Theorem gpg5grlic 48721

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 12886	. . . 4 ⊢ 5 ∈ (ℤ≥‘3)
2		3z 12606	. . . . . . 7 ⊢ 3 ∈ ℤ
3		1lt3 12395	. . . . . . 7 ⊢ 1 < 3
4		eluz2b1 12922	. . . . . . 7 ⊢ (3 ∈ (ℤ≥‘2) ↔ (3 ∈ ℤ ∧ 1 < 3))
5	2, 3, 4	mpbir2an 721	. . . . . 6 ⊢ 3 ∈ (ℤ≥‘2)
6		fzo1lb 13721	. . . . . 6 ⊢ (1 ∈ (1..^3) ↔ 3 ∈ (ℤ≥‘2))
7	5, 6	mpbir 233	. . . . 5 ⊢ 1 ∈ (1..^3)
8		ceil5half3 47945	. . . . . . 7 ⊢ (⌈‘(5 / 2)) = 3
9	8	eqcomi 2773	. . . . . 6 ⊢ 3 = (⌈‘(5 / 2))
10	9	oveq2i 7409	. . . . 5 ⊢ (1..^3) = (1..^(⌈‘(5 / 2)))
11	7, 10	eleqtri 2862	. . . 4 ⊢ 1 ∈ (1..^(⌈‘(5 / 2)))
12		gpgusgra 48684	. . . 4 ⊢ ((5 ∈ (ℤ≥‘3) ∧ 1 ∈ (1..^(⌈‘(5 / 2)))) → (5 gPetersenGr 1) ∈ USGraph)
13	1, 11, 12	mp2an 702	. . 3 ⊢ (5 gPetersenGr 1) ∈ USGraph
14		pglem 48718	. . . 4 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
15		gpgusgra 48684	. . . 4 ⊢ ((5 ∈ (ℤ≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) → (5 gPetersenGr 2) ∈ USGraph)
16	1, 14, 15	mp2an 702	. . 3 ⊢ (5 gPetersenGr 2) ∈ USGraph
17		2eluzge1 12885	. . . . . 6 ⊢ 2 ∈ (ℤ≥‘1)
18		eluzfz1 13538	. . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → 1 ∈ (1...2))
19	17, 18	ax-mp 5	. . . . 5 ⊢ 1 ∈ (1...2)
20		gpg5order 48687	. . . . 5 ⊢ (1 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 1))) = ;10)
21	19, 20	ax-mp 5	. . . 4 ⊢ (♯‘(Vtx‘(5 gPetersenGr 1))) = ;10
22		eluzfz2 13539	. . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → 2 ∈ (1...2))
23	17, 22	ax-mp 5	. . . . 5 ⊢ 2 ∈ (1...2)
24		gpg5order 48687	. . . . 5 ⊢ (2 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10)
25	23, 24	ax-mp 5	. . . 4 ⊢ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10
26		eqtr3 2786	. . . . 5 ⊢ (((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 ∧ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10) → (♯‘(Vtx‘(5 gPetersenGr 1))) = (♯‘(Vtx‘(5 gPetersenGr 2))))
27		fvex 6882	. . . . . . 7 ⊢ (Vtx‘(5 gPetersenGr 1)) ∈ V
28		10nn0 12712	. . . . . . 7 ⊢ ;10 ∈ ℕ0
29		hashvnfin 14375	. . . . . . 7 ⊢ (((Vtx‘(5 gPetersenGr 1)) ∈ V ∧ ;10 ∈ ℕ0) → ((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 → (Vtx‘(5 gPetersenGr 1)) ∈ Fin))
30	27, 28, 29	mp2an 702	. . . . . 6 ⊢ ((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 → (Vtx‘(5 gPetersenGr 1)) ∈ Fin)
31		fvex 6882	. . . . . . 7 ⊢ (Vtx‘(5 gPetersenGr 2)) ∈ V
32		hashvnfin 14375	. . . . . . 7 ⊢ (((Vtx‘(5 gPetersenGr 2)) ∈ V ∧ ;10 ∈ ℕ0) → ((♯‘(Vtx‘(5 gPetersenGr 2))) = ;10 → (Vtx‘(5 gPetersenGr 2)) ∈ Fin))
33	31, 28, 32	mp2an 702	. . . . . 6 ⊢ ((♯‘(Vtx‘(5 gPetersenGr 2))) = ;10 → (Vtx‘(5 gPetersenGr 2)) ∈ Fin)
34		hashen 14362	. . . . . 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 605	. . . . 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 234	. . . 4 ⊢ (((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 ∧ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10) → (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2)))
