Theorem gpg5grlic 49395

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.) (Proof shortened by AV, 22-Nov-2025.)

Assertion

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

Proof of Theorem gpg5grlic

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		5eluz3 13034	. . . 4 ⊢ 5 ∈ (ℤ≥‘3)
2		3z 12753	. . . . . . 7 ⊢ 3 ∈ ℤ
3		1lt3 12542	. . . . . . 7 ⊢ 1 < 3
4		eluz2b1 13070	. . . . . . 7 ⊢ (3 ∈ (ℤ≥‘2) ↔ (3 ∈ ℤ ∧ 1 < 3))
5	2, 3, 4	mpbir2an 724	. . . . . 6 ⊢ 3 ∈ (ℤ≥‘2)
6		fzo1lb 13872	. . . . . 6 ⊢ (1 ∈ (1..^3) ↔ 3 ∈ (ℤ≥‘2))
7	5, 6	mpbir 234	. . . . 5 ⊢ 1 ∈ (1..^3)
8		ceil5half3 48619	. . . . . . 7 ⊢ (⌈‘(5 / 2)) = 3
9	8	eqcomi 2772	. . . . . 6 ⊢ 3 = (⌈‘(5 / 2))
10	9	oveq2i 7451	. . . . 5 ⊢ (1..^3) = (1..^(⌈‘(5 / 2)))
11	7, 10	eleqtri 2862	. . . 4 ⊢ 1 ∈ (1..^(⌈‘(5 / 2)))
12		gpgusgra 49358	. . . 4 ⊢ ((5 ∈ (ℤ≥‘3) ∧ 1 ∈ (1..^(⌈‘(5 / 2)))) → (5 gPetersenGr 1) ∈ USGraph)
13	1, 11, 12	mp2an 705	. . 3 ⊢ (5 gPetersenGr 1) ∈ USGraph
14		pglem 49392	. . . 4 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
15		gpgusgra 49358	. . . 4 ⊢ ((5 ∈ (ℤ≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) → (5 gPetersenGr 2) ∈ USGraph)
16	1, 14, 15	mp2an 705	. . 3 ⊢ (5 gPetersenGr 2) ∈ USGraph
17		2eluzge1 13033	. . . . . 6 ⊢ 2 ∈ (ℤ≥‘1)
18		eluzfz1 13688	. . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → 1 ∈ (1...2))
19	17, 18	ax-mp 5	. . . . 5 ⊢ 1 ∈ (1...2)
20		gpg5order 49361	. . . . 5 ⊢ (1 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 1))) = ;10)
21	19, 20	ax-mp 5	. . . 4 ⊢ (♯‘(Vtx‘(5 gPetersenGr 1))) = ;10
22		eluzfz2 13689	. . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → 2 ∈ (1...2))
23	17, 22	ax-mp 5	. . . . 5 ⊢ 2 ∈ (1...2)
24		gpg5order 49361	. . . . 5 ⊢ (2 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10)
25	23, 24	ax-mp 5	. . . 4 ⊢ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10
26		eqtr3 2785	. . . . 5 ⊢ (((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 ∧ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10) → (♯‘(Vtx‘(5 gPetersenGr 1))) = (♯‘(Vtx‘(5 gPetersenGr 2))))
27		fvex 6918	. . . . . . 7 ⊢ (Vtx‘(5 gPetersenGr 1)) ∈ V
28		10nn0 12860	. . . . . . 7 ⊢ ;10 ∈ ℕ0
29		hashvnfin 14528	. . . . . . 7 ⊢ (((Vtx‘(5 gPetersenGr 1)) ∈ V ∧ ;10 ∈ ℕ0) → ((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 → (Vtx‘(5 gPetersenGr 1)) ∈ Fin))
30	27, 28, 29	mp2an 705	. . . . . 6 ⊢ ((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 → (Vtx‘(5 gPetersenGr 1)) ∈ Fin)
31		fvex 6918	. . . . . . 7 ⊢ (Vtx‘(5 gPetersenGr 2)) ∈ V
32		hashvnfin 14528	. . . . . . 7 ⊢ (((Vtx‘(5 gPetersenGr 2)) ∈ V ∧ ;10 ∈ ℕ0) → ((♯‘(Vtx‘(5 gPetersenGr 2))) = ;10 → (Vtx‘(5 gPetersenGr 2)) ∈ Fin))
33	31, 28, 32	mp2an 705	. . . . . 6 ⊢ ((♯‘(Vtx‘(5 gPetersenGr 2))) = ;10 → (Vtx‘(5 gPetersenGr 2)) ∈ Fin)
34		hashen 14515	. . . . . 6 ⊢ (((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))))
35	30, 33, 34	syl2an 608	. . . . 5 ⊢ (((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 ∧ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10) → ((♯‘(Vtx‘(5 gPetersenGr 1))) = (♯‘(Vtx‘(5 gPetersenGr 2))) ↔ (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2))))
