gpgvtxedg1 - Mathbox for Alexander van der Vekens

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Theorem gpgvtxedg1 48310

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) ∧ (1^st ‘𝑋) = 1 ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))

Proof of Theorem gpgvtxedg1 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gpgusgra 48303	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
2		gpgedgvtx0.j	. . . . . . 7 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
3	2	eleq2i 2828	. . . . . 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 2827	. . . . 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) ∧ (1^st ‘𝑋) = 1 ∧ {𝑋, 𝑌} ∈ 𝐸) → 𝐺 ∈ USGraph)
9		simp3 1138	. . 3 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘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 29270	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
13	8, 9, 12	syl2anc 584	. 2 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
14		eqid 2736	. . . . . . . 8 ⊢ (0..^𝑁) = (0..^𝑁)
15	14, 2, 5, 10	gpgedgel 48296	. . . . . . 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) ∧ (1^st ‘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) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
18	17	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))
19		opex 5412	. . . . . . . . . . 11 ⊢ ⟨0, 𝑦⟩ ∈ V
20		opex 5412	. . . . . . . . . . 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 4809	. . . . . . . . . 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) ∧ (1^st ‘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 11126	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
25		vex 3444	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
26	24, 25	op1std 7943	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (1^st ‘𝑋) = 0)
27	26	eqeq1d 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨0, 𝑦⟩ → ((1^st ‘𝑋) = 1 ↔ 0 = 1))
28		eqcom 2743	. . . . . . . . . . . . . . . 16 ⊢ (0 = 1 ↔ 1 = 0)
29	27, 28	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨0, 𝑦⟩ → ((1^st ‘𝑋) = 1 ↔ 1 = 0))
30		ax-1ne0 11095	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 0
31		eqneqall 2943	. . . . . . . . . . . . . . . . 17 ⊢ (1 = 0 → (1 ≠ 0 → (𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
32	31	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 0 → (1 = 0 → (𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
33	30, 32	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (1 = 0 → (𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
34	29, 33	sylbid 240	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨0, 𝑦⟩ → ((1^st ‘𝑋) = 1 → (𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
35	34	com12 32	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑋) = 1 → (𝑋 = ⟨0, 𝑦⟩ → (𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
36	35	impd 410	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑋) = 1 → ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
37		ovex 7391	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 + 1) mod 𝑁) ∈ V
38	24, 37	op1std 7943	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (1^st ‘𝑋) = 0)
39	38	eqeq1d 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → ((1^st ‘𝑋) = 1 ↔ 0 = 1))
40	39, 28	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → ((1^st ‘𝑋) = 1 ↔ 1 = 0))
41		eqneqall 2943	. . . . . . . . . . . . . . . . 17 ⊢ (1 = 0 → (1 ≠ 0 → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
42	41	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 0 → (1 = 0 → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
43	30, 42	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (1 = 0 → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
44	40, 43	sylbid 240	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → ((1^st ‘𝑋) = 1 → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
45	44	com12 32	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑋) = 1 → (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (𝑌 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
46	45	impd 410	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑋) = 1 → ((𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
47	36, 46	jaod 859	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑋) = 1 → (((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩) ∨ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
48	47	3ad2ant2 1134	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘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, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
49	48	adantr 480	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘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, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
50	23, 49	sylbid 240	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ({𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨0, ((𝑦 + 1) mod 𝑁)⟩} → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
51		opex 5412	. . . . . . . . . . 11 ⊢ ⟨1, 𝑦⟩ ∈ V
52	19, 51	pm3.2i 470	. . . . . . . . . 10 ⊢ (⟨0, 𝑦⟩ ∈ V ∧ ⟨1, 𝑦⟩ ∈ V)
53		preq12bg 4809	. . . . . . . . . 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) ∧ (1^st ‘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 2943	. . . . . . . . . . . . . . . . 17 ⊢ (1 = 0 → (1 ≠ 0 → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
56	55	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 0 → (1 = 0 → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
57	30, 56	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (1 = 0 → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
58	29, 57	sylbid 240	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨0, 𝑦⟩ → ((1^st ‘𝑋) = 1 → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
59	58	com12 32	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑋) = 1 → (𝑋 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
60	59	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑋 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
61	60	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → (𝑋 = ⟨0, 𝑦⟩ → (𝑌 = ⟨1, 𝑦⟩ → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))))
62	61	impd 410	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
63		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → 𝑌 = ⟨0, 𝑦⟩)
64		1ex 11128	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ V
65	64, 25	op2ndd 7944	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
66	65	eqcomd 2742	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨1, 𝑦⟩ → 𝑦 = (2^nd ‘𝑋))
67	66	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → 𝑦 = (2^nd ‘𝑋))
68	67	opeq2d 4836	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → ⟨0, 𝑦⟩ = ⟨0, (2^nd ‘𝑋)⟩)
69	63, 68	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → 𝑌 = ⟨0, (2^nd ‘𝑋)⟩)
70	69	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → 𝑌 = ⟨0, (2^nd ‘𝑋)⟩)
71	70	3mix2d 1338	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))
72	71	ex 412	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
73	62, 72	jaod 859	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → (((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) ∨ (𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩)) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
