Theorem gpgvtxedg1 48095

Description: The edges starting at an inside vertex 𝑋 in a generalized Petersen graph 𝐺. (Contributed by AV, 2-Sep-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
gpgvtxedg1	⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))

Proof of Theorem gpgvtxedg1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gpgusgra 48088	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
2		gpgedgvtx0.j	. . . . . . 7 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
3	2	eleq2i 2823	. . . . . 6 ⊢ (𝐾 ∈ 𝐽 ↔ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
4	3	anbi2i 623	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ↔ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))))
5		gpgedgvtx0.g	. . . . . 6 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
6	5	eleq1i 2822	. . . . 5 ⊢ (𝐺 ∈ USGraph ↔ (𝑁 gPetersenGr 𝐾) ∈ USGraph)
7	1, 4, 6	3imtr4i 292	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → 𝐺 ∈ USGraph)
8	7	3ad2ant1 1133	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ {𝑋, 𝑌} ∈ 𝐸) → 𝐺 ∈ USGraph)
9		simp3 1138	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ {𝑋, 𝑌} ∈ 𝐸) → {𝑋, 𝑌} ∈ 𝐸)
10		gpgedgvtx0.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
11		gpgedgvtx0.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
12	10, 11	usgrpredgv 29170	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
13	8, 9, 12	syl2anc 584	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
14		eqid 2731	. . . . . . . 8 ⊢ (0..^𝑁) = (0..^𝑁)
15	14, 2, 5, 10	gpgedgel 48081	. . . . . . 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 1133	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (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 1138	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
18	17	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
19		opex 5399	. . . . . . . . . . 11 ⊢ ⟨0, 𝑦⟩ ∈ V
20		opex 5399	. . . . . . . . . . 11 ⊢ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ V
21	19, 20	pm3.2i 470	. . . . . . . . . 10 ⊢ (⟨0, 𝑦⟩ ∈ V ∧ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ V)
22		preq12bg 4800	. . . . . . . . . 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 586	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (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		c0ex 11101	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
25		vex 3440	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
26	24, 25	op1std 7926	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (1st ‘𝑋) = 0)
27	26	eqeq1d 2733	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨0, 𝑦⟩ → ((1st ‘𝑋) = 1 ↔ 0 = 1))
28		eqcom 2738	. . . . . . . . . . . . . . . 16 ⊢ (0 = 1 ↔ 1 = 0)
29	27, 28	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨0, 𝑦⟩ → ((1st ‘𝑋) = 1 ↔ 1 = 0))
30		ax-1ne0 11070	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 0
31		eqneqall 2939	. . . . . . . . . . . . . . . . 17 ⊢ (1 = 0 → (1 ≠ 0 → (𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
32	31	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 0 → (1 = 0 → (𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
33	30, 32	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (1 = 0 → (𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
34	29, 33	sylbid 240	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨0, 𝑦⟩ → ((1st ‘𝑋) = 1 → (𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
35	34	com12 32	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑋) = 1 → (𝑋 = ⟨0, 𝑦⟩ → (𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
36	35	impd 410	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑋) = 1 → ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
37		ovex 7374	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 + 1) mod 𝑁) ∈ V
38	24, 37	op1std 7926	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (1st ‘𝑋) = 0)
39	38	eqeq1d 2733	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → ((1st ‘𝑋) = 1 ↔ 0 = 1))
40	39, 28	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → ((1st ‘𝑋) = 1 ↔ 1 = 0))
41		eqneqall 2939	. . . . . . . . . . . . . . . . 17 ⊢ (1 = 0 → (1 ≠ 0 → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
42	41	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 0 → (1 = 0 → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
43	30, 42	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (1 = 0 → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
44	40, 43	sylbid 240	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → ((1st ‘𝑋) = 1 → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
45	44	com12 32	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑋) = 1 → (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
46	45	impd 410	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑋) = 1 → ((𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
47	36, 46	jaod 859	. . . . . . . . . . 11 ⊢ ((1st ‘𝑋) = 1 → (((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) ∨ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
48	47	3ad2ant2 1134	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) ∨ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
49	48	adantr 480	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (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 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
