gpgedg2ov - Mathbox for Alexander van der Vekens

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Theorem gpgedg2ov 48097

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 4680	. . . . . 6 ⊢ {⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} = {⟨0, 𝑋⟩, ⟨0, ((𝑌 − 1) mod 𝑁)⟩}
2	1	eleq1i 2822	. . . . 5 ⊢ ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∈ 𝐸 ↔ {⟨0, 𝑋⟩, ⟨0, ((𝑌 − 1) mod 𝑁)⟩} ∈ 𝐸)
3		uzuzle35 12780	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘5) → 𝑁 ∈ (ℤ_≥‘3))
4	3	anim1i 615	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) → (𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽))
5	4	adantr 480	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽))
6	5	adantr 480	. . . . . . 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 11101	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
8	7	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ 𝐼 → 0 ∈ V)
9	8	anim1i 615	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝐼) → (0 ∈ V ∧ 𝑋 ∈ 𝐼))
10	9	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (0 ∈ V ∧ 𝑋 ∈ 𝐼))
11		op1stg 7928	. . . . . . . . . 10 ⊢ ((0 ∈ V ∧ 𝑋 ∈ 𝐼) → (1^st ‘⟨0, 𝑋⟩) = 0)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (1^st ‘⟨0, 𝑋⟩) = 0)
13	12	adantl 481	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (1^st ‘⟨0, 𝑋⟩) = 0)
14	13	adantr 480	. . . . . . 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 484	. . . . . . 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 2731	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
19		gpgedgiov.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
20	16, 17, 18, 19	gpgvtxedg0 48094	. . . . . . 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 1373	. . . . . 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 412	. . . . 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 242	. . . 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 480	. . . . . . . 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 480	. . . . . . . 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 484	. . . . . . . 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 48094	. . . . . . . 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 1373	. . . . . . 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 412	. . . . . 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 7374	. . . . . . . . . . . . 13 ⊢ ((𝑌 + 1) mod 𝑁) ∈ V
31	7, 30	opth 5411	. . . . . . . . . . . 12 ⊢ (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ↔ (0 = 0 ∧ ((𝑌 + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)))
32	3	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) → 𝑁 ∈ (ℤ_≥‘3))
33	32	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → 𝑁 ∈ (ℤ_≥‘3))
34		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼))
35	34	ancomd 461	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝐼))
36		1zzd 12498	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → 1 ∈ ℤ)
37		gpgedgiov.i	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐼 = (0..^𝑁)
38	37	modaddid 13809	. . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (((𝑌 + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁) ↔ 𝑌 = 𝑋))
40	39	biimpa 476	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ((𝑌 + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)) → 𝑌 = 𝑋)
41	40	eqcomd 2737	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ ((𝑌 + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)) → 𝑋 = 𝑌)
42	41	ex 412	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (((𝑌 + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁) → 𝑋 = 𝑌))
43	42	adantld 490	. . . . . . . . . . . 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 242	. . . . . . . . . . 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 406	. . . . . . . . . 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 25	. . . . . . . . 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 412	. . . . . . . 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 5411	. . . . . . . . . . . 12 ⊢ (⟨0, ((𝑌 + 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ ↔ (0 = 1 ∧ ((𝑌 + 1) mod 𝑁) = 𝑋))
