Theorem gpgvtxedg0 48712

Description: The edges starting at an outside vertex 𝑋 in a generalized Petersen graph 𝐺. (Contributed by AV, 30-Aug-2025.)

Hypotheses

Ref	Expression
gpgedgvtx0.j	⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
gpgedgvtx0.g	⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
gpgedgvtx0.v	⊢ 𝑉 = (Vtx‘𝐺)
gpgedgvtx0.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
gpgvtxedg0	⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))

Proof of Theorem gpgvtxedg0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gpgusgra 48706	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
2		gpgedgvtx0.j	. . . . . . 7 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
3	2	eleq2i 2861	. . . . . 6 ⊢ (𝐾 ∈ 𝐽 ↔ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
4	3	anbi2i 634	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ↔ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))))
5		gpgedgvtx0.g	. . . . . 6 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
6	5	eleq1i 2860	. . . . 5 ⊢ (𝐺 ∈ USGraph ↔ (𝑁 gPetersenGr 𝐾) ∈ USGraph)
7	1, 4, 6	3imtr4i 295	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → 𝐺 ∈ USGraph)
8	7	3ad2ant1 1149	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ {𝑋, 𝑌} ∈ 𝐸) → 𝐺 ∈ USGraph)
9		simp3 1154	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ {𝑋, 𝑌} ∈ 𝐸) → {𝑋, 𝑌} ∈ 𝐸)
10		gpgedgvtx0.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
11		gpgedgvtx0.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
12	10, 11	usgrpredgv 29484	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
13	8, 9, 12	syl2anc 595	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
14		eqid 2769	. . . . . . . 8 ⊢ (0..^𝑁) = (0..^𝑁)
15	14, 2, 5, 10	gpgedgel 48699	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → ({𝑋, 𝑌} ∈ 𝐸 ↔ ∃𝑦 ∈ (0..^𝑁)({𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨0, ((𝑦 + 1) mod 𝑁)⟩} ∨ {𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨1, 𝑦⟩} ∨ {𝑋, 𝑌} = {⟨1, 𝑦⟩, ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩})))
16	15	3ad2ant1 1149	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ({𝑋, 𝑌} ∈ 𝐸 ↔ ∃𝑦 ∈ (0..^𝑁)({𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨0, ((𝑦 + 1) mod 𝑁)⟩} ∨ {𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨1, 𝑦⟩} ∨ {𝑋, 𝑌} = {⟨1, 𝑦⟩, ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩})))
17		simp3 1154	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
18	17	adantr 485	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
19		opex 5443	. . . . . . . . . . 11 ⊢ ⟨0, 𝑦⟩ ∈ V
20		opex 5443	. . . . . . . . . . 11 ⊢ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ V
21	19, 20	pm3.2i 475	. . . . . . . . . 10 ⊢ (⟨0, 𝑦⟩ ∈ V ∧ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ V)
22		preq12bg 4819	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (⟨0, 𝑦⟩ ∈ V ∧ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ V)) → ({𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨0, ((𝑦 + 1) mod 𝑁)⟩} ↔ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) ∨ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩))))
23	18, 21, 22	sylancl 597	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ({𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨0, ((𝑦 + 1) mod 𝑁)⟩} ↔ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) ∨ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩))))
24		simpr 489	. . . . . . . . . . . . 13 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) → 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩)
25		c0ex 11196	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ V
26		vex 3467	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑦 ∈ V
27	25, 26	op2ndd 7993	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (2nd ‘𝑋) = 𝑦)
28	27	eqcomd 2775	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨0, 𝑦⟩ → 𝑦 = (2nd ‘𝑋))
29	28	oveq1d 7423	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (𝑦 + 1) = ((2nd ‘𝑋) + 1))
30	29	oveq1d 7423	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨0, 𝑦⟩ → ((𝑦 + 1) mod 𝑁) = (((2nd ‘𝑋) + 1) mod 𝑁))
31	30	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) → ((𝑦 + 1) mod 𝑁) = (((2nd ‘𝑋) + 1) mod 𝑁))
32	31	opeq2d 4846	. . . . . . . . . . . . 13 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) → ⟨0, ((𝑦 + 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩)
33	24, 32	eqtrd 2804	. . . . . . . . . . . 12 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) → 𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩)
34	33	3mix1d 1353	. . . . . . . . . . 11 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))
35	34	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
36		elfzoelz 13683	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℤ)
37	36	zred 12696	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℝ)
38		1red 11205	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0..^𝑁) → 1 ∈ ℝ)
39	37, 38	readdcld 11234	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 + 1) ∈ ℝ)
40		elfzo0 13725	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0..^𝑁) ↔ (𝑦 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑦 < 𝑁))
41		nnrp 13024	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+)
42	41	3ad2ant2 1150	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑦 < 𝑁) → 𝑁 ∈ ℝ+)
43	40, 42	sylbi 220	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑁 ∈ ℝ+)
44		modsubmod 13961	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 + 1) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁) = (((𝑦 + 1) − 1) mod 𝑁))
45	39, 38, 43, 44	syl3anc 1396	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0..^𝑁) → ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁) = (((𝑦 + 1) − 1) mod 𝑁))
46	36	zcnd 12697	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℂ)
47		pncan1 11634	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℂ → ((𝑦 + 1) − 1) = 𝑦)
48	46, 47	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0..^𝑁) → ((𝑦 + 1) − 1) = 𝑦)
