Theorem gpg5grlic 48098

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 12784	. . . 4 ⊢ 5 ∈ (ℤ≥‘3)
2		3z 12508	. . . . . . 7 ⊢ 3 ∈ ℤ
3		1lt3 12296	. . . . . . 7 ⊢ 1 < 3
4		eluz2b1 12820	. . . . . . 7 ⊢ (3 ∈ (ℤ≥‘2) ↔ (3 ∈ ℤ ∧ 1 < 3))
5	2, 3, 4	mpbir2an 711	. . . . . 6 ⊢ 3 ∈ (ℤ≥‘2)
6		fzo1lb 13616	. . . . . 6 ⊢ (1 ∈ (1..^3) ↔ 3 ∈ (ℤ≥‘2))
7	5, 6	mpbir 231	. . . . 5 ⊢ 1 ∈ (1..^3)
8		ceil5half3 47344	. . . . . . 7 ⊢ (⌈‘(5 / 2)) = 3
9	8	eqcomi 2738	. . . . . 6 ⊢ 3 = (⌈‘(5 / 2))
10	9	oveq2i 7360	. . . . 5 ⊢ (1..^3) = (1..^(⌈‘(5 / 2)))
11	7, 10	eleqtri 2826	. . . 4 ⊢ 1 ∈ (1..^(⌈‘(5 / 2)))
12		gpgusgra 48061	. . . 4 ⊢ ((5 ∈ (ℤ≥‘3) ∧ 1 ∈ (1..^(⌈‘(5 / 2)))) → (5 gPetersenGr 1) ∈ USGraph)
13	1, 11, 12	mp2an 692	. . 3 ⊢ (5 gPetersenGr 1) ∈ USGraph
14		pglem 48095	. . . 4 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
15		gpgusgra 48061	. . . 4 ⊢ ((5 ∈ (ℤ≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) → (5 gPetersenGr 2) ∈ USGraph)
16	1, 14, 15	mp2an 692	. . 3 ⊢ (5 gPetersenGr 2) ∈ USGraph
17		2eluzge1 12783	. . . . . 6 ⊢ 2 ∈ (ℤ≥‘1)
18		eluzfz1 13434	. . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → 1 ∈ (1...2))
19	17, 18	ax-mp 5	. . . . 5 ⊢ 1 ∈ (1...2)
20		gpg5order 48064	. . . . 5 ⊢ (1 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 1))) = ;10)
21	19, 20	ax-mp 5	. . . 4 ⊢ (♯‘(Vtx‘(5 gPetersenGr 1))) = ;10
22		eluzfz2 13435	. . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → 2 ∈ (1...2))
23	17, 22	ax-mp 5	. . . . 5 ⊢ 2 ∈ (1...2)
24		gpg5order 48064	. . . . 5 ⊢ (2 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10)
25	23, 24	ax-mp 5	. . . 4 ⊢ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10
26		eqtr3 2751	. . . . 5 ⊢ (((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 ∧ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10) → (♯‘(Vtx‘(5 gPetersenGr 1))) = (♯‘(Vtx‘(5 gPetersenGr 2))))
27		fvex 6835	. . . . . . 7 ⊢ (Vtx‘(5 gPetersenGr 1)) ∈ V
28		10nn0 12609	. . . . . . 7 ⊢ ;10 ∈ ℕ0
29		hashvnfin 14267	. . . . . . 7 ⊢ (((Vtx‘(5 gPetersenGr 1)) ∈ V ∧ ;10 ∈ ℕ0) → ((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 → (Vtx‘(5 gPetersenGr 1)) ∈ Fin))
30	27, 28, 29	mp2an 692	. . . . . 6 ⊢ ((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 → (Vtx‘(5 gPetersenGr 1)) ∈ Fin)
31		fvex 6835	. . . . . . 7 ⊢ (Vtx‘(5 gPetersenGr 2)) ∈ V
32		hashvnfin 14267	. . . . . . 7 ⊢ (((Vtx‘(5 gPetersenGr 2)) ∈ V ∧ ;10 ∈ ℕ0) → ((♯‘(Vtx‘(5 gPetersenGr 2))) = ;10 → (Vtx‘(5 gPetersenGr 2)) ∈ Fin))
33	31, 28, 32	mp2an 692	. . . . . 6 ⊢ ((♯‘(Vtx‘(5 gPetersenGr 2))) = ;10 → (Vtx‘(5 gPetersenGr 2)) ∈ Fin)
34		hashen 14254	. . . . . 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 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))))
36	26, 35	mpbid 232	. . . 4 ⊢ (((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 ∧ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10) → (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2)))