49		0ne1 12191	. . . . . . . . . . . . . . 15 ⊢ 0 ≠ 1
50		eqneqall 2939	. . . . . . . . . . . . . . 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 20	. . . . . . . . . . . . . 14 ⊢ (0 = 1 → (((𝑌 + 1) mod 𝑁) = 𝑋 → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌)))
52	51	imp 406	. . . . . . . . . . . . 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 242	. . . . . . . . . . 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 406	. . . . . . . . . 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 2743	. . . . . . . . . . . . 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 2739	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 7374	. . . . . . . . . . . . . 14 ⊢ ((𝑌 − 1) mod 𝑁) ∈ V
60	7, 59	opth 5411	. . . . . . . . . . . . 13 ⊢ (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩ ↔ (0 = 0 ∧ ((𝑌 − 1) mod 𝑁) = ((𝑌 + 1) mod 𝑁)))
61		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 𝑌 ∈ 𝐼)
62	37	modm1nep1 47396	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑌 ∈ 𝐼) → ((𝑌 − 1) mod 𝑁) ≠ ((𝑌 + 1) mod 𝑁))
63	32, 61, 62	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ((𝑌 − 1) mod 𝑁) ≠ ((𝑌 + 1) mod 𝑁))
64		eqneqall 2939	. . . . . . . . . . . . . . . 16 ⊢ (((𝑌 − 1) mod 𝑁) = ((𝑌 + 1) mod 𝑁) → (((𝑌 − 1) mod 𝑁) ≠ ((𝑌 + 1) mod 𝑁) → 𝑋 = 𝑌))
65	64	com12 32	. . . . . . . . . . . . . . 15 ⊢ (((𝑌 − 1) mod 𝑁) ≠ ((𝑌 + 1) mod 𝑁) → (((𝑌 − 1) mod 𝑁) = ((𝑌 + 1) mod 𝑁) → 𝑋 = 𝑌))
66	63, 65	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (((𝑌 − 1) mod 𝑁) = ((𝑌 + 1) mod 𝑁) → 𝑋 = 𝑌))
67	66	adantld 490	. . . . . . . . . . . . 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 242	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 240	. . . . . . . . . 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 865	. . . . . . . . . . . . . 14 ⊢ (0 ≠ 1 ∨ ((𝑌 + 1) mod 𝑁) ≠ 𝑋)
72	7, 30	opthne 5417	. . . . . . . . . . . . . 14 ⊢ (⟨0, ((𝑌 + 1) mod 𝑁)⟩ ≠ ⟨1, 𝑋⟩ ↔ (0 ≠ 1 ∨ ((𝑌 + 1) mod 𝑁) ≠ 𝑋))
73	71, 72	mpbir 231	. . . . . . . . . . . . 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 2939	. . . . . . . . . . . . 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 32	. . . . . . . . . . . 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 17	. . . . . . . . . . 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 406	. . . . . . . . . 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 1431	. . . . . . . . 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 412	. . . . . . . 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 5411	. . . . . . . . . . . 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 5411	. . . . . . . . . . . . . . 15 ⊢ (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ↔ (0 = 0 ∧ ((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)))
83		simpll 766	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → 𝑁 ∈ (ℤ_≥‘5))
84		simprl 770	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → 𝑋 ∈ 𝐼)
85		simprr 772	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → 𝑌 ∈ 𝐼)
86	37	modm1p1ne 47401	. . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . 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 2939	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑌 + 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁) → (((𝑌 + 1) mod 𝑁) ≠ ((𝑋 − 1) mod 𝑁) → 𝑋 = 𝑌))
89	88	com12 32	. . . . . . . . . . . . . . . . . 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 47	. . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . 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 86	. . . . . . . . . . . . 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 490	. . . . . . . . . . . 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 242	. . . . . . . . . . 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 406	. . . . . . . . . 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 865	. . . . . . . . . . . . 13 ⊢ (0 ≠ 1 ∨ ((𝑌 − 1) mod 𝑁) ≠ 𝑋)
99	7, 59	opthne 5417	. . . . . . . . . . . . 13 ⊢ (⟨0, ((𝑌 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑋⟩ ↔ (0 ≠ 1 ∨ ((𝑌 − 1) mod 𝑁) ≠ 𝑋))
100	98, 99	mpbir 231	. . . . . . . . . . . 12 ⊢ ⟨0, ((𝑌 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑋⟩
101		eqneqall 2939	. . . . . . . . . . . 12 ⊢ (⟨0, ((𝑌 − 1) mod 𝑁)⟩ = ⟨1, 𝑋⟩ → (⟨0, ((𝑌 − 1) mod 𝑁)⟩ ≠ ⟨1, 𝑋⟩ → 𝑋 = 𝑌))
102	100, 101	mpi 20	. . . . . . . . . . 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 2743	. . . . . . . . . . . . 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 2739	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 480	. . . . . . . . . . 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 240	. . . . . . . . . 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 1431	. . . . . . . . 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 412	. . . . . . . 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 1431	. . . . . . 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 7929	. . . . . . . . . . 11 ⊢ ((0 ∈ V ∧ 𝑋 ∈ 𝐼) → (2^nd ‘⟨0, 𝑋⟩) = 𝑋)
113	7, 112	mpan 690	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝐼 → (2^nd ‘⟨0, 𝑋⟩) = 𝑋)
114	113	adantr 480	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (2^nd ‘⟨0, 𝑋⟩) = 𝑋)
115	114	adantl 481	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (2^nd ‘⟨0, 𝑋⟩) = 𝑋)
116		oveq1 7348	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → ((2^nd ‘⟨0, 𝑋⟩) + 1) = (𝑋 + 1))
117	116	oveq1d 7356	. . . . . . . . . . . 12 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁))
118	117	opeq2d 4827	. . . . . . . . . . 11 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨0, (((2^nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩)
119	118	eqeq2d 2742	. . . . . . . . . 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 4821	. . . . . . . . . . 11 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨1, (2^nd ‘⟨0, 𝑋⟩)⟩ = ⟨1, 𝑋⟩)
121	120	eqeq2d 2742	. . . . . . . . . 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 7348	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → ((2^nd ‘⟨0, 𝑋⟩) − 1) = (𝑋 − 1))
123	122	oveq1d 7356	. . . . . . . . . . . 12 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁))
124	123	opeq2d 4827	. . . . . . . . . . 11 ⊢ ((2^nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨0, (((2^nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩)
125	124	eqeq2d 2742	. . . . . . . . . 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 1437	. . . . . . . . 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 2742	. . . . . . . . . . 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 2742	. . . . . . . . . . 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 2742	. . . . . . . . . . 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 1437	. . . . . . . . . 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 341	. . . . . . . . 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 344	. . . . . . . 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 17	. . . . . . 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 257	. . . . . 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 47	. . . . 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 86	. . . 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 47	. . 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 410	. 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 2731	. . . . . . . . 9 ⊢ 0 = 0
140	139	orci 865	. . . . . . . 8 ⊢ (0 = 0 ∨ 0 = 1)
141	61, 140	jctil 519	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → ((0 = 0 ∨ 0 = 1) ∧ 𝑌 ∈ 𝐼))
142	141	adantl 481	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ((0 = 0 ∨ 0 = 1) ∧ 𝑌 ∈ 𝐼))
143	37, 16, 17, 18	opgpgvtx 48086	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) → (⟨0, 𝑌⟩ ∈ (Vtx‘𝐺) ↔ ((0 = 0 ∨ 0 = 1) ∧ 𝑌 ∈ 𝐼)))
144	4, 143	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) → (⟨0, 𝑌⟩ ∈ (Vtx‘𝐺) ↔ ((0 = 0 ∨ 0 = 1) ∧ 𝑌 ∈ 𝐼)))
145	144	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (⟨0, 𝑌⟩ ∈ (Vtx‘𝐺) ↔ ((0 = 0 ∨ 0 = 1) ∧ 𝑌 ∈ 𝐼)))
146	142, 145	mpbird 257	. . . . 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 7928	. . . . . . 7 ⊢ ((0 ∈ V ∧ 𝑌 ∈ 𝐼) → (1^st ‘⟨0, 𝑌⟩) = 0)
149	147, 148	sylan 580	. . . . . 6 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (1^st ‘⟨0, 𝑌⟩) = 0)
150	149	adantl 481	. . . . 5 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (1^st ‘⟨0, 𝑌⟩) = 0)