49	48	oveq1d 7423	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0..^𝑁) → (((𝑦 + 1) − 1) mod 𝑁) = (𝑦 mod 𝑁))
50		zmodidfzoimp 13930	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 mod 𝑁) = 𝑦)
51	45, 49, 50	3eqtrrd 2809	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 = ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁))
52	51	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 = ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁))
53	52	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → 𝑦 = ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁))
54	53	opeq2d 4846	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → ⟨0, 𝑦⟩ = ⟨0, ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁)⟩)
55		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → 𝑌 = ⟨0, 𝑦⟩)
56		ovex 7441	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 + 1) mod 𝑁) ∈ V
57	25, 56	op2ndd 7993	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (2nd ‘𝑋) = ((𝑦 + 1) mod 𝑁))
58	57	oveq1d 7423	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → ((2nd ‘𝑋) − 1) = (((𝑦 + 1) mod 𝑁) − 1))
59	58	oveq1d 7423	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (((2nd ‘𝑋) − 1) mod 𝑁) = ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁))
60	59	opeq2d 4846	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁)⟩)
61	60	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁)⟩)
62	55, 61	eqeq12d 2785	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → (𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ↔ ⟨0, 𝑦⟩ = ⟨0, ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁)⟩))
63	62	adantl 486	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → (𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ↔ ⟨0, 𝑦⟩ = ⟨0, ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁)⟩))
64	54, 63	mpbird 260	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)
65	64	3mix3d 1355	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))
66	65	ex 417	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
67	35, 66	jaod 872	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → (((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) ∨ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
68	23, 67	sylbid 243	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ({𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨0, ((𝑦 + 1) mod 𝑁)⟩} → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
69		opex 5443	. . . . . . . . . . 11 ⊢ ⟨1, 𝑦⟩ ∈ V
70	19, 69	pm3.2i 475	. . . . . . . . . 10 ⊢ (⟨0, 𝑦⟩ ∈ V ∧ ⟨1, 𝑦⟩ ∈ V)
71		preq12bg 4819	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (⟨0, 𝑦⟩ ∈ V ∧ ⟨1, 𝑦⟩ ∈ V)) → ({𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨1, 𝑦⟩} ↔ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) ∨ (𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩))))
72	18, 70, 71	sylancl 597	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ({𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨1, 𝑦⟩} ↔ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) ∨ (𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩))))
73		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → 𝑌 = ⟨1, 𝑦⟩)
74	28	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → 𝑦 = (2nd ‘𝑋))
75	74	opeq2d 4846	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → ⟨1, 𝑦⟩ = ⟨1, (2nd ‘𝑋)⟩)
76	73, 75	eqtrd 2804	. . . . . . . . . . . . 13 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → 𝑌 = ⟨1, (2nd ‘𝑋)⟩)
77	76	adantl 486	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩)) → 𝑌 = ⟨1, (2nd ‘𝑋)⟩)
78	77	3mix2d 1354	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩)) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))
79	78	ex 417	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
80		1ex 11199	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
81	80, 26	op1std 7992	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (1st ‘𝑋) = 1)
82	81	eqeq1d 2771	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨1, 𝑦⟩ → ((1st ‘𝑋) = 0 ↔ 1 = 0))
83		ax-1ne0 11165	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 0
84		eqneqall 2975	. . . . . . . . . . . . . . . . 17 ⊢ (1 = 0 → (1 ≠ 0 → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
85	84	com12 33	. . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 0 → (1 = 0 → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
86	83, 85	mp1i 14	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (1 = 0 → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
87	82, 86	sylbid 243	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨1, 𝑦⟩ → ((1st ‘𝑋) = 0 → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
88	87	com12 33	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑋) = 0 → (𝑋 = ⟨1, 𝑦⟩ → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
89	88	3ad2ant2 1150	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑋 = ⟨1, 𝑦⟩ → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
90	89	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → (𝑋 = ⟨1, 𝑦⟩ → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
91	90	impd 415	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
92	79, 91	jaod 872	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → (((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) ∨ (𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
93	72, 92	sylbid 243	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ({𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨1, 𝑦⟩} → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
94		opex 5443	. . . . . . . . . . 11 ⊢ ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∈ V
95	69, 94	pm3.2i 475	. . . . . . . . . 10 ⊢ (⟨1, 𝑦⟩ ∈ V ∧ ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∈ V)
96		preq12bg 4819	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (⟨1, 𝑦⟩ ∈ V ∧ ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∈ V)) → ({𝑋, 𝑌} = {⟨1, 𝑦⟩, ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩} ↔ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) ∨ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩))))