36	26, 35	mpbid 235	. . . 4 ⊢ (((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 ∧ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10) → (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2)))
37	21, 25, 36	mp2an 705	. . 3 ⊢ (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2))
38	13, 16, 37	3pm3.2i 1358	. 2 ⊢ ((5 gPetersenGr 1) ∈ USGraph ∧ (5 gPetersenGr 2) ∈ USGraph ∧ (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2)))
39		eqid 2763	. . . . 5 ⊢ (5 gPetersenGr 1) = (5 gPetersenGr 1)
40	39	gpg5gricstgr3 49391	. . . 4 ⊢ ((1 ∈ (1...2) ∧ 𝑣 ∈ (Vtx‘(5 gPetersenGr 1))) → ((5 gPetersenGr 1) ISubGr ((5 gPetersenGr 1) ClNeighbVtx 𝑣)) ≃𝑔𝑟 (StarGr‘3))
41	19, 40	mpan 703	. . 3 ⊢ (𝑣 ∈ (Vtx‘(5 gPetersenGr 1)) → ((5 gPetersenGr 1) ISubGr ((5 gPetersenGr 1) ClNeighbVtx 𝑣)) ≃𝑔𝑟 (StarGr‘3))
42	41	rgen 3082	. 2 ⊢ ∀𝑣 ∈ (Vtx‘(5 gPetersenGr 1))((5 gPetersenGr 1) ISubGr ((5 gPetersenGr 1) ClNeighbVtx 𝑣)) ≃𝑔𝑟 (StarGr‘3)
43		eqid 2763	. . . . 5 ⊢ (5 gPetersenGr 2) = (5 gPetersenGr 2)
44	43	gpg5gricstgr3 49391	. . . 4 ⊢ ((2 ∈ (1...2) ∧ 𝑤 ∈ (Vtx‘(5 gPetersenGr 2))) → ((5 gPetersenGr 2) ISubGr ((5 gPetersenGr 2) ClNeighbVtx 𝑤)) ≃𝑔𝑟 (StarGr‘3))
45	23, 44	mpan 703	. . 3 ⊢ (𝑤 ∈ (Vtx‘(5 gPetersenGr 2)) → ((5 gPetersenGr 2) ISubGr ((5 gPetersenGr 2) ClNeighbVtx 𝑤)) ≃𝑔𝑟 (StarGr‘3))
46	45	rgen 3082	. 2 ⊢ ∀𝑤 ∈ (Vtx‘(5 gPetersenGr 2))((5 gPetersenGr 2) ISubGr ((5 gPetersenGr 2) ClNeighbVtx 𝑤)) ≃𝑔𝑟 (StarGr‘3)
47		3nn0 12648	. . 3 ⊢ 3 ∈ ℕ0
48		eqid 2763	. . 3 ⊢ (Vtx‘(5 gPetersenGr 1)) = (Vtx‘(5 gPetersenGr 1))
49		eqid 2763	. . 3 ⊢ (Vtx‘(5 gPetersenGr 2)) = (Vtx‘(5 gPetersenGr 2))
50	47, 48, 49	clnbgr3stgrgrlic 49321	. 2 ⊢ ((((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))
51	38, 42, 46, 50	mp3an 1491	1 ⊢ (5 gPetersenGr 1) ≃𝑙𝑔𝑟 (5 gPetersenGr 2)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   class class class wbr 5106  ‘cfv 6555  (class class class)co 7440   ≈ cen 8995  Fincfn 8998  0cc0 11224  1c1 11225   < clt 11367   / cdiv 11997  2c2 12421  3c3 12422  5c5 12424  ℕ₀cn0 12630  ℤcz 12717  ;cdc 12838  ℤ_≥cuz 12989  ...cfz 13663  ..^cfzo 13812  ⌈cceil 13955  ♯chash 14498  Vtxcvtx 29601  USGraphcusgr 29757   ClNeighbVtx cclnbgr 49119   ISubGr cisubgr 49161   ≃_𝑔𝑟 cgric 49177  StarGrcstgr 49252   ≃_𝑙𝑔𝑟 cgrlic 49278   gPetersenGr cgpg 49341

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-pre-sup 11302

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-sup 9458  df-inf 9459  df-dju 10006  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-div 11998  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-xnn0 12704  df-z 12718  df-dec 12839  df-uz 12990  df-rp 13145  df-ico 13506  df-fz 13664  df-fzo 13813  df-fl 13956  df-ceil 13957  df-mod 14034  df-seq 14169  df-exp 14229  df-hash 14499  df-cj 15292  df-re 15293  df-im 15294  df-sqrt 15428  df-abs 15429  df-dvds 16449  df-struct 17351  df-slot 17386  df-ndx 17398  df-base 17414  df-edgf 29594  df-vtx 29603  df-iedg 29604  df-edg 29653  df-uhgr 29663  df-ushgr 29664  df-upgr 29687  df-umgr 29688  df-uspgr 29758  df-usgr 29759  df-subgr 29876  df-nbgr 29941  df-clnbgr 49120  df-isubgr 49162  df-grim 49179  df-gric 49182  df-stgr 49253  df-grlim 49279  df-grlic 49282  df-gpg 49342

This theorem is used by:  lgricngricex  49430