gpgvtxedg1 - Mathbox for Alexander van der Vekens

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Theorem gpgvtxedg1 48045

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 48038	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
2		gpgedgvtx0.j	. . . . . . 7 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
3	2	eleq2i 2821	. . . . . 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 2820	. . . . 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 29130	. . 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 2730	. . . . . . . 8 ⊢ (0..^𝑁) = (0..^𝑁)
15	14, 2, 5, 10	gpgedgel 48031	. . . . . . 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 5426	. . . . . . . . . . 11 ⊢ ⟨0, 𝑦⟩ ∈ V
20		opex 5426	. . . . . . . . . . 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 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 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 11174	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
25		vex 3454	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
26	24, 25	op1std 7980	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (1^st ‘𝑋) = 0)
27	26	eqeq1d 2732	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨0, 𝑦⟩ → ((1^st ‘𝑋) = 1 ↔ 0 = 1))
28		eqcom 2737	. . . . . . . . . . . . . . . 16 ⊢ (0 = 1 ↔ 1 = 0)
29	27, 28	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨0, 𝑦⟩ → ((1^st ‘𝑋) = 1 ↔ 1 = 0))
30		ax-1ne0 11143	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 0
31		eqneqall 2937	. . . . . . . . . . . . . . . . 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 7422	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 + 1) mod 𝑁) ∈ V
38	24, 37	op1std 7980	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨0, ((𝑦 + 1) mod 𝑁)⟩ → (1^st ‘𝑋) = 0)
39	38	eqeq1d 2732	. . . . . . . . . . . . . . . 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 2937	. . . . . . . . . . . . . . . . 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 5426	. . . . . . . . . . 11 ⊢ ⟨1, 𝑦⟩ ∈ V
52	19, 51	pm3.2i 470	. . . . . . . . . 10 ⊢ (⟨0, 𝑦⟩ ∈ V ∧ ⟨1, 𝑦⟩ ∈ V)
53		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⟩))))
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 2937	. . . . . . . . . . . . . . . . 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 11176	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ V
65	64, 25	op2ndd 7981	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
66	65	eqcomd 2736	. . . . . . . . . . . . . . . 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 4846	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑌 = ⟨0, 𝑦⟩) → ⟨0, 𝑦⟩ = ⟨0, (2^nd ‘𝑋)⟩)
69	63, 68	eqtrd 2765	. . . . . . . . . . . . 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 5426	. . . . . . . . . . 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 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⟩))))
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 7404	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (𝑦 + 𝐾) = ((2^nd ‘𝑋) + 𝐾))
81	80	oveq1d 7404	. . . . . . . . . . . . . . 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 4846	. . . . . . . . . . . . 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 2765	. . . . . . . . . . . 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 13626	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℤ)
88	87	zred 12644	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℝ)
89	88	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 ∈ ℝ)
90		elfzoelz 13626	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐾 ∈ (1..^(⌈‘(𝑁 / 2))) → 𝐾 ∈ ℤ)
91	90	zred 12644	. . . . . . . . . . . . . . . . . . . . . . . 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 11209	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → (𝑦 + 𝐾) ∈ ℝ)
95		elfzo0 13667	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^𝑁) ↔ (𝑦 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 𝑦 < 𝑁))
96		nnrp 12969	. . . . . . . . . . . . . . . . . . . . . . . 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 13900	. . . . . . . . . . . . . . . . . . . . 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 12645	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℂ)
103	102	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 ∈ ℂ)
104	93	recnd 11208	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → 𝐾 ∈ ℂ)
105	103, 104	pncand 11540	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑦 + 𝐾) − 𝐾) = 𝑦)
106	105	oveq1d 7404	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ 𝐽 ∧ 𝑦 ∈ (0..^𝑁)) → (((𝑦 + 𝐾) − 𝐾) mod 𝑁) = (𝑦 mod 𝑁))
107		zmodidfzoimp 13869	. . . . . . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . . . . 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 4846	. . . . . . . . . . . . 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 7422	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 + 𝐾) mod 𝑁) ∈ V
118	64, 117	op2ndd 7981	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (2^nd ‘𝑋) = ((𝑦 + 𝐾) mod 𝑁))
119	118	oveq1d 7404	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → ((2^nd ‘𝑋) − 𝐾) = (((𝑦 + 𝐾) mod 𝑁) − 𝐾))
120	119	oveq1d 7404	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨1, ((𝑦 + 𝐾) mod 𝑁)⟩ → (((2^nd ‘𝑋) − 𝐾) mod 𝑁) = ((((𝑦 + 𝐾) mod 𝑁) − 𝐾) mod 𝑁))
121	120	opeq2d 4846	. . . . . . . . . . . . . . . 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 2746	. . . . . . . . . . . . . 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 3135	. . . . . 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 1540 ∈ wcel 2109 ≠ wne 2926 ∃wrex 3054 Vcvv 3450 {cpr 4593 ⟨cop 4597 class class class wbr 5109 ‘cfv 6513 (class class class)co 7389 1^st c1st 7968 2^nd c2nd 7969 ℂcc 11072 ℝcr 11073 0cc0 11074 1c1 11075 + caddc 11077 < clt 11214 − cmin 11411 / cdiv 11841 ℕcn 12187 2c2 12242 3c3 12243 ℕ₀cn0 12448 ℤ_≥cuz 12799 ℝ⁺crp 12957 ..^cfzo 13621 ⌈cceil 13759 mod cmo 13837 Vtxcvtx 28929 Edgcedg 28980 USGraphcusgr 29082 gPetersenGr cgpg 48021

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-oadd 8440 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-sup 9399 df-inf 9400 df-dju 9860 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-xnn0 12522 df-z 12536 df-dec 12656 df-uz 12800 df-rp 12958 df-fz 13475 df-fzo 13622 df-fl 13760 df-ceil 13761 df-mod 13838 df-hash 14302 df-dvds 16229 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17186 df-edgf 28922 df-vtx 28931 df-iedg 28932 df-edg 28981 df-umgr 29016 df-usgr 29084 df-gpg 48022

This theorem is referenced by: gpgedg2iv 48048 gpgnbgrvtx1 48056 pgnioedg1 48088 pgnioedg2 48089 pgnioedg3 48090 pgnioedg4 48091 pgnioedg5 48092 pgnbgreunbgrlem5lem1 48100 pgnbgreunbgrlem5lem2 48101

W3C validator