩))
29	28	ex 418	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸 → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩)))
30		ovex 7452	. . . . . . . . . . . . 13 ⊢ ((𝑌 + 1) mod 𝑁) ∈ V
31	7, 30	opth 5460	. . . . . . . . . . . 12 ⊢ (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ↔ (0 = 0 ∧ ((𝑌 + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)))
32	3	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) → 𝑁 ∈ (ℤ_≥‘3))
33	32	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → 𝑁 ∈ (ℤ_≥‘3))
34		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼))
35	34	ancomd 467	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝐼))
36		1zzd 12640	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → 1 ∈ ℤ)
37		gpgedgiov.i	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐼 = (0..^𝑁)
38	37	modaddid 13961	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ (𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝐼) ∧ 1 ∈ ℤ) → (((𝑌 + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁) ↔ 𝑌 = 𝑋))
39	33, 35, 36, 38	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (((𝑌 + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁) ↔ 𝑌 = 𝑋))
40	39	biimpa 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ((𝑌 + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)) → 𝑌 = 𝑋)
41	40	eqcomd 2771	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ((𝑌 + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)) → 𝑋 = 𝑌)
42	41	ex 418	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (((𝑌 + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁) → 𝑋 = 𝑌))
43	42	adantld 496	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ((0 = 0 ∧ ((𝑌 + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)) → 𝑋 = 𝑌))
44	31, 43	biimtrid 245	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌))
45	44	imp 412	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩) → 𝑋 = 𝑌)
46	45	a1d 26	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩) → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → 𝑋 = 𝑌))
47	46	ex 418	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → 𝑋 = 𝑌)))
48	7, 30	opth 5460	. . . . . . . . . . . 12 ⊢ (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ↔ (0 = 1 ∧ ((𝑌 + 1) mod 𝑁) = 𝑋))
49		0ne1 12327	. . . . . . . . . . . . . . 15 ⊢ 0 ≠ 1
50		eqneqall 2971	. . . . . . . . . . . . . . 15 ⊢ (0 = 1 → (0 ≠ 1 → (((𝑌 + 1) mod 𝑁) = 𝑋 → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌))))
51	49, 50	mpi 21	. . . . . . . . . . . . . 14 ⊢ (0 = 1 → (((𝑌 + 1) mod 𝑁) = 𝑋 → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌)))
52	51	imp 412	. . . . . . . . . . . . 13 ⊢ ((0 = 1 ∧ ((𝑌 + 1) mod 𝑁) = 𝑋) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌))
53	52	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ((0 = 1 ∧ ((𝑌 + 1) mod 𝑁) = 𝑋) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌)))
54	48, 53	biimtrid 245	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌)))
55	54	imp 412	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌))
56		eqeq2 2777	. . . . . . . . . . . . 13 ⊢ (⟨1, 𝑋⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩ → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ↔ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩))
57	56	eqcoms 2773	. . . . . . . . . . . 12 ⊢ (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ↔ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩))
58	57	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ↔ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩))
59		ovex 7452	. . . . . . . . . . . . . 14 ⊢ ((𝑌 − 1) mod 𝑁) ∈ V
60	7, 59	opth 5460	. . . . . . . . . . . . 13 ⊢ (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩ ↔ (0 = 0 ∧ ((𝑌 − 1) mod 𝑁) = ((𝑌 + 1) mod 𝑁)))
61		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 𝑌 ∈ 𝐼)
62	37	modm1nep1 48166	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑌 ∈ 𝐼) → ((𝑌 − 1) mod 𝑁) ≠ ((𝑌 + 1) mod 𝑁))
63	32, 61, 62	syl2an 608	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ((𝑌 − 1) mod 𝑁) ≠ ((𝑌 + 1) mod 𝑁))
64		eqneqall 2971	. . . . . . . . . . . . . . . 16 ⊢ (((𝑌 − 1) mod 𝑁) = ((𝑌 + 1) mod 𝑁) → (((𝑌 − 1) mod 𝑁) ≠ ((𝑌 + 1) mod 𝑁) → 𝑋 = 𝑌))
65	64	com12 33	. . . . . . . . . . . . . . 15 ⊢ (((𝑌 − 1) mod 𝑁) ≠ ((𝑌 + 1) mod 𝑁) → (((𝑌 − 1) mod 𝑁) = ((𝑌 + 1) mod 𝑁) → 𝑋 = 𝑌))