74	54, 73	sylbid 240	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ({𝑋, 𝑌} = {⟨0, 𝑦⟩, ⟨1, 𝑦⟩} → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
75		opex 5412	. . . . . . . . . . 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 4809	. . . . . . . . . 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) ∧ (1^st ‘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 7373	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (𝑦 + 𝐾) = ((2^nd ‘𝑋) + 𝐾))
81	80	oveq1d 7373	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨1, 𝑦⟩ → ((𝑦 + 𝐾) mod 𝑁) = (((2^nd ‘𝑋) + 𝐾) mod 𝑁))
82	81	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) → ((𝑦 + 𝐾) mod 𝑁) = (((2^nd ‘𝑋) + 𝐾) mod 𝑁))
83	82	opeq2d 4836	. . . . . . . . . . . . 13 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) → ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩)
84	79, 83	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) → 𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩)
85	84	3mix1d 1337	. . . . . . . . . . 11 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))
86	85	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
87		elfzoelz 13575	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℤ)
88	87	zred 12596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℝ)
89	88	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 ∈ ℝ)
90		elfzoelz 13575	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐾 ∈ (1..^(⌈‘(𝑁 / 2))) → 𝐾 ∈ ℤ)
91	90	zred 12596	. . . . . . . . . . . . . . . . . . . . . . . 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 11161	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → (𝑦 + 𝐾) ∈ ℝ)
95		elfzo0 13616	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^𝑁) ↔ (𝑦 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 𝑦 < 𝑁))
96		nnrp 12917	. . . . . . . . . . . . . . . . . . . . . . . 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 13852	. . . . . . . . . . . . . . . . . . . . 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 12597	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℂ)
103	102	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 ∈ ℂ)
104	93	recnd 11160	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → 𝐾 ∈ ℂ)
105	103, 104	pncand 11493	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑦 + 𝐾) − 𝐾) = 𝑦)
106	105	oveq1d 7373	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → (((𝑦 + 𝐾) − 𝐾) mod 𝑁) = (𝑦 mod 𝑁))
107		zmodidfzoimp 13821	. . . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . . 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) ∧ (1^st ‘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) ∧ (1^st ‘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) ∧ (1^st ‘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 4836	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘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 7391	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 + 𝐾) mod 𝑁) ∈ V
118	64, 117	op2ndd 7944	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (2^nd ‘𝑋) = ((𝑦 + 𝐾) mod 𝑁))
119	118	oveq1d 7373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → ((2^nd ‘𝑋) − 𝐾) = (((𝑦 + 𝐾) mod 𝑁) − 𝐾))
120	119	oveq1d 7373	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (((2^nd ‘𝑋) − 𝐾) mod 𝑁) = ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁))
121	120	opeq2d 4836	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → ⟨1, (((2^nd ‘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, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩ = ⟨1, ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁)⟩)
123	116, 122	eqeq12d 2752	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → (𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ↔ ⟨1, 𝑦⟩ = ⟨1, ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁)⟩))
124	123	adantl 481	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩)) → (𝑌 = ⟨1, (((2^nd ‘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) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩)) → 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)
126	125	3mix3d 1339	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) ∧ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩)) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))
127	126	ex 412	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ ∧ 𝑌 = ⟨1, 𝑦⟩) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
128	86, 127	jaod 859	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘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, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
129	78, 128	sylbid 240	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑦 ∈ (0..^𝑁)) → ({𝑋, 𝑌} = {⟨1, 𝑦⟩, ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩} → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
130	50, 74, 129	3jaod 1431	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘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, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
131	130	rexlimdva 3137	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘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, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
132	16, 131	sylbid 240	. . . . 5 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ({𝑋, 𝑌} ∈ 𝐸 → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
133	132	3exp 1119	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) → ((1^st ‘𝑋) = 1 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ({𝑋, 𝑌} ∈ 𝐸 → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))))
134	133	com34 91	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) → ((1^st ‘𝑋) = 1 → ({𝑋, 𝑌} ∈ 𝐸 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))))
135	134	3imp 1110	. 2 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ {𝑋, 𝑌} ∈ 𝐸) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
136	13, 135	mpd 15	1 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1^st ‘𝑋) = 1 ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑌 = ⟨1, (((2^nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2^nd ‘𝑋) − 𝐾) mod 𝑁)⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∃wrex 3060 Vcvv 3440 {cpr 4582 ⟨cop 4586 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 1^st c1st 7931 2^nd c2nd 7932 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 < clt 11166 − cmin 11364 / cdiv 11794 ℕcn 12145 2c2 12200 3c3 12201 ℕ₀cn0 12401 ℤ_≥cuz 12751 ℝ⁺crp 12905 ..^cfzo 13570 ⌈cceil 13711 mod cmo 13789 Vtxcvtx 29069 Edgcedg 29120 USGraphcusgr 29222 gPetersenGr cgpg 48286

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-oadd 8401 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-rp 12906 df-fz 13424 df-fzo 13571 df-fl 13712 df-ceil 13713 df-mod 13790 df-hash 14254 df-dvds 16180 df-struct 17074 df-slot 17109 df-ndx 17121 df-base 17137 df-edgf 29062 df-vtx 29071 df-iedg 29072 df-edg 29121 df-umgr 29156 df-usgr 29224 df-gpg 48287

This theorem is referenced by: gpgedg2iv 48313 gpgnbgrvtx1 48321 pgnioedg1 48354 pgnioedg2 48355 pgnioedg3 48356 pgnioedg4 48357 pgnioedg5 48358 pgnbgreunbgrlem5lem1 48366 pgnbgreunbgrlem5lem2 48367

W3C validator