37	21, 25, 36	mp2an 702	. . 3 ⊢ (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2))
38	13, 16, 37	3pm3.2i 1354	. 2 ⊢ ((5 gPetersenGr 1) ∈ USGraph ∧ (5 gPetersenGr 2) ∈ USGraph ∧ (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2)))
39		eqid 2764	. . . . 5 ⊢ (5 gPetersenGr 1) = (5 gPetersenGr 1)
40	39	gpg5gricstgr3 48717	. . . 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 700	. . 3 ⊢ (𝑣 ∈ (Vtx‘(5 gPetersenGr 1)) → ((5 gPetersenGr 1) ISubGr ((5 gPetersenGr 1) ClNeighbVtx 𝑣)) ≃𝑔𝑟 (StarGr‘3))
42	41	rgen 3080	. 2 ⊢ ∀𝑣 ∈ (Vtx‘(5 gPetersenGr 1))((5 gPetersenGr 1) ISubGr ((5 gPetersenGr 1) ClNeighbVtx 𝑣)) ≃𝑔𝑟 (StarGr‘3)
43		eqid 2764	. . . . 5 ⊢ (5 gPetersenGr 2) = (5 gPetersenGr 2)
44	43	gpg5gricstgr3 48717	. . . 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 700	. . 3 ⊢ (𝑤 ∈ (Vtx‘(5 gPetersenGr 2)) → ((5 gPetersenGr 2) ISubGr ((5 gPetersenGr 2) ClNeighbVtx 𝑤)) ≃𝑔𝑟 (StarGr‘3))
46	45	rgen 3080	. 2 ⊢ ∀𝑤 ∈ (Vtx‘(5 gPetersenGr 2))((5 gPetersenGr 2) ISubGr ((5 gPetersenGr 2) ClNeighbVtx 𝑤)) ≃𝑔𝑟 (StarGr‘3)
47		3nn0 12501	. . 3 ⊢ 3 ∈ ℕ0
48		eqid 2764	. . 3 ⊢ (Vtx‘(5 gPetersenGr 1)) = (Vtx‘(5 gPetersenGr 1))
49		eqid 2764	. . 3 ⊢ (Vtx‘(5 gPetersenGr 2)) = (Vtx‘(5 gPetersenGr 2))
50	47, 48, 49	clnbgr3stgrgrlic 48647	. 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 1484	1 ⊢ (5 gPetersenGr 1) ≃𝑙𝑔𝑟 (5 gPetersenGr 2)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078  Vcvv 3456   class class class wbr 5102  ‘cfv 6523  (class class class)co 7398   ≈ cen 8926  Fincfn 8929  0cc0 11075  1c1 11076   < clt 11218   / cdiv 11846  2c2 12274  3c3 12275  5c5 12277  ℕ₀cn0 12483  ℤcz 12570  ;cdc 12690  ℤ_≥cuz 12841  ...cfz 13514  ..^cfzo 13661  ⌈cceil 13803  ♯chash 14345  Vtxcvtx 29199  USGraphcusgr 29352   ClNeighbVtx cclnbgr 48445   ISubGr cisubgr 48487   ≃_𝑔𝑟 cgric 48503  StarGrcstgr 48578   ≃_𝑙𝑔𝑟 cgrlic 48604   gPetersenGr cgpg 48667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-oadd 8443  df-er 8680  df-map 8812  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-sup 9390  df-inf 9391  df-dju 9861  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-div 11847  df-nn 12213  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12484  df-xnn0 12557  df-z 12571  df-dec 12691  df-uz 12842  df-rp 12996  df-ico 13357  df-fz 13515  df-fzo 13662  df-fl 13804  df-ceil 13805  df-mod 13882  df-seq 14017  df-exp 14077  df-hash 14346  df-cj 15128  df-re 15129  df-im 15130  df-sqrt 15264  df-abs 15265  df-dvds 16289  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17248  df-edgf 29192  df-vtx 29201  df-iedg 29202  df-edg 29251  df-uhgr 29261  df-ushgr 29262  df-upgr 29285  df-umgr 29286  df-uspgr 29353  df-usgr 29354  df-subgr 29471  df-nbgr 29536  df-clnbgr 48446  df-isubgr 48488  df-grim 48505  df-gric 48508  df-stgr 48579  df-grlim 48605  df-grlic 48608  df-gpg 48668

This theorem is referenced by:  lgricngricex  48756