66	63, 65	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (((𝑌 − 1) mod 𝑁) = ((𝑌 + 1) mod 𝑁) → 𝑋 = 𝑌))
67	66	adantld 496	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ((0 = 0 ∧ ((𝑌 − 1) mod 𝑁) = ((𝑌 + 1) mod 𝑁)) → 𝑋 = 𝑌))
68	60, 67	biimtrid 245	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌))
69	68	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌))
70	58, 69	sylbid 243	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ → 𝑋 = 𝑌))
71	49	orci 879	. . . . . . . . . . . . . 14 ⊢ (0 ≠ 1 ∨ ((𝑌 + 1) mod 𝑁) ≠ 𝑋)
72	7, 30	opthne 5466	. . . . . . . . . . . . . 14 ⊢ (⟨0, ((𝑌 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑋⟩ ↔ (0 ≠ 1 ∨ ((𝑌 + 1) mod 𝑁) ≠ 𝑋))
73	71, 72	mpbir 234	. . . . . . . . . . . . 13 ⊢ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑋⟩
74	73	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ⟨0, ((𝑌 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑋⟩)
75		eqneqall 2971	. . . . . . . . . . . . 13 ⊢ (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑋⟩ → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ → 𝑋 = 𝑌)))
76	75	com12 33	. . . . . . . . . . . 12 ⊢ (⟨0, ((𝑌 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑋⟩ → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ → 𝑋 = 𝑌)))
77	74, 76	syl 18	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ → 𝑋 = 𝑌)))
78	77	imp 412	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ → 𝑋 = 𝑌))
79	55, 70, 78	3jaod 1456	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩) → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → 𝑋 = 𝑌))
80	79	ex 418	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → 𝑋 = 𝑌)))
81	7, 30	opth 5460	. . . . . . . . . . . 12 ⊢ (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ↔ (0 = 0 ∧ ((𝑌 + 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁)))
82	7, 59	opth 5460	. . . . . . . . . . . . . . 15 ⊢ (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ↔ (0 = 0 ∧ ((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)))
83		simpll 779	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → 𝑁 ∈ (ℤ_≥‘5))
84		simprl 783	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → 𝑋 ∈ 𝐼)
85		simprr 785	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → 𝑌 ∈ 𝐼)
86	37	modm1p1ne 48171	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁) → ((𝑌 + 1) mod 𝑁) ≠ ((𝑋 − 1) mod 𝑁)))
87	83, 84, 85, 86	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁) → ((𝑌 + 1) mod 𝑁) ≠ ((𝑋 − 1) mod 𝑁)))
88		eqneqall 2971	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑌 + 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁) → (((𝑌 + 1) mod 𝑁) ≠ ((𝑋 − 1) mod 𝑁) → 𝑋 = 𝑌))
89	88	com12 33	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑌 + 1) mod 𝑁) ≠ ((𝑋 − 1) mod 𝑁) → (((𝑌 + 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁) → 𝑋 = 𝑌))
90	89	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (((𝑌 + 1) mod 𝑁) ≠ ((𝑋 − 1) mod 𝑁) → (((𝑌 + 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁) → 𝑋 = 𝑌)))
91	87, 90	syld 48	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁) → (((𝑌 + 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁) → 𝑋 = 𝑌)))
92	91	adantld 496	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ((0 = 0 ∧ ((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)) → (((𝑌 + 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁) → 𝑋 = 𝑌)))
93	82, 92	biimtrid 245	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → (((𝑌 + 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁) → 𝑋 = 𝑌)))
94	93	com23 87	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (((𝑌 + 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌)))
95	94	adantld 496	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ((0 = 0 ∧ ((𝑌 + 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁)) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌)))
96	81, 95	biimtrid 245	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌)))
97	96	imp 412	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌))
98	49	orci 879	. . . . . . . . . . . . 13 ⊢ (0 ≠ 1 ∨ ((𝑌 − 1) mod 𝑁) ≠ 𝑋)
99	7, 59	opthne 5466	. . . . . . . . . . . . 13 ⊢ (⟨0, ((𝑌 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑋⟩ ↔ (0 ≠ 1 ∨ ((𝑌 − 1) mod 𝑁) ≠ 𝑋))
100	98, 99	mpbir 234	. . . . . . . . . . . 12 ⊢ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑋⟩
101		eqneqall 2971	. . . . . . . . . . . 12 ⊢ (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑋⟩ → 𝑋 = 𝑌))
102	100, 101	mpi 21	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ → 𝑋 = 𝑌)
103	102	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ → 𝑋 = 𝑌))
104		eqeq2 2777	. . . . . . . . . . . . 13 ⊢ (⟨0, ((𝑋 − 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩ → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ↔ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩))
105	104	eqcoms 2773	. . . . . . . . . . . 12 ⊢ (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ↔ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩))
106	105	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ↔ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩))
107	68	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌))
108	106, 107	sylbid 243	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ → 𝑋 = 𝑌))
109	97, 103, 108	3jaod 1456	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → 𝑋 = 𝑌))
110	109	ex 418	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → 𝑋 = 𝑌)))
111	47, 80, 110	3jaod 1456	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ((⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → 𝑋 = 𝑌)))
112		op2ndg 8005	. . . . . . . . . . 11 ⊢ ((0 ∈ V ∧ 𝑋 ∈ 𝐼) → (2^nd ‘⟨0, 𝑋⟩) = 𝑋)
113	7, 112	mpan 703	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝐼 → (2^nd ‘⟨0, 𝑋⟩) = 𝑋)
114	113	adantr 486	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (2^nd ‘⟨0, 𝑋⟩) = 𝑋)
115	114	adantl 487	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (2^nd ‘⟨0, 𝑋⟩) = 𝑋)
116		oveq1 7426	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → ((2^nd ‘⟨0, 𝑋⟩) + 1) = (𝑋 + 1))
117	116	oveq1d 7434	. . . . . . . . . . . 12 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁))
118	117	opeq2d 4847	. . . . . . . . . . 11 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩)
119	118	eqeq2d 2776	. . . . . . . . . 10 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ↔ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩))
120		opeq2 4841	. . . . . . . . . . 11 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ = ⟨1, 𝑋⟩)
121	120	eqeq2d 2776	. . . . . . . . . 10 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ↔ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩))
122		oveq1 7426	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → ((2^nd ‘⟨0, 𝑋⟩) − 1) = (𝑋 − 1))
123	122	oveq1d 7434	. . . . . . . . . . . 12 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁))
124	123	opeq2d 4847	. . . . . . . . . . 11 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩)
125	124	eqeq2d 2776	. . . . . . . . . 10 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩ ↔ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩))
126	119, 121, 125	3orbi123d 1463	. . . . . . . . 9 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → ((⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) ↔ (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩)))
127	118	eqeq2d 2776	. . . . . . . . . . 11 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ↔ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩))
128	120	eqeq2d 2776	. . . . . . . . . . 11 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ↔ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩))
129	124	eqeq2d 2776	. . . . . . . . . . 11 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩ ↔ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩))
130	127, 128, 129	3orbi123d 1463	. . . . . . . . . 10 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) ↔ (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩)))
131	130	imbi1d 344	. . . . . . . . 9 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → (((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) → 𝑋 = 𝑌) ↔ ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → 𝑋 = 𝑌)))
132	126, 131	imbi12d 347	. . . . . . . 8 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → (((⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) → 𝑋 = 𝑌)) ↔ ((⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → 𝑋 = 𝑌))))
133	115, 132	syl 18	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (((⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) → 𝑋 = 𝑌)) ↔ ((⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → 𝑋 = 𝑌))))
