Metamath Proof Explorer


Theorem gpg5grlic

Description: The two generalized Petersen graphs G(N,K) of order 10 ( N = 5 ), which are the Petersen graph G(5,2) and the 5-prism G(5,1), are locally isomorphic. (Contributed by AV, 29-Sep-2025)

Ref Expression
Assertion gpg5grlic ( 5 gPetersenGr 1 ) ≃𝑙𝑔𝑟 ( 5 gPetersenGr 2 )

Proof

Step Hyp Ref Expression
1 5eluz3 5 ∈ ( ℤ ‘ 3 )
2 3z 3 ∈ ℤ
3 1lt3 1 < 3
4 eluz2b1 ( 3 ∈ ( ℤ ‘ 2 ) ↔ ( 3 ∈ ℤ ∧ 1 < 3 ) )
5 2 3 4 mpbir2an 3 ∈ ( ℤ ‘ 2 )
6 fzo1lb ( 1 ∈ ( 1 ..^ 3 ) ↔ 3 ∈ ( ℤ ‘ 2 ) )
7 5 6 mpbir 1 ∈ ( 1 ..^ 3 )
8 ceil5half3 ( ⌈ ‘ ( 5 / 2 ) ) = 3
9 8 eqcomi 3 = ( ⌈ ‘ ( 5 / 2 ) )
10 9 oveq2i ( 1 ..^ 3 ) = ( 1 ..^ ( ⌈ ‘ ( 5 / 2 ) ) )
11 7 10 eleqtri 1 ∈ ( 1 ..^ ( ⌈ ‘ ( 5 / 2 ) ) )
12 gpgusgra ( ( 5 ∈ ( ℤ ‘ 3 ) ∧ 1 ∈ ( 1 ..^ ( ⌈ ‘ ( 5 / 2 ) ) ) ) → ( 5 gPetersenGr 1 ) ∈ USGraph )
13 1 11 12 mp2an ( 5 gPetersenGr 1 ) ∈ USGraph
14 2nn 2 ∈ ℕ
15 3nn 3 ∈ ℕ
16 2lt3 2 < 3
17 elfzo1 ( 2 ∈ ( 1 ..^ 3 ) ↔ ( 2 ∈ ℕ ∧ 3 ∈ ℕ ∧ 2 < 3 ) )
18 14 15 16 17 mpbir3an 2 ∈ ( 1 ..^ 3 )
19 18 10 eleqtri 2 ∈ ( 1 ..^ ( ⌈ ‘ ( 5 / 2 ) ) )
20 gpgusgra ( ( 5 ∈ ( ℤ ‘ 3 ) ∧ 2 ∈ ( 1 ..^ ( ⌈ ‘ ( 5 / 2 ) ) ) ) → ( 5 gPetersenGr 2 ) ∈ USGraph )
21 1 19 20 mp2an ( 5 gPetersenGr 2 ) ∈ USGraph
22 2eluzge1 2 ∈ ( ℤ ‘ 1 )
23 eluzfz1 ( 2 ∈ ( ℤ ‘ 1 ) → 1 ∈ ( 1 ... 2 ) )
24 22 23 ax-mp 1 ∈ ( 1 ... 2 )
25 gpg5order ( 1 ∈ ( 1 ... 2 ) → ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ) = 1 0 )
26 24 25 ax-mp ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ) = 1 0
27 eluzfz2 ( 2 ∈ ( ℤ ‘ 1 ) → 2 ∈ ( 1 ... 2 ) )
28 22 27 ax-mp 2 ∈ ( 1 ... 2 )
29 gpg5order ( 2 ∈ ( 1 ... 2 ) → ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ) = 1 0 )
30 28 29 ax-mp ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ) = 1 0
31 eqtr3 ( ( ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ) = 1 0 ∧ ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ) = 1 0 ) → ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ) = ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ) )
32 fvex ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ∈ V
33 10nn0 1 0 ∈ ℕ0
34 hashvnfin ( ( ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ∈ V ∧ 1 0 ∈ ℕ0 ) → ( ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ) = 1 0 → ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ∈ Fin ) )
35 32 33 34 mp2an ( ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ) = 1 0 → ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ∈ Fin )
36 fvex ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ∈ V
37 hashvnfin ( ( ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ∈ V ∧ 1 0 ∈ ℕ0 ) → ( ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ) = 1 0 → ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ∈ Fin ) )
38 36 33 37 mp2an ( ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ) = 1 0 → ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ∈ Fin )
