Theorem gpgov 48006

Description: The generalized Petersen graph GPG(N,K). (Contributed by AV, 26-Aug-2025.)

Hypotheses

Ref	Expression
gpgov.j	⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
gpgov.i	⊢ 𝐼 = (0..^𝑁)

Assertion

Ref	Expression
gpgov	⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (𝑁 gPetersenGr 𝐾) = {⟨(Base‘ndx), ({0, 1} × 𝐼)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})})⟩})

Distinct variable groups:   𝑒,𝐼,𝑥   𝑒,𝐾,𝑥   𝑒,𝑁,𝑥

Allowed substitution hints:   𝐽(𝑥,𝑒)

Proof of Theorem gpgov

Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prex 5387	. 2 ⊢ {⟨(Base‘ndx), ({0, 1} × 𝐼)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})})⟩} ∈ V
2		oveq2 7377	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁))
3		gpgov.i	. . . . . . . 8 ⊢ 𝐼 = (0..^𝑁)
4	2, 3	eqtr4di 2782	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = 𝐼)
5	4	xpeq2d 5661	. . . . . 6 ⊢ (𝑛 = 𝑁 → ({0, 1} × (0..^𝑛)) = ({0, 1} × 𝐼))
6	5	opeq2d 4840	. . . . 5 ⊢ (𝑛 = 𝑁 → ⟨(Base‘ndx), ({0, 1} × (0..^𝑛))⟩ = ⟨(Base‘ndx), ({0, 1} × 𝐼)⟩)
7	6	adantr 480	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ⟨(Base‘ndx), ({0, 1} × (0..^𝑛))⟩ = ⟨(Base‘ndx), ({0, 1} × 𝐼)⟩)
8	5	pweqd 4576	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → 𝒫 ({0, 1} × (0..^𝑛)) = 𝒫 ({0, 1} × 𝐼))
9	8	adantr 480	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → 𝒫 ({0, 1} × (0..^𝑛)) = 𝒫 ({0, 1} × 𝐼))
10	4	rexeqdv 3297	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (∃𝑥 ∈ (0..^𝑛)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩}) ↔ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩})))
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (∃𝑥 ∈ (0..^𝑛)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩}) ↔ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩})))
12		oveq2 7377	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → ((𝑥 + 1) mod 𝑛) = ((𝑥 + 1) mod 𝑁))
13	12	opeq2d 4840	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → ⟨0, ((𝑥 + 1) mod 𝑛)⟩ = ⟨0, ((𝑥 + 1) mod 𝑁)⟩)
14	13	preq2d 4700	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩})
15	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩})
16	15	eqeq2d 2740	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ↔ 𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩}))
17		biidd 262	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩}))
18		oveq2 7377	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐾 → (𝑥 + 𝑘) = (𝑥 + 𝐾))
19	18	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (𝑥 + 𝑘) = (𝑥 + 𝐾))
20		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → 𝑛 = 𝑁)
21	19, 20	oveq12d 7387	. . . . . . . . . . . . 13 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((𝑥 + 𝑘) mod 𝑛) = ((𝑥 + 𝐾) mod 𝑁))
22	21	opeq2d 4840	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩ = ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩)
23	22	preq2d 4700	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})
24	23	eqeq2d 2740	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩} ↔ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}))
25	16, 17, 24	3orbi123d 1437	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ((𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩}) ↔ (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})))
26	25	rexbidv 3157	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩}) ↔ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})))
27	11, 26	bitrd 279	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → (∃𝑥 ∈ (0..^𝑛)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩}) ↔ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})))
28	9, 27	rabeqbidv 3421	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑛)) ∣ ∃𝑥 ∈ (0..^𝑛)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩})} = {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})})
29	28	reseq2d 5939	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑛)) ∣ ∃𝑥 ∈ (0..^𝑛)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩})}) = ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})}))
30	29	opeq2d 4840	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑛)) ∣ ∃𝑥 ∈ (0..^𝑛)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩})})⟩ = ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})})⟩)
31	7, 30	preq12d 4701	. . 3 ⊢ ((𝑛 = 𝑁 ∧ 𝑘 = 𝐾) → {⟨(Base‘ndx), ({0, 1} × (0..^𝑛))⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑛)) ∣ ∃𝑥 ∈ (0..^𝑛)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩})})⟩} = {⟨(Base‘ndx), ({0, 1} × 𝐼)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})})⟩})
32		fvoveq1 7392	. . . . 5 ⊢ (𝑛 = 𝑁 → (⌈‘(𝑛 / 2)) = (⌈‘(𝑁 / 2)))
33	32	oveq2d 7385	. . . 4 ⊢ (𝑛 = 𝑁 → (1..^(⌈‘(𝑛 / 2))) = (1..^(⌈‘(𝑁 / 2))))
34		gpgov.j	. . . 4 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
35	33, 34	eqtr4di 2782	. . 3 ⊢ (𝑛 = 𝑁 → (1..^(⌈‘(𝑛 / 2))) = 𝐽)
36		df-gpg 48005	. . 3 ⊢ gPetersenGr = (𝑛 ∈ ℕ, 𝑘 ∈ (1..^(⌈‘(𝑛 / 2))) ↦ {⟨(Base‘ndx), ({0, 1} × (0..^𝑛))⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑛)) ∣ ∃𝑥 ∈ (0..^𝑛)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩})})⟩})
37	31, 35, 36	ovmpox 7522	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽 ∧ {⟨(Base‘ndx), ({0, 1} × 𝐼)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})})⟩} ∈ V) → (𝑁 gPetersenGr 𝐾) = {⟨(Base‘ndx), ({0, 1} × 𝐼)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})})⟩})
38	1, 37	mp3an3 1452	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (𝑁 gPetersenGr 𝐾) = {⟨(Base‘ndx), ({0, 1} × 𝐼)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})})⟩})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3402  Vcvv 3444  𝒫 cpw 4559  {cpr 4587  ⟨cop 4591   I cid 5525   × cxp 5629   ↾ cres 5633  ‘cfv 6499  (class class class)co 7369  0cc0 11044  1c1 11045   + caddc 11047   / cdiv 11811  ℕcn 12162  2c2 12217  ..^cfzo 13591  ⌈cceil 13729   mod cmo 13807  ndxcnx 17139  Basecbs 17155  .efcedgf 28891   gPetersenGr cgpg 48004

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 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-gpg 48005

This theorem is referenced by:  gpgvtx  48007  gpgiedg  48008