97	18, 95, 96	sylancl 597	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ({𝑋, 𝑌} = {⟨1, 𝑦⟩, ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩} ↔ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) ∨ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩))))
98		eqneqall 2975	. . . . . . . . . . . . . . . . 17 ⊢ (1 = 0 → (1 ≠ 0 → (𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
99	98	com12 33	. . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 0 → (1 = 0 → (𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
100	83, 99	mp1i 14	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (1 = 0 → (𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
101	82, 100	sylbid 243	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨1, 𝑦⟩ → ((1st ‘𝑋) = 0 → (𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
102	101	com12 33	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑋) = 0 → (𝑋 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
103	102	impd 415	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑋) = 0 → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
104		ovex 7441	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 + 𝐾) mod 𝑁) ∈ V
105	80, 104	op1std 7992	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (1st ‘𝑋) = 1)
106	105	eqeq1d 2771	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → ((1st ‘𝑋) = 0 ↔ 1 = 0))
107		eqneqall 2975	. . . . . . . . . . . . . . . . 17 ⊢ (1 = 0 → (1 ≠ 0 → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
108	107	com12 33	. . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 0 → (1 = 0 → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
109	83, 108	mp1i 14	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (1 = 0 → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
110	106, 109	sylbid 243	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → ((1st ‘𝑋) = 0 → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
111	110	com12 33	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑋) = 0 → (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))))
112	111	impd 415	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑋) = 0 → ((𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
113	103, 112	jaod 872	. . . . . . . . . . 11 ⊢ ((1st ‘𝑋) = 0 → (((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) ∨ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩)) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
114	113	3ad2ant2 1150	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) ∨ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩)) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
115	114	adantr 485	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → (((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) ∨ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩)) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
116	97, 115	sylbid 243	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ({𝑋, 𝑌} = {⟨1, 𝑦⟩, ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩} → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
117	68, 93, 116	3jaod 1454	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → (({𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨0, ((𝑦 + 1) mod 𝑁)⟩} ∨ {𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨1, 𝑦⟩} ∨ {𝑋, 𝑌} = {⟨1, 𝑦⟩, ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩}) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
118	117	rexlimdva 3172	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (∃𝑦 ∈ (0..^𝑁)({𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨0, ((𝑦 + 1) mod 𝑁)⟩} ∨ {𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨1, 𝑦⟩} ∨ {𝑋, 𝑌} = {⟨1, 𝑦⟩, ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩}) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
119	16, 118	sylbid 243	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ({𝑋, 𝑌} ∈ 𝐸 → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
120	119	3exp 1135	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → ((1st ‘𝑋) = 0 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ({𝑋, 𝑌} ∈ 𝐸 → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))))
121	120	com34 92	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → ((1st ‘𝑋) = 0 → ({𝑋, 𝑌} ∈ 𝐸 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))))
122	121	3imp 1126	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ {𝑋, 𝑌} ∈ 𝐸) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
123	13, 122	mpd 16	1 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∨ w3o 1100   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095  Vcvv 3463  {cpr 4593  ⟨cop 4597   class class class wbr 5110  ‘cfv 6534  (class class class)co 7408  1^st c1st 7980  2^nd c2nd 7981  ℂcc 11094  ℝcr 11095  0cc0 11096  1c1 11097   + caddc 11099   < clt 11239   − cmin 11437   / cdiv 11867  ℕcn 12229  2c2 12291  3c3 12292  ℕ₀cn0 12500  ℤ_≥cuz 12858  ℝ⁺crp 13012  ..^cfzo 13678  ⌈cceil 13820   mod cmo 13898  Vtxcvtx 29283  Edgcedg 29334  USGraphcusgr 29436   gPetersenGr cgpg 48689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173  ax-pre-sup 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-dju 9883  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-div 11868  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12501  df-xnn0 12574  df-z 12588  df-dec 12708  df-uz 12859  df-rp 13013  df-fz 13532  df-fzo 13679  df-fl 13821  df-ceil 13822  df-mod 13899  df-hash 14363  df-dvds 16307  df-struct 17203  df-slot 17238  df-ndx 17250  df-base 17266  df-edgf 29276  df-vtx 29285  df-iedg 29286  df-edg 29335  df-umgr 29370  df-usgr 29438  df-gpg 48690

This theorem is referenced by:  gpgedgiov  48714  gpgedg2ov  48715  gpgnbgrvtx0  48723  pgnbgreunbgrlem2lem1  48763  pgnbgreunbgrlem2lem2  48764  pgnbgreunbgrlem2lem3  48765  pgnbgreunbgrlem5lem3  48771