134	111, 133	mpbird 260	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ((⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) → 𝑋 = 𝑌)))
135	29, 134	syld 48	. . . . 5 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸 → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) → 𝑋 = 𝑌)))
136	135	com23 87	. . . 4 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ((⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) → ({⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸 → 𝑋 = 𝑌)))
137	23, 136	syld 48	. . 3 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∈ 𝐸 → ({⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸 → 𝑋 = 𝑌)))
138	137	impd 416	. 2 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∈ 𝐸 ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸) → 𝑋 = 𝑌))
139		eqid 2765	. . . . . . . . 9 ⊢ 0 = 0
140	139	orci 879	. . . . . . . 8 ⊢ (0 = 0 ∨ 0 = 1)
141	61, 140	jctil 529	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → ((0 = 0 ∨ 0 = 1) ∧ 𝑌 ∈ 𝐼))
142	141	adantl 487	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ((0 = 0 ∨ 0 = 1) ∧ 𝑌 ∈ 𝐼))
143	37, 16, 17, 18	opgpgvtx 48878	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) → (⟨0, 𝑌⟩ ∈ (Vtx‘𝐺) ↔ ((0 = 0 ∨ 0 = 1) ∧ 𝑌 ∈ 𝐼)))
144	4, 143	syl 18	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) → (⟨0, 𝑌⟩ ∈ (Vtx‘𝐺) ↔ ((0 = 0 ∨ 0 = 1) ∧ 𝑌 ∈ 𝐼)))
145	144	adantr 486	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (⟨0, 𝑌⟩ ∈ (Vtx‘𝐺) ↔ ((0 = 0 ∨ 0 = 1) ∧ 𝑌 ∈ 𝐼)))
146	142, 145	mpbird 260	. . . . 5 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ⟨0, 𝑌⟩ ∈ (Vtx‘𝐺))
147	7	a1i 11	. . . . . . 7 ⊢ (𝑋 ∈ 𝐼 → 0 ∈ V)
148		op1stg 8004	. . . . . . 7 ⊢ ((0 ∈ V ∧ 𝑌 ∈ 𝐼) → (1^st ‘⟨0, 𝑌⟩) = 0)
149	147, 148	sylan 592	. . . . . 6 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (1^st ‘⟨0, 𝑌⟩) = 0)
150	149	adantl 487	. . . . 5 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (1^st ‘⟨0, 𝑌⟩) = 0)
151	16, 17, 18, 19	gpgedgvtx0 48884	. . . . 5 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (⟨0, 𝑌⟩ ∈ (Vtx‘𝐺) ∧ (1^st ‘⟨0, 𝑌⟩) = 0)) → ({⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨1, (2^nd ‘⟨0, 𝑌⟩)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩} ∈ 𝐸))
152	5, 146, 150, 151	syl12anc 850	. . . 4 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨1, (2^nd ‘⟨0, 𝑌⟩)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩} ∈ 𝐸))
153		op2ndg 8005	. . . . . . . . . . . 12 ⊢ ((0 ∈ V ∧ 𝑌 ∈ 𝐼) → (2^nd ‘⟨0, 𝑌⟩) = 𝑌)
154	7, 153	mpan 703	. . . . . . . . . . 11 ⊢ (𝑌 ∈ 𝐼 → (2^nd ‘⟨0, 𝑌⟩) = 𝑌)
155		oveq1 7426	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → ((2^nd ‘⟨0, 𝑌⟩) − 1) = (𝑌 − 1))
156	155	oveq1d 7434	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁) = ((𝑌 − 1) mod 𝑁))
157	156	opeq2d 4847	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩ = ⟨0, ((𝑌 − 1) mod 𝑁)⟩)
158	157	preq2d 4708	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → {⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩} = {⟨0, 𝑌⟩, ⟨0, ((𝑌 − 1) mod 𝑁)⟩})
159		prcom 4700	. . . . . . . . . . . . . 14 ⊢ {⟨0, 𝑌⟩, ⟨0, ((𝑌 − 1) mod 𝑁)⟩} = {⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩}
160	158, 159	eqtrdi 2816	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → {⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩} = {⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩})
161	160	eleq1d 2850	. . . . . . . . . . . 12 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → ({⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩} ∈ 𝐸 ↔ {⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩} ∈ 𝐸))
162		oveq1 7426	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → ((2^nd ‘⟨0, 𝑌⟩) + 1) = (𝑌 + 1))
163	162	oveq1d 7434	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁) = ((𝑌 + 1) mod 𝑁))
164	163	opeq2d 4847	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → ⟨0, (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩)
165	164	preq2d 4708	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → {⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁)⟩} = {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩})
166	165	eleq1d 2850	. . . . . . . . . . . 12 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → ({⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁)⟩} ∈ 𝐸 ↔ {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸))
