Theorem gpg5grlic 48041

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 12899	. . . 4 ⊢ 5 ∈ (ℤ≥‘3)
2		3z 12623	. . . . . . 7 ⊢ 3 ∈ ℤ
3		1lt3 12411	. . . . . . 7 ⊢ 1 < 3
4		eluz2b1 12933	. . . . . . 7 ⊢ (3 ∈ (ℤ≥‘2) ↔ (3 ∈ ℤ ∧ 1 < 3))
5	2, 3, 4	mpbir2an 711	. . . . . 6 ⊢ 3 ∈ (ℤ≥‘2)
6		fzo1lb 13728	. . . . . 6 ⊢ (1 ∈ (1..^3) ↔ 3 ∈ (ℤ≥‘2))
7	5, 6	mpbir 231	. . . . 5 ⊢ 1 ∈ (1..^3)
8		ceil5half3 47317	. . . . . . 7 ⊢ (⌈‘(5 / 2)) = 3
9	8	eqcomi 2744	. . . . . 6 ⊢ 3 = (⌈‘(5 / 2))
10	9	oveq2i 7414	. . . . 5 ⊢ (1..^3) = (1..^(⌈‘(5 / 2)))
11	7, 10	eleqtri 2832	. . . 4 ⊢ 1 ∈ (1..^(⌈‘(5 / 2)))
12		gpgusgra 48009	. . . 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 12311	. . . . . 6 ⊢ 2 ∈ ℕ
15		3nn 12317	. . . . . 6 ⊢ 3 ∈ ℕ
16		2lt3 12410	. . . . . 6 ⊢ 2 < 3
17		elfzo1 13727	. . . . . 6 ⊢ (2 ∈ (1..^3) ↔ (2 ∈ ℕ ∧ 3 ∈ ℕ ∧ 2 < 3))
18	14, 15, 16, 17	mpbir3an 1342	. . . . 5 ⊢ 2 ∈ (1..^3)
19	18, 10	eleqtri 2832	. . . 4 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
20		gpgusgra 48009	. . . 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 12908	. . . . . 6 ⊢ 2 ∈ (ℤ≥‘1)
23		eluzfz1 13546	. . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → 1 ∈ (1...2))
24	22, 23	ax-mp 5	. . . . 5 ⊢ 1 ∈ (1...2)
25		gpg5order 48012	. . . . 5 ⊢ (1 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 1))) = ;10)
26	24, 25	ax-mp 5	. . . 4 ⊢ (♯‘(Vtx‘(5 gPetersenGr 1))) = ;10
27		eluzfz2 13547	. . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → 2 ∈ (1...2))
28	22, 27	ax-mp 5	. . . . 5 ⊢ 2 ∈ (1...2)
29		gpg5order 48012	. . . . 5 ⊢ (2 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10)
30	28, 29	ax-mp 5	. . . 4 ⊢ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10
31		eqtr3 2757	. . . . 5 ⊢ (((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 ∧ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10) → (♯‘(Vtx‘(5 gPetersenGr 1))) = (♯‘(Vtx‘(5 gPetersenGr 2))))
32		fvex 6888	. . . . . . 7 ⊢ (Vtx‘(5 gPetersenGr 1)) ∈ V
33		10nn0 12724	. . . . . . 7 ⊢ ;10 ∈ ℕ0
34		hashvnfin 14376	. . . . . . 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 6888	. . . . . . 7 ⊢ (Vtx‘(5 gPetersenGr 2)) ∈ V
37		hashvnfin 14376	. . . . . . 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 14363	. . . . . 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 1340	. 2 ⊢ ((5 gPetersenGr 1) ∈ USGraph ∧ (5 gPetersenGr 2) ∈ USGraph ∧ (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2)))
44		eqid 2735	. . . . 5 ⊢ (5 gPetersenGr 1) = (5 gPetersenGr 1)
45	44	gpg5gricstgr3 48040	. . . 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 3053	. 2 ⊢ ∀𝑣 ∈ (Vtx‘(5 gPetersenGr 1))((5 gPetersenGr 1) ISubGr ((5 gPetersenGr 1) ClNeighbVtx 𝑣)) ≃𝑔𝑟 (StarGr‘3)
48		eqid 2735	. . . . 5 ⊢ (5 gPetersenGr 2) = (5 gPetersenGr 2)
49	48	gpg5gricstgr3 48040	. . . 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 3053	. 2 ⊢ ∀𝑤 ∈ (Vtx‘(5 gPetersenGr 2))((5 gPetersenGr 2) ISubGr ((5 gPetersenGr 2) ClNeighbVtx 𝑤)) ≃𝑔𝑟 (StarGr‘3)
52		3nn0 12517	. . 3 ⊢ 3 ∈ ℕ0
53		eqid 2735	. . 3 ⊢ (Vtx‘(5 gPetersenGr 1)) = (Vtx‘(5 gPetersenGr 1))
54		eqid 2735	. . 3 ⊢ (Vtx‘(5 gPetersenGr 2)) = (Vtx‘(5 gPetersenGr 2))
55	52, 53, 54	clnbgr3stgrgrlic 47972	. 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 1463	1 ⊢ (5 gPetersenGr 1) ≃𝑙𝑔𝑟 (5 gPetersenGr 2)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  Vcvv 3459   class class class wbr 5119  ‘cfv 6530  (class class class)co 7403   ≈ cen 8954  Fincfn 8957  0cc0 11127  1c1 11128   < clt 11267   / cdiv 11892  ℕcn 12238  2c2 12293  3c3 12294  5c5 12296  ℕ₀cn0 12499  ℤcz 12586  ;cdc 12706  ℤ_≥cuz 12850  ...cfz 13522  ..^cfzo 13669  ⌈cceil 13806  ♯chash 14346  Vtxcvtx 28921  USGraphcusgr 29074   ClNeighbVtx cclnbgr 47780   ISubGr cisubgr 47821   ≃_𝑔𝑟 cgric 47837  StarGrcstgr 47911   ≃_𝑙𝑔𝑟 cgrlic 47937   gPetersenGr cgpg 47992

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-2o 8479  df-oadd 8482  df-er 8717  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9452  df-inf 9453  df-dju 9913  df-card 9951  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-div 11893  df-nn 12239  df-2 12301  df-3 12302  df-4 12303  df-5 12304  df-6 12305  df-7 12306  df-8 12307  df-9 12308  df-n0 12500  df-xnn0 12573  df-z 12587  df-dec 12707  df-uz 12851  df-rp 13007  df-ico 13366  df-fz 13523  df-fzo 13670  df-fl 13807  df-ceil 13808  df-mod 13885  df-seq 14018  df-exp 14078  df-hash 14347  df-cj 15116  df-re 15117  df-im 15118  df-sqrt 15252  df-abs 15253  df-dvds 16271  df-struct 17164  df-slot 17199  df-ndx 17211  df-base 17227  df-edgf 28914  df-vtx 28923  df-iedg 28924  df-edg 28973  df-uhgr 28983  df-ushgr 28984  df-upgr 29007  df-umgr 29008  df-uspgr 29075  df-usgr 29076  df-subgr 29193  df-nbgr 29258  df-clnbgr 47781  df-isubgr 47822  df-grim 47839  df-gric 47842  df-stgr 47912  df-grlim 47938  df-grlic 47941  df-gpg 47993

This theorem is referenced by: (None)