Theorem gpgvtxedg0 49428

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 49422	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
2		gpgedgvtx0.j	. . . . . . 7 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
3	2	eleq2i 2856	. . . . . 6 ⊢ (𝐾 ∈ 𝐽 ↔ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
4	3	anbi2i 635	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ↔ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))))
5		gpgedgvtx0.g	. . . . . 6 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
6	5	eleq1i 2855	. . . . 5 ⊢ (𝐺 ∈ USGraph ↔ (𝑁 gPetersenGr 𝐾) ∈ USGraph)
7	1, 4, 6	3imtr4i 295	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → 𝐺 ∈ USGraph)
8	7	3ad2ant1 1151	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ {𝑋, 𝑌} ∈ 𝐸) → 𝐺 ∈ USGraph)
9		simp3 1156	. . 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 29824	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
13	8, 9, 12	syl2anc 596	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
14		eqid 2763	. . . . . . . 8 ⊢ (0..^𝑁) = (0..^𝑁)
15	14, 2, 5, 10	gpgedgel 49415	. . . . . . 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 1151	. . . . . 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 1156	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
18	17	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
19		opex 5437	. . . . . . . . . . 11 ⊢ ⟨0, 𝑦⟩ ∈ V
20		opex 5437	. . . . . . . . . . 11 ⊢ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ V
21	19, 20	pm3.2i 476	. . . . . . . . . 10 ⊢ (⟨0, 𝑦⟩ ∈ V ∧ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ V)
22		preq12bg 4811	. . . . . . . . . 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 598	. . . . . . . . 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 490	. . . . . . . . . . . . 13 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) → 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩)
25		c0ex 11337	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ V
26		vex 3458	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑦 ∈ V
27	25, 26	op2ndd 8044	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (2nd ‘𝑋) = 𝑦)
28	27	eqcomd 2769	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨0, 𝑦⟩ → 𝑦 = (2nd ‘𝑋))
29	28	oveq1d 7464	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (𝑦 + 1) = ((2nd ‘𝑋) + 1))
30	29	oveq1d 7464	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨0, 𝑦⟩ → ((𝑦 + 1) mod 𝑁) = (((2nd ‘𝑋) + 1) mod 𝑁))
31	30	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) → ((𝑦 + 1) mod 𝑁) = (((2nd ‘𝑋) + 1) mod 𝑁))
32	31	opeq2d 4850	. . . . . . . . . . . . 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 2798	. . . . . . . . . . . 12 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) → 𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩)
34	33	3mix1d 1355	. . . . . . . . . . 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 13831	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℤ)
37	36	zred 12841	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℝ)
38		1red 11346	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0..^𝑁) → 1 ∈ ℝ)
39	37, 38	readdcld 11375	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 + 1) ∈ ℝ)
40		elfzo0 13873	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0..^𝑁) ↔ (𝑦 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑦 < 𝑁))
41		nnrp 13170	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+)
42	41	3ad2ant2 1152	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑦 < 𝑁) → 𝑁 ∈ ℝ+)
43	40, 42	sylbi 220	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑁 ∈ ℝ+)
44		modsubmod 14110	. . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0..^𝑁) → ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁) = (((𝑦 + 1) − 1) mod 𝑁))
46	36	zcnd 12842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℂ)
47		pncan1 11777	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℂ → ((𝑦 + 1) − 1) = 𝑦)
48	46, 47	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0..^𝑁) → ((𝑦 + 1) − 1) = 𝑦)
49	48	oveq1d 7464	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0..^𝑁) → (((𝑦 + 1) − 1) mod 𝑁) = (𝑦 mod 𝑁))
50		zmodidfzoimp 14079	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 mod 𝑁) = 𝑦)
51	45, 49, 50	3eqtrrd 2803	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 = ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁))
52	51	adantl 487	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 4850	. . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → 𝑌 = ⟨0, 𝑦⟩)
56		ovex 7482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 + 1) mod 𝑁) ∈ V
57	25, 56	op2ndd 8044	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (2nd ‘𝑋) = ((𝑦 + 1) mod 𝑁))
58	57	oveq1d 7464	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → ((2nd ‘𝑋) − 1) = (((𝑦 + 1) mod 𝑁) − 1))
59	58	oveq1d 7464	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (((2nd ‘𝑋) − 1) mod 𝑁) = ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁))
60	59	opeq2d 4850	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, ((((𝑦 + 1) mod 𝑁) − 1) mod 𝑁)⟩)
61	60	adantr 486	. . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . 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 487	. . . . . . . . . . . . 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 1357	. . . . . . . . . . 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 418	. . . . . . . . . 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 873	. . . . . . . . 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 5437	. . . . . . . . . . 11 ⊢ ⟨1, 𝑦⟩ ∈ V
70	19, 69	pm3.2i 476	. . . . . . . . . 10 ⊢ (⟨0, 𝑦⟩ ∈ V ∧ ⟨1, 𝑦⟩ ∈ V)
71		preq12bg 4811	. . . . . . . . . 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 598	. . . . . . . . 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 490	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → 𝑌 = ⟨1, 𝑦⟩)
74	28	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → 𝑦 = (2nd ‘𝑋))
75	74	opeq2d 4850	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → ⟨1, 𝑦⟩ = ⟨1, (2nd ‘𝑋)⟩)
76	73, 75	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → 𝑌 = ⟨1, (2nd ‘𝑋)⟩)
77	76	adantl 487	. . . . . . . . . . . 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 1356	. . . . . . . . . . 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 418	. . . . . . . . . 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 11340	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
81	80, 26	op1std 8043	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (1st ‘𝑋) = 1)
82	81	eqeq1d 2765	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨1, 𝑦⟩ → ((1st ‘𝑋) = 0 ↔ 1 = 0))
83		ax-1ne0 11306	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 0
84		eqneqall 2970	. . . . . . . . . . . . . . . . 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 1152	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 416	. . . . . . . . . 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 873	. . . . . . . . 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 5437	. . . . . . . . . . 11 ⊢ ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∈ V
95	69, 94	pm3.2i 476	. . . . . . . . . 10 ⊢ (⟨1, 𝑦⟩ ∈ V ∧ ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∈ V)
96		preq12bg 4811	. . . . . . . . . 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 598	. . . . . . . . 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 2970	. . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . 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 7482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 + 𝐾) mod 𝑁) ∈ V
105	80, 104	op1std 8043	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (1st ‘𝑋) = 1)
106	105	eqeq1d 2765	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → ((1st ‘𝑋) = 0 ↔ 1 = 0))
107		eqneqall 2970	. . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . 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 873	. . . . . . . . . . 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 1152	. . . . . . . . . 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 486	. . . . . . . . 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 1457	. . . . . . 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 3167	. . . . . 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 1137	. . . 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 1128	. 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)⟩))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090  Vcvv 3454  {cpr 4590  ⟨cop 4838   class class class wbr 5106  ‘cfv 6564  (class class class)co 7449  1^st c1st 8031  2^nd c2nd 8032  ℂcc 11235  ℝcr 11236  0cc0 11237  1c1 11238   + caddc 11240   < clt 11380   − cmin 11578   / cdiv 12011  ℕcn 12373  2c2 12435  3c3 12436  ℕ₀cn0 12644  ℤ_≥cuz 13003  ℝ⁺crp 13158  ..^cfzo 13826  ⌈cceil 13969   mod cmo 14047  Vtxcvtx 29620  Edgcedg 29671  USGraphcusgr 29776   gPetersenGr cgpg 49405

