Theorem gpg5grlic 48005

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.)

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 12909	. . . 4 ⊢ 5 ∈ (ℤ≥‘3)
2		3z 12633	. . . . . . 7 ⊢ 3 ∈ ℤ
3		1lt3 12421	. . . . . . 7 ⊢ 1 < 3
4		eluz2b1 12943	. . . . . . 7 ⊢ (3 ∈ (ℤ≥‘2) ↔ (3 ∈ ℤ ∧ 1 < 3))
5	2, 3, 4	mpbir2an 711	. . . . . 6 ⊢ 3 ∈ (ℤ≥‘2)
6		fzo1lb 13735	. . . . . 6 ⊢ (1 ∈ (1..^3) ↔ 3 ∈ (ℤ≥‘2))
7	5, 6	mpbir 231	. . . . 5 ⊢ 1 ∈ (1..^3)
8		ceil5half3 47300	. . . . . . 7 ⊢ (⌈‘(5 / 2)) = 3
9	8	eqcomi 2743	. . . . . 6 ⊢ 3 = (⌈‘(5 / 2))
10	9	oveq2i 7424	. . . . 5 ⊢ (1..^3) = (1..^(⌈‘(5 / 2)))
11	7, 10	eleqtri 2831	. . . 4 ⊢ 1 ∈ (1..^(⌈‘(5 / 2)))
12		gpgusgra 47970	. . . 4 ⊢ ((5 ∈ (ℤ≥‘3) ∧ 1 ∈ (1..^(⌈‘(5 / 2)))) → (5 gPetersenGr 1) ∈ USGraph)
13	1, 11, 12	mp2an 692	. . 3 ⊢ (5 gPetersenGr 1) ∈ USGraph
14		2nn 12321	. . . . . 6 ⊢ 2 ∈ ℕ
15		3nn 12327	. . . . . 6 ⊢ 3 ∈ ℕ
16		2lt3 12420	. . . . . 6 ⊢ 2 < 3
17		elfzo1 13734	. . . . . 6 ⊢ (2 ∈ (1..^3) ↔ (2 ∈ ℕ ∧ 3 ∈ ℕ ∧ 2 < 3))
18	14, 15, 16, 17	mpbir3an 1341	. . . . 5 ⊢ 2 ∈ (1..^3)
19	18, 10	eleqtri 2831	. . . 4 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
20		gpgusgra 47970	. . . 4 ⊢ ((5 ∈ (ℤ≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) → (5 gPetersenGr 2) ∈ USGraph)
21	1, 19, 20	mp2an 692	. . 3 ⊢ (5 gPetersenGr 2) ∈ USGraph
22		2eluzge1 12918	. . . . . 6 ⊢ 2 ∈ (ℤ≥‘1)
23		eluzfz1 13553	. . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → 1 ∈ (1...2))
24	22, 23	ax-mp 5	. . . . 5 ⊢ 1 ∈ (1...2)
25		gpg5order 47972	. . . . 5 ⊢ (1 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 1))) = ;10)
26	24, 25	ax-mp 5	. . . 4 ⊢ (♯‘(Vtx‘(5 gPetersenGr 1))) = ;10
27		eluzfz2 13554	. . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → 2 ∈ (1...2))
28	22, 27	ax-mp 5	. . . . 5 ⊢ 2 ∈ (1...2)
29		gpg5order 47972	. . . . 5 ⊢ (2 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10)
30	28, 29	ax-mp 5	. . . 4 ⊢ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10
31		eqtr3 2756	. . . . 5 ⊢ (((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 ∧ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10) → (♯‘(Vtx‘(5 gPetersenGr 1))) = (♯‘(Vtx‘(5 gPetersenGr 2))))
32		fvex 6899	. . . . . . 7 ⊢ (Vtx‘(5 gPetersenGr 1)) ∈ V
33		10nn0 12734	. . . . . . 7 ⊢ ;10 ∈ ℕ0
34		hashvnfin 14381	. . . . . . 7 ⊢ (((Vtx‘(5 gPetersenGr 1)) ∈ V ∧ ;10 ∈ ℕ0) → ((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 → (Vtx‘(5 gPetersenGr 1)) ∈ Fin))
35	32, 33, 34	mp2an 692	. . . . . 6 ⊢ ((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 → (Vtx‘(5 gPetersenGr 1)) ∈ Fin)
36		fvex 6899	. . . . . . 7 ⊢ (Vtx‘(5 gPetersenGr 2)) ∈ V
37		hashvnfin 14381	. . . . . . 7 ⊢ (((Vtx‘(5 gPetersenGr 2)) ∈ V ∧ ;10 ∈ ℕ0) → ((♯‘(Vtx‘(5 gPetersenGr 2))) = ;10 → (Vtx‘(5 gPetersenGr 2)) ∈ Fin))
38	36, 33, 37	mp2an 692	. . . . . 6 ⊢ ((♯‘(Vtx‘(5 gPetersenGr 2))) = ;10 → (Vtx‘(5 gPetersenGr 2)) ∈ Fin)
39		hashen 14368	. . . . . 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))))
40	35, 38, 39	syl2an 596	. . . . 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))))
41	31, 40	mpbid 232	. . . 4 ⊢ (((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 ∧ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10) → (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2)))
