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Theorem gpg5grlic 47974
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.
StepHypRef Expression
1 5eluz3 12924 . . . 4 5 ∈ (ℤ‘3)
2 3z 12647 . . . . . . 7 3 ∈ ℤ
3 1lt3 12436 . . . . . . 7 1 < 3
4 eluz2b1 12958 . . . . . . 7 (3 ∈ (ℤ‘2) ↔ (3 ∈ ℤ ∧ 1 < 3))
52, 3, 4mpbir2an 711 . . . . . 6 3 ∈ (ℤ‘2)
6 fzo1lb 13749 . . . . . 6 (1 ∈ (1..^3) ↔ 3 ∈ (ℤ‘2))
75, 6mpbir 231 . . . . 5 1 ∈ (1..^3)
8 ceil5half3 47279 . . . . . . 7 (⌈‘(5 / 2)) = 3
98eqcomi 2743 . . . . . 6 3 = (⌈‘(5 / 2))
109oveq2i 7441 . . . . 5 (1..^3) = (1..^(⌈‘(5 / 2)))
117, 10eleqtri 2836 . . . 4 1 ∈ (1..^(⌈‘(5 / 2)))
12 gpgusgra 47946 . . . 4 ((5 ∈ (ℤ‘3) ∧ 1 ∈ (1..^(⌈‘(5 / 2)))) → (5 gPetersenGr 1) ∈ USGraph)
131, 11, 12mp2an 692 . . 3 (5 gPetersenGr 1) ∈ USGraph
14 2nn 12336 . . . . . 6 2 ∈ ℕ
15 3nn 12342 . . . . . 6 3 ∈ ℕ
16 2lt3 12435 . . . . . 6 2 < 3
17 elfzo1 13748 . . . . . 6 (2 ∈ (1..^3) ↔ (2 ∈ ℕ ∧ 3 ∈ ℕ ∧ 2 < 3))
1814, 15, 16, 17mpbir3an 1340 . . . . 5 2 ∈ (1..^3)
1918, 10eleqtri 2836 . . . 4 2 ∈ (1..^(⌈‘(5 / 2)))
20 gpgusgra 47946 . . . 4 ((5 ∈ (ℤ‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) → (5 gPetersenGr 2) ∈ USGraph)
211, 19, 20mp2an 692 . . 3 (5 gPetersenGr 2) ∈ USGraph
22 2eluzge1 12933 . . . . . 6 2 ∈ (ℤ‘1)
23 eluzfz1 13567 . . . . . 6 (2 ∈ (ℤ‘1) → 1 ∈ (1...2))
2422, 23ax-mp 5 . . . . 5 1 ∈ (1...2)
25 gpg5order 47948 . . . . 5 (1 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 1))) = 10)
2624, 25ax-mp 5 . . . 4 (♯‘(Vtx‘(5 gPetersenGr 1))) = 10
27 eluzfz2 13568 . . . . . 6 (2 ∈ (ℤ‘1) → 2 ∈ (1...2))
2822, 27ax-mp 5 . . . . 5 2 ∈ (1...2)
29 gpg5order 47948 . . . . 5 (2 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 2))) = 10)
3028, 29ax-mp 5 . . . 4 (♯‘(Vtx‘(5 gPetersenGr 2))) = 10
31 eqtr3 2760 . . . . 5 (((♯‘(Vtx‘(5 gPetersenGr 1))) = 10 ∧ (♯‘(Vtx‘(5 gPetersenGr 2))) = 10) → (♯‘(Vtx‘(5 gPetersenGr 1))) = (♯‘(Vtx‘(5 gPetersenGr 2))))
32 fvex 6919 . . . . . . 7 (Vtx‘(5 gPetersenGr 1)) ∈ V
33 10nn0 12748 . . . . . . 7 10 ∈ ℕ0
34 hashvnfin 14395 . . . . . . 7 (((Vtx‘(5 gPetersenGr 1)) ∈ V ∧ 10 ∈ ℕ0) → ((♯‘(Vtx‘(5 gPetersenGr 1))) = 10 → (Vtx‘(5 gPetersenGr 1)) ∈ Fin))
3532, 33, 34mp2an 692 . . . . . 6 ((♯‘(Vtx‘(5 gPetersenGr 1))) = 10 → (Vtx‘(5 gPetersenGr 1)) ∈ Fin)
36 fvex 6919 . . . . . . 7 (Vtx‘(5 gPetersenGr 2)) ∈ V
37 hashvnfin 14395 . . . . . . 7 (((Vtx‘(5 gPetersenGr 2)) ∈ V ∧ 10 ∈ ℕ0) → ((♯‘(Vtx‘(5 gPetersenGr 2))) = 10 → (Vtx‘(5 gPetersenGr 2)) ∈ Fin))
3836, 33, 37mp2an 692 . . . . . 6 ((♯‘(Vtx‘(5 gPetersenGr 2))) = 10 → (Vtx‘(5 gPetersenGr 2)) ∈ Fin)
39 hashen 14382 . . . . . 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))))