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781  ax-cnex 11293  ax-resscn 11294  ax-1cn 11295  ax-icn 11296  ax-addcl 11297  ax-addrcl 11298  ax-mulcl 11299  ax-mulrcl 11300  ax-mulcom 11301  ax-addass 11302  ax-mulass 11303  ax-distr 11304  ax-i2m1 11305  ax-1ne0 11306  ax-1rid 11307  ax-rnegex 11308  ax-rrecex 11309  ax-cnre 11310  ax-pre-lttri 11311  ax-pre-lttrn 11312  ax-pre-ltadd 11313  ax-pre-mulgt0 11314  ax-pre-sup 11315

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-om 7910  df-1st 8033  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-1o 8512  df-oadd 8516  df-er 8753  df-en 9010  df-dom 9011  df-sdom 9012  df-fin 9013  df-sup 9469  df-inf 9470  df-dju 10017  df-card 10055  df-pnf 11382  df-mnf 11383  df-xr 11384  df-ltxr 11385  df-le 11386  df-sub 11580  df-neg 11581  df-div 12012  df-nn 12374  df-2 12443  df-3 12444  df-4 12445  df-5 12446  df-6 12447  df-7 12448  df-8 12449  df-9 12450  df-n0 12645  df-xnn0 12718  df-z 12732  df-dec 12853  df-uz 13004  df-rp 13159  df-fz 13678  df-fzo 13827  df-fl 13970  df-ceil 13971  df-mod 14048  df-hash 14514  df-dvds 16464  df-struct 17366  df-slot 17401  df-ndx 17413  df-base 17429  df-edgf 29613  df-vtx 29622  df-iedg 29623  df-edg 29672  df-umgr 29707  df-usgr 29778  df-gpg 49406

This theorem is used by:  gpgedgiov  49430  gpgedg2ov  49431  gpgnbgrvtx0  49439  pgnbgreunbgrlem2lem1  49479  pgnbgreunbgrlem2lem2  49480  pgnbgreunbgrlem2lem3  49481  pgnbgreunbgrlem5lem3  49487