50	23, 49	sylbid 240	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ({𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨0, ((𝑦 + 1) mod 𝑁)⟩} → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
51		opex 5399	. . . . . . . . . . 11 ⊢ ⟨1, 𝑦⟩ ∈ V
52	19, 51	pm3.2i 470	. . . . . . . . . 10 ⊢ (⟨0, 𝑦⟩ ∈ V ∧ ⟨1, 𝑦⟩ ∈ V)
53		preq12bg 4800	. . . . . . . . . 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⟩))))
54	18, 52, 53	sylancl 586	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (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⟩))))
55		eqneqall 2939	. . . . . . . . . . . . . . . . 17 ⊢ (1 = 0 → (1 ≠ 0 → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
56	55	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 0 → (1 = 0 → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
57	30, 56	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (1 = 0 → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
58	29, 57	sylbid 240	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨0, 𝑦⟩ → ((1st ‘𝑋) = 1 → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
59	58	com12 32	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑋) = 1 → (𝑋 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
60	59	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑋 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
61	60	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → (𝑋 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
62	61	impd 410	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
63		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → 𝑌 = ⟨0, 𝑦⟩)
64		1ex 11103	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ V
65	64, 25	op2ndd 7927	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (2nd ‘𝑋) = 𝑦)
66	65	eqcomd 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨1, 𝑦⟩ → 𝑦 = (2nd ‘𝑋))
67	66	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → 𝑦 = (2nd ‘𝑋))
68	67	opeq2d 4827	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → ⟨0, 𝑦⟩ = ⟨0, (2nd ‘𝑋)⟩)
69	63, 68	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → 𝑌 = ⟨0, (2nd ‘𝑋)⟩)
70	69	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → 𝑌 = ⟨0, (2nd ‘𝑋)⟩)
71	70	3mix2d 1338	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))
72	71	ex 412	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
73	62, 72	jaod 859	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → (((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) ∨ (𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
74	54, 73	sylbid 240	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ({𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨1, 𝑦⟩} → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
75		opex 5399	. . . . . . . . . . 11 ⊢ ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∈ V
76	51, 75	pm3.2i 470	. . . . . . . . . 10 ⊢ (⟨1, 𝑦⟩ ∈ V ∧ ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∈ V)
77		preq12bg 4800	. . . . . . . . . 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⟩))))
78	18, 76, 77	sylancl 586	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (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⟩))))
79		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) → 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩)
80	66	oveq1d 7356	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (𝑦 + 𝐾) = ((2nd ‘𝑋) + 𝐾))
81	80	oveq1d 7356	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨1, 𝑦⟩ → ((𝑦 + 𝐾) mod 𝑁) = (((2nd ‘𝑋) + 𝐾) mod 𝑁))
82	81	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) → ((𝑦 + 𝐾) mod 𝑁) = (((2nd ‘𝑋) + 𝐾) mod 𝑁))
83	82	opeq2d 4827	. . . . . . . . . . . . 13 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) → ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩)
84	79, 83	eqtrd 2766	. . . . . . . . . . . 12 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) → 𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩)
85	84	3mix1d 1337	. . . . . . . . . . 11 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))
86	85	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
87		elfzoelz 13554	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℤ)
88	87	zred 12572	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℝ)
89	88	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 ∈ ℝ)
90		elfzoelz 13554	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐾 ∈ (1..^(⌈‘(𝑁 / 2))) → 𝐾 ∈ ℤ)
91	90	zred 12572	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐾 ∈ (1..^(⌈‘(𝑁 / 2))) → 𝐾 ∈ ℝ)
92	3, 91	sylbi 217	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 ∈ 𝐽 → 𝐾 ∈ ℝ)
93	92	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → 𝐾 ∈ ℝ)
94	89, 93	readdcld 11136	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → (𝑦 + 𝐾) ∈ ℝ)
95		elfzo0 13595	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^𝑁) ↔ (𝑦 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑦 < 𝑁))
96		nnrp 12897	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+)
97	96	3ad2ant2 1134	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑦 < 𝑁) → 𝑁 ∈ ℝ+)
98	95, 97	sylbi 217	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑁 ∈ ℝ+)
99	98	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → 𝑁 ∈ ℝ+)
100		modsubmod 13831	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 + 𝐾) ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁) = (((𝑦 + 𝐾) − 𝐾) mod 𝑁))
101	94, 93, 99, 100	syl3anc 1373	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁) = (((𝑦 + 𝐾) − 𝐾) mod 𝑁))