39 hashen ( ( ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ∈ Fin ∧ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ∈ Fin ) → ( ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ) = ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ) ↔ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ≈ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ) )
40 35 38 39 syl2an ( ( ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ) = 1 0 ∧ ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ) = 1 0 ) → ( ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ) = ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ) ↔ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ≈ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ) )
41 31 40 mpbid ( ( ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ) = 1 0 ∧ ( ♯ ‘ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ) = 1 0 ) → ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ≈ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) )
42 26 30 41 mp2an ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ≈ ( Vtx ‘ ( 5 gPetersenGr 2 ) )
43 13 21 42 3pm3.2i ( ( 5 gPetersenGr 1 ) ∈ USGraph ∧ ( 5 gPetersenGr 2 ) ∈ USGraph ∧ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ≈ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) )
44 eqid ( 5 gPetersenGr 1 ) = ( 5 gPetersenGr 1 )
45 44 gpg5gricstgr3 ( ( 1 ∈ ( 1 ... 2 ) ∧ 𝑣 ∈ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ) → ( ( 5 gPetersenGr 1 ) ISubGr ( ( 5 gPetersenGr 1 ) ClNeighbVtx 𝑣 ) ) ≃𝑔𝑟 ( StarGr ‘ 3 ) )
46 24 45 mpan ( 𝑣 ∈ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) → ( ( 5 gPetersenGr 1 ) ISubGr ( ( 5 gPetersenGr 1 ) ClNeighbVtx 𝑣 ) ) ≃𝑔𝑟 ( StarGr ‘ 3 ) )
47 46 rgen 𝑣 ∈ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ( ( 5 gPetersenGr 1 ) ISubGr ( ( 5 gPetersenGr 1 ) ClNeighbVtx 𝑣 ) ) ≃𝑔𝑟 ( StarGr ‘ 3 )
48 eqid ( 5 gPetersenGr 2 ) = ( 5 gPetersenGr 2 )
49 48 gpg5gricstgr3 ( ( 2 ∈ ( 1 ... 2 ) ∧ 𝑤 ∈ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ) → ( ( 5 gPetersenGr 2 ) ISubGr ( ( 5 gPetersenGr 2 ) ClNeighbVtx 𝑤 ) ) ≃𝑔𝑟 ( StarGr ‘ 3 ) )
50 28 49 mpan ( 𝑤 ∈ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) → ( ( 5 gPetersenGr 2 ) ISubGr ( ( 5 gPetersenGr 2 ) ClNeighbVtx 𝑤 ) ) ≃𝑔𝑟 ( StarGr ‘ 3 ) )
51 50 rgen 𝑤 ∈ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ( ( 5 gPetersenGr 2 ) ISubGr ( ( 5 gPetersenGr 2 ) ClNeighbVtx 𝑤 ) ) ≃𝑔𝑟 ( StarGr ‘ 3 )
52 3nn0 3 ∈ ℕ0
53 eqid ( Vtx ‘ ( 5 gPetersenGr 1 ) ) = ( Vtx ‘ ( 5 gPetersenGr 1 ) )
54 eqid ( Vtx ‘ ( 5 gPetersenGr 2 ) ) = ( Vtx ‘ ( 5 gPetersenGr 2 ) )
55 52 53 54 clnbgr3stgrgrlic ( ( ( ( 5 gPetersenGr 1 ) ∈ USGraph ∧ ( 5 gPetersenGr 2 ) ∈ USGraph ∧ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ≈ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ) ∧ ∀ 𝑣 ∈ ( Vtx ‘ ( 5 gPetersenGr 1 ) ) ( ( 5 gPetersenGr 1 ) ISubGr ( ( 5 gPetersenGr 1 ) ClNeighbVtx 𝑣 ) ) ≃𝑔𝑟 ( StarGr ‘ 3 ) ∧ ∀ 𝑤 ∈ ( Vtx ‘ ( 5 gPetersenGr 2 ) ) ( ( 5 gPetersenGr 2 ) ISubGr ( ( 5 gPetersenGr 2 ) ClNeighbVtx 𝑤 ) ) ≃𝑔𝑟 ( StarGr ‘ 3 ) ) → ( 5 gPetersenGr 1 ) ≃𝑙𝑔𝑟 ( 5 gPetersenGr 2 ) )
56 43 47 51 55 mp3an ( 5 gPetersenGr 1 ) ≃𝑙𝑔𝑟 ( 5 gPetersenGr 2 )