167	161, 166	anbi12d 644	. . . . . . . . . . 11 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → (({⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁)⟩} ∈ 𝐸) ↔ ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸)))
168	154, 167	syl 18	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝐼 → (({⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁)⟩} ∈ 𝐸) ↔ ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸)))
169	168	biimpcd 252	. . . . . . . . 9 ⊢ (({⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁)⟩} ∈ 𝐸) → (𝑌 ∈ 𝐼 → ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸)))
170	169	ancoms 464	. . . . . . . 8 ⊢ (({⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩} ∈ 𝐸) → (𝑌 ∈ 𝐼 → ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸)))
171	170	3adant2 1149	. . . . . . 7 ⊢ (({⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨1, (2^nd ‘⟨0, 𝑌⟩)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩} ∈ 𝐸) → (𝑌 ∈ 𝐼 → ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸)))
172	171	com12 33	. . . . . 6 ⊢ (𝑌 ∈ 𝐼 → (({⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨1, (2^nd ‘⟨0, 𝑌⟩)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩} ∈ 𝐸) → ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸)))
173	172	adantl 487	. . . . 5 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (({⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨1, (2^nd ‘⟨0, 𝑌⟩)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩} ∈ 𝐸) → ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸)))
174	173	adantl 487	. . . 4 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (({⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨1, (2^nd ‘⟨0, 𝑌⟩)⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩} ∈ 𝐸) → ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸)))
175	152, 174	mpd 16	. . 3 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸))
176		opeq2 4841	. . . . . 6 ⊢ (𝑋 = 𝑌 → ⟨0, 𝑋⟩ = ⟨0, 𝑌⟩)
177	176	preq2d 4708	. . . . 5 ⊢ (𝑋 = 𝑌 → {⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} = {⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩})
178	177	eleq1d 2850	. . . 4 ⊢ (𝑋 = 𝑌 → ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∈ 𝐸 ↔ {⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩} ∈ 𝐸))
179	176	preq1d 4707	. . . . 5 ⊢ (𝑋 = 𝑌 → {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} = {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩})
180	179	eleq1d 2850	. . . 4 ⊢ (𝑋 = 𝑌 → ({⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸 ↔ {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸))
181	178, 180	anbi12d 644	. . 3 ⊢ (𝑋 = 𝑌 → (({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∈ 𝐸 ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸) ↔ ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩} ∈ 𝐸 ∧ {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸)))
182	175, 181	syl5ibrcom 250	. 2 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋 = 𝑌 → ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∈ 𝐸 ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸)))
183	138, 182	impbid 215	1 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∈ 𝐸 ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸) ↔ 𝑋 = 𝑌))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 Vcvv 3457 {cpr 4593 ⟨cop 4597 ‘cfv 6540 (class class class)co 7419 1^st c1st 7990 2^nd c2nd 7991 0cc0 11115 1c1 11116 + caddc 11118 − cmin 11456 / cdiv 11886 2c2 12310 3c3 12311 5c5 12313 ℤcz 12606 ℤ_≥cuz 12878 ..^cfzo 13699 ⌈cceil 13842 mod cmo 13920 Vtxcvtx 29401 Edgcedg 29452 gPetersenGr cgpg 48863

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-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193

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-int 4915 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 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-oadd 8463 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-inf 9410 df-dju 9903 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-xnn0 12593 df-z 12607 df-dec 12728 df-uz 12879 df-rp 13033 df-ico 13394 df-fz 13552 df-fzo 13700 df-fl 13843 df-ceil 13844 df-mod 13921 df-hash 14385 df-dvds 16333 df-struct 17229 df-slot 17264 df-ndx 17276 df-base 17292 df-edgf 29394 df-vtx 29403 df-iedg 29404 df-edg 29453 df-umgr 29488 df-usgr 29559 df-gpg 48864

This theorem is used by: pgnbgreunbgrlem1 48936

W3C validator