4035, 38, 39syl2an 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))))
4131, 40mpbid 232 . . . 4 (((♯‘(Vtx‘(5 gPetersenGr 1))) = 10 ∧ (♯‘(Vtx‘(5 gPetersenGr 2))) = 10) → (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2)))
4226, 30, 41mp2an 692 . . 3 (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2))
4313, 21, 423pm3.2i 1338 . 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)
4544gpg5gricstgr3 47973 . . . 4 ((1 ∈ (1...2) ∧ 𝑣 ∈ (Vtx‘(5 gPetersenGr 1))) → ((5 gPetersenGr 1) ISubGr ((5 gPetersenGr 1) ClNeighbVtx 𝑣)) ≃𝑔𝑟 (StarGr‘3))
4624, 45mpan 690 . . 3 (𝑣 ∈ (Vtx‘(5 gPetersenGr 1)) → ((5 gPetersenGr 1) ISubGr ((5 gPetersenGr 1) ClNeighbVtx 𝑣)) ≃𝑔𝑟 (StarGr‘3))
4746rgen 3060 . 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)
4948gpg5gricstgr3 47973 . . . 4 ((2 ∈ (1...2) ∧ 𝑤 ∈ (Vtx‘(5 gPetersenGr 2))) → ((5 gPetersenGr 2) ISubGr ((5 gPetersenGr 2) ClNeighbVtx 𝑤)) ≃𝑔𝑟 (StarGr‘3))
5028, 49mpan 690 . . 3 (𝑤 ∈ (Vtx‘(5 gPetersenGr 2)) → ((5 gPetersenGr 2) ISubGr ((5 gPetersenGr 2) ClNeighbVtx 𝑤)) ≃𝑔𝑟 (StarGr‘3))
5150rgen 3060 . 2 𝑤 ∈ (Vtx‘(5 gPetersenGr 2))((5 gPetersenGr 2) ISubGr ((5 gPetersenGr 2) ClNeighbVtx 𝑤)) ≃𝑔𝑟 (StarGr‘3)
52 3nn0 12541 . . 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))
5552, 53, 54clnbgr3stgrgrlic 47914 . 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))
5643, 47, 51, 55mp3an 1460 1 (5 gPetersenGr 1) ≃𝑙𝑔𝑟 (5 gPetersenGr 2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477   class class class wbr 5147  cfv 6562  (class class class)co 7430  cen 8980  Fincfn 8983  0cc0 11152  1c1 11153   < clt 11292   / cdiv 11917  cn 12263  2c2 12318  3c3 12319  5c5 12321  0cn0 12523  cz 12610  cdc 12730  cuz 12875  ...cfz 13543  ..^cfzo 13690  cceil 13827  chash 14365  Vtxcvtx 29027  USGraphcusgr 29180   ClNeighbVtx cclnbgr 47742   ISubGr cisubgr 47783  𝑔𝑟 cgric 47799  StarGrcstgr 47853  𝑙𝑔𝑟 cgrlic 47879   gPetersenGr cgpg 47934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-xnn0 12597  df-z 12611  df-dec 12731  df-uz 12876  df-rp 13032  df-ico 13389  df-fz 13544  df-fzo 13691  df-fl 13828  df-ceil 13829  df-mod 13906  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-dvds 16287  df-struct 17180  df-slot 17215  df-ndx 17227  df-base 17245  df-edgf 29018  df-vtx 29029  df-iedg 29030  df-edg 29079  df-uhgr 29089  df-ushgr 29090  df-upgr 29113  df-umgr 29114  df-uspgr 29181  df-usgr 29182  df-subgr 29299  df-nbgr 29364  df-clnbgr 47743  df-isubgr 47784  df-grim 47801  df-gric 47804  df-stgr 47854  df-grlim 47880  df-grlic 47883  df-gpg 47935
This theorem is referenced by: (None)
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