37	21, 25, 36	mp2an 692	. . 3 ⊢ (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2))
38	13, 16, 37	3pm3.2i 1340	. 2 ⊢ ((5 gPetersenGr 1) ∈ USGraph ∧ (5 gPetersenGr 2) ∈ USGraph ∧ (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2)))
39		eqid 2729	. . . . 5 ⊢ (5 gPetersenGr 1) = (5 gPetersenGr 1)
40	39	gpg5gricstgr3 48094	. . . 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 690	. . 3 ⊢ (𝑣 ∈ (Vtx‘(5 gPetersenGr 1)) → ((5 gPetersenGr 1) ISubGr ((5 gPetersenGr 1) ClNeighbVtx 𝑣)) ≃𝑔𝑟 (StarGr‘3))
42	41	rgen 3046	. 2 ⊢ ∀𝑣 ∈ (Vtx‘(5 gPetersenGr 1))((5 gPetersenGr 1) ISubGr ((5 gPetersenGr 1) ClNeighbVtx 𝑣)) ≃𝑔𝑟 (StarGr‘3)
43		eqid 2729	. . . . 5 ⊢ (5 gPetersenGr 2) = (5 gPetersenGr 2)
44	43	gpg5gricstgr3 48094	. . . 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 690	. . 3 ⊢ (𝑤 ∈ (Vtx‘(5 gPetersenGr 2)) → ((5 gPetersenGr 2) ISubGr ((5 gPetersenGr 2) ClNeighbVtx 𝑤)) ≃𝑔𝑟 (StarGr‘3))
46	45	rgen 3046	. 2 ⊢ ∀𝑤 ∈ (Vtx‘(5 gPetersenGr 2))((5 gPetersenGr 2) ISubGr ((5 gPetersenGr 2) ClNeighbVtx 𝑤)) ≃𝑔𝑟 (StarGr‘3)
47		3nn0 12402	. . 3 ⊢ 3 ∈ ℕ0
48		eqid 2729	. . 3 ⊢ (Vtx‘(5 gPetersenGr 1)) = (Vtx‘(5 gPetersenGr 1))
49		eqid 2729	. . 3 ⊢ (Vtx‘(5 gPetersenGr 2)) = (Vtx‘(5 gPetersenGr 2))
50	47, 48, 49	clnbgr3stgrgrlic 48024	. 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 1463	1 ⊢ (5 gPetersenGr 1) ≃𝑙𝑔𝑟 (5 gPetersenGr 2)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3436   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349   ≈ cen 8869  Fincfn 8872  0cc0 11009  1c1 11010   < clt 11149   / cdiv 11777  2c2 12183  3c3 12184  5c5 12186  ℕ₀cn0 12384  ℤcz 12471  ;cdc 12591  ℤ_≥cuz 12735  ...cfz 13410  ..^cfzo 13557  ⌈cceil 13695  ♯chash 14237  Vtxcvtx 28945  USGraphcusgr 29098   ClNeighbVtx cclnbgr 47822   ISubGr cisubgr 47864   ≃_𝑔𝑟 cgric 47880  StarGrcstgr 47955   ≃_𝑙𝑔𝑟 cgrlic 47981   gPetersenGr cgpg 48044

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-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

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-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-inf 9333  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-xnn0 12458  df-z 12472  df-dec 12592  df-uz 12736  df-rp 12894  df-ico 13254  df-fz 13411  df-fzo 13558  df-fl 13696  df-ceil 13697  df-mod 13774  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-dvds 16164  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-edgf 28938  df-vtx 28947  df-iedg 28948  df-edg 28997  df-uhgr 29007  df-ushgr 29008  df-upgr 29031  df-umgr 29032  df-uspgr 29099  df-usgr 29100  df-subgr 29217  df-nbgr 29282  df-clnbgr 47823  df-isubgr 47865  df-grim 47882  df-gric 47885  df-stgr 47956  df-grlim 47982  df-grlic 47985  df-gpg 48045

This theorem is referenced by:  lgricngricex  48133