42	26, 30, 41	mp2an 692	. . 3 ⊢ (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2))
43	13, 21, 42	3pm3.2i 1339	. 2 ⊢ ((5 gPetersenGr 1) ∈ USGraph ∧ (5 gPetersenGr 2) ∈ USGraph ∧ (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2)))
44		eqid 2734	. . . . 5 ⊢ (5 gPetersenGr 1) = (5 gPetersenGr 1)
45	44	gpg5gricstgr3 48004	. . . 4 ⊢ ((1 ∈ (1...2) ∧ 𝑣 ∈ (Vtx‘(5 gPetersenGr 1))) → ((5 gPetersenGr 1) ISubGr ((5 gPetersenGr 1) ClNeighbVtx 𝑣)) ≃𝑔𝑟 (StarGr‘3))
46	24, 45	mpan 690	. . 3 ⊢ (𝑣 ∈ (Vtx‘(5 gPetersenGr 1)) → ((5 gPetersenGr 1) ISubGr ((5 gPetersenGr 1) ClNeighbVtx 𝑣)) ≃𝑔𝑟 (StarGr‘3))
47	46	rgen 3052	. 2 ⊢ ∀𝑣 ∈ (Vtx‘(5 gPetersenGr 1))((5 gPetersenGr 1) ISubGr ((5 gPetersenGr 1) ClNeighbVtx 𝑣)) ≃𝑔𝑟 (StarGr‘3)
48		eqid 2734	. . . . 5 ⊢ (5 gPetersenGr 2) = (5 gPetersenGr 2)
49	48	gpg5gricstgr3 48004	. . . 4 ⊢ ((2 ∈ (1...2) ∧ 𝑤 ∈ (Vtx‘(5 gPetersenGr 2))) → ((5 gPetersenGr 2) ISubGr ((5 gPetersenGr 2) ClNeighbVtx 𝑤)) ≃𝑔𝑟 (StarGr‘3))
50	28, 49	mpan 690	. . 3 ⊢ (𝑤 ∈ (Vtx‘(5 gPetersenGr 2)) → ((5 gPetersenGr 2) ISubGr ((5 gPetersenGr 2) ClNeighbVtx 𝑤)) ≃𝑔𝑟 (StarGr‘3))
51	50	rgen 3052	. 2 ⊢ ∀𝑤 ∈ (Vtx‘(5 gPetersenGr 2))((5 gPetersenGr 2) ISubGr ((5 gPetersenGr 2) ClNeighbVtx 𝑤)) ≃𝑔𝑟 (StarGr‘3)
52		3nn0 12527	. . 3 ⊢ 3 ∈ ℕ0
53		eqid 2734	. . 3 ⊢ (Vtx‘(5 gPetersenGr 1)) = (Vtx‘(5 gPetersenGr 1))
54		eqid 2734	. . 3 ⊢ (Vtx‘(5 gPetersenGr 2)) = (Vtx‘(5 gPetersenGr 2))
55	52, 53, 54	clnbgr3stgrgrlic 47937	. 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))
56	43, 47, 51, 55	mp3an 1462	1 ⊢ (5 gPetersenGr 1) ≃𝑙𝑔𝑟 (5 gPetersenGr 2)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050  Vcvv 3463   class class class wbr 5123  ‘cfv 6541  (class class class)co 7413   ≈ cen 8964  Fincfn 8967  0cc0 11137  1c1 11138   < clt 11277   / cdiv 11902  ℕcn 12248  2c2 12303  3c3 12304  5c5 12306  ℕ₀cn0 12509  ℤcz 12596  ;cdc 12716  ℤ_≥cuz 12860  ...cfz 13529  ..^cfzo 13676  ⌈cceil 13813  ♯chash 14351  Vtxcvtx 28941  USGraphcusgr 29094   ClNeighbVtx cclnbgr 47763   ISubGr cisubgr 47804   ≃_𝑔𝑟 cgric 47820  StarGrcstgr 47876   ≃_𝑙𝑔𝑟 cgrlic 47902   gPetersenGr cgpg 47957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214  ax-pre-sup 11215

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4888  df-int 4927  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-oadd 8492  df-er 8727  df-map 8850  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-sup 9464  df-inf 9465  df-dju 9923  df-card 9961  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-div 11903  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12510  df-xnn0 12583  df-z 12597  df-dec 12717  df-uz 12861  df-rp 13017  df-ico 13375  df-fz 13530  df-fzo 13677  df-fl 13814  df-ceil 13815  df-mod 13892  df-seq 14025  df-exp 14085  df-hash 14352  df-cj 15120  df-re 15121  df-im 15122  df-sqrt 15256  df-abs 15257  df-dvds 16273  df-struct 17166  df-slot 17201  df-ndx 17213  df-base 17230  df-edgf 28934  df-vtx 28943  df-iedg 28944  df-edg 28993  df-uhgr 29003  df-ushgr 29004  df-upgr 29027  df-umgr 29028  df-uspgr 29095  df-usgr 29096  df-subgr 29213  df-nbgr 29278  df-clnbgr 47764  df-isubgr 47805  df-grim 47822  df-gric 47825  df-stgr 47877  df-grlim 47903  df-grlic 47906  df-gpg 47958

This theorem is referenced by: (None)