102	87	zcnd 12573	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℂ)
103	102	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 ∈ ℂ)
104	93	recnd 11135	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → 𝐾 ∈ ℂ)
105	103, 104	pncand 11468	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑦 + 𝐾) − 𝐾) = 𝑦)
106	105	oveq1d 7356	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → (((𝑦 + 𝐾) − 𝐾) mod 𝑁) = (𝑦 mod 𝑁))
107		zmodidfzoimp 13800	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 mod 𝑁) = 𝑦)
108	107	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → (𝑦 mod 𝑁) = 𝑦)
109	101, 106, 108	3eqtrrd 2771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 = ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁))
110	109	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ 𝐽 → (𝑦 ∈ (0..^𝑁) → 𝑦 = ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁)))
111	110	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑦 ∈ (0..^𝑁) → 𝑦 = ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁)))
112	111	3ad2ant1 1133	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑦 ∈ (0..^𝑁) → 𝑦 = ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁)))
113	112	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 = ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁))
114	113	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩)) → 𝑦 = ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁))
115	114	opeq2d 4827	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩)) → ⟨1, 𝑦⟩ = ⟨1, ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁)⟩)
116		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → 𝑌 = ⟨1, 𝑦⟩)
117		ovex 7374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 + 𝐾) mod 𝑁) ∈ V
118	64, 117	op2ndd 7927	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (2nd ‘𝑋) = ((𝑦 + 𝐾) mod 𝑁))
119	118	oveq1d 7356	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → ((2nd ‘𝑋) − 𝐾) = (((𝑦 + 𝐾) mod 𝑁) − 𝐾))
120	119	oveq1d 7356	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (((2nd ‘𝑋) − 𝐾) mod 𝑁) = ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁))
121	120	opeq2d 4827	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ = ⟨1, ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁)⟩)
122	121	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ = ⟨1, ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁)⟩)
123	116, 122	eqeq12d 2747	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → (𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ↔ ⟨1, 𝑦⟩ = ⟨1, ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁)⟩))
124	123	adantl 481	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩)) → (𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ↔ ⟨1, 𝑦⟩ = ⟨1, ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁)⟩))
125	115, 124	mpbird 257	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩)) → 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)
126	125	3mix3d 1339	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩)) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))
127	126	ex 412	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
128	86, 127	jaod 859	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (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 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
129	78, 128	sylbid 240	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ({𝑋, 𝑌} = {⟨1, 𝑦⟩, ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩} → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
130	50, 74, 129	3jaod 1431	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (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 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
131	130	rexlimdva 3133	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (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 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
132	16, 131	sylbid 240	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ({𝑋, 𝑌} ∈ 𝐸 → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
133	132	3exp 1119	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → ((1st ‘𝑋) = 1 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ({𝑋, 𝑌} ∈ 𝐸 → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))))
134	133	com34 91	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → ((1st ‘𝑋) = 1 → ({𝑋, 𝑌} ∈ 𝐸 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))))
135	134	3imp 1110	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ {𝑋, 𝑌} ∈ 𝐸) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
136	13, 135	mpd 15	1 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056  Vcvv 3436  {cpr 4573  ⟨cop 4577   class class class wbr 5086  ‘cfv 6476  (class class class)co 7341  1^st c1st 7914  2^nd c2nd 7915  ℂcc 10999  ℝcr 11000  0cc0 11001  1c1 11002   + caddc 11004   < clt 11141   − cmin 11339   / cdiv 11769  ℕcn 12120  2c2 12175  3c3 12176  ℕ₀cn0 12376  ℤ_≥cuz 12727  ℝ⁺crp 12885  ..^cfzo 13549  ⌈cceil 13690   mod cmo 13768  Vtxcvtx 28969  Edgcedg 29020  USGraphcusgr 29122   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-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:  gpgedg2iv  48098  gpgnbgrvtx1  48106  pgnioedg1  48139  pgnioedg2  48140  pgnioedg3  48141  pgnioedg4  48142  pgnioedg5  48143  pgnbgreunbgrlem5lem1  48151  pgnbgreunbgrlem5lem2  48152