151	16, 17, 18, 19	gpgedgvtx0 48092	. . . . 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 836	. . . 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 7929	. . . . . . . . . . . 12 ⊢ ((0 ∈ V ∧ 𝑌 ∈ 𝐼) → (2^nd ‘⟨0, 𝑌⟩) = 𝑌)
154	7, 153	mpan 690	. . . . . . . . . . 11 ⊢ (𝑌 ∈ 𝐼 → (2^nd ‘⟨0, 𝑌⟩) = 𝑌)
155		oveq1 7348	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → ((2^nd ‘⟨0, 𝑌⟩) − 1) = (𝑌 − 1))
156	155	oveq1d 7356	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁) = ((𝑌 − 1) mod 𝑁))
157	156	opeq2d 4827	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → ⟨0, (((2^nd ‘⟨0, 𝑌⟩) − 1) mod 𝑁)⟩ = ⟨0, ((𝑌 − 1) mod 𝑁)⟩)
158	157	preq2d 4688	. . . . . . . . . . . . . 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 4680	. . . . . . . . . . . . . 14 ⊢ {⟨0, 𝑌⟩, ⟨0, ((𝑌 − 1) mod 𝑁)⟩} = {⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩}
160	158, 159	eqtrdi 2782	. . . . . . . . . . . . 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 2816	. . . . . . . . . . . 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 7348	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → ((2^nd ‘⟨0, 𝑌⟩) + 1) = (𝑌 + 1))
163	162	oveq1d 7356	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁) = ((𝑌 + 1) mod 𝑁))
164	163	opeq2d 4827	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘⟨0, 𝑌⟩) = 𝑌 → ⟨0, (((2^nd ‘⟨0, 𝑌⟩) + 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩)
165	164	preq2d 4688	. . . . . . . . . . . . 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 2816	. . . . . . . . . . . 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 632	. . . . . . . . . . 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 17	. . . . . . . . . 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 249	. . . . . . . . 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 458	. . . . . . . 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 1131	. . . . . . 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 32	. . . . . 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 481	. . . . 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 481	. . . 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 15	. . 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 4821	. . . . . 6 ⊢ (𝑋 = 𝑌 → ⟨0, 𝑋⟩ = ⟨0, 𝑌⟩)
177	176	preq2d 4688	. . . . 5 ⊢ (𝑋 = 𝑌 → {⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} = {⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩})
178	177	eleq1d 2816	. . . 4 ⊢ (𝑋 = 𝑌 → ({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∈ 𝐸 ↔ {⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑌⟩} ∈ 𝐸))
179	176	preq1d 4687	. . . . 5 ⊢ (𝑋 = 𝑌 → {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} = {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩})
180	179	eleq1d 2816	. . . 4 ⊢ (𝑋 = 𝑌 → ({⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸 ↔ {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸))
181	178, 180	anbi12d 632	. . 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 247	. 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 212	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

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 Vcvv 3436 {cpr 4573 ⟨cop 4577 ‘cfv 6476 (class class class)co 7341 1^st c1st 7914 2^nd c2nd 7915 0cc0 11001 1c1 11002 + caddc 11004 − cmin 11339 / cdiv 11769 2c2 12175 3c3 12176 5c5 12178 ℤcz 12463 ℤ_≥cuz 12727 ..^cfzo 13549 ⌈cceil 13690 mod cmo 13768 Vtxcvtx 28969 Edgcedg 29020 gPetersenGr cgpg 48071

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-oadd 8384 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-sup 9321 df-inf 9322 df-dju 9789 df-card 9827 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-xnn0 12450 df-z 12464 df-dec 12584 df-uz 12728 df-rp 12886 df-ico 13246 df-fz 13403 df-fzo 13550 df-fl 13691 df-ceil 13692 df-mod 13769 df-hash 14233 df-dvds 16159 df-struct 17053 df-slot 17088 df-ndx 17100 df-base 17116 df-edgf 28962 df-vtx 28971 df-iedg 28972 df-edg 29021 df-umgr 29056 df-usgr 29124 df-gpg 48072

This theorem is referenced by: pgnbgreunbgrlem1 48144

W3C validator