Theorem gpg5grlic 48078

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 12820	. . . 4 ⊢ 5 ∈ (ℤ≥‘3)
2		3z 12544	. . . . . . 7 ⊢ 3 ∈ ℤ
3		1lt3 12332	. . . . . . 7 ⊢ 1 < 3
4		eluz2b1 12856	. . . . . . 7 ⊢ (3 ∈ (ℤ≥‘2) ↔ (3 ∈ ℤ ∧ 1 < 3))
5	2, 3, 4	mpbir2an 711	. . . . . 6 ⊢ 3 ∈ (ℤ≥‘2)
6		fzo1lb 13652	. . . . . 6 ⊢ (1 ∈ (1..^3) ↔ 3 ∈ (ℤ≥‘2))
7	5, 6	mpbir 231	. . . . 5 ⊢ 1 ∈ (1..^3)
8		ceil5half3 47335	. . . . . . 7 ⊢ (⌈‘(5 / 2)) = 3
9	8	eqcomi 2738	. . . . . 6 ⊢ 3 = (⌈‘(5 / 2))
10	9	oveq2i 7380	. . . . 5 ⊢ (1..^3) = (1..^(⌈‘(5 / 2)))
11	7, 10	eleqtri 2826	. . . 4 ⊢ 1 ∈ (1..^(⌈‘(5 / 2)))
12		gpgusgra 48042	. . . 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 48076	. . . 4 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
15		gpgusgra 48042	. . . 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 12819	. . . . . 6 ⊢ 2 ∈ (ℤ≥‘1)
18		eluzfz1 13470	. . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → 1 ∈ (1...2))
19	17, 18	ax-mp 5	. . . . 5 ⊢ 1 ∈ (1...2)
20		gpg5order 48045	. . . . 5 ⊢ (1 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 1))) = ;10)
21	19, 20	ax-mp 5	. . . 4 ⊢ (♯‘(Vtx‘(5 gPetersenGr 1))) = ;10
22		eluzfz2 13471	. . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → 2 ∈ (1...2))
23	17, 22	ax-mp 5	. . . . 5 ⊢ 2 ∈ (1...2)
24		gpg5order 48045	. . . . 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 6853	. . . . . . 7 ⊢ (Vtx‘(5 gPetersenGr 1)) ∈ V
28		10nn0 12645	. . . . . . 7 ⊢ ;10 ∈ ℕ0
29		hashvnfin 14303	. . . . . . 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 6853	. . . . . . 7 ⊢ (Vtx‘(5 gPetersenGr 2)) ∈ V
32		hashvnfin 14303	. . . . . . 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 14290	. . . . . 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 48075	. . . 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 48075	. . . 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 12438	. . 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 48005	. 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 3444   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369   ≈ cen 8892  Fincfn 8895  0cc0 11046  1c1 11047   < clt 11186   / cdiv 11813  2c2 12219  3c3 12220  5c5 12222  ℕ₀cn0 12420  ℤcz 12507  ;cdc 12627  ℤ_≥cuz 12771  ...cfz 13446  ..^cfzo 13593  ⌈cceil 13731  ♯chash 14273  Vtxcvtx 28977  USGraphcusgr 29130   ClNeighbVtx cclnbgr 47813   ISubGr cisubgr 47854   ≃_𝑔𝑟 cgric 47870  StarGrcstgr 47944   ≃_𝑙𝑔𝑟 cgrlic 47970   gPetersenGr cgpg 48025

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11102  ax-resscn 11103  ax-1cn 11104  ax-icn 11105  ax-addcl 11106  ax-addrcl 11107  ax-mulcl 11108  ax-mulrcl 11109  ax-mulcom 11110  ax-addass 11111  ax-mulass 11112  ax-distr 11113  ax-i2m1 11114  ax-1ne0 11115  ax-1rid 11116  ax-rnegex 11117  ax-rrecex 11118  ax-cnre 11119  ax-pre-lttri 11120  ax-pre-lttrn 11121  ax-pre-ltadd 11122  ax-pre-mulgt0 11123  ax-pre-sup 11124

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-inf 9370  df-dju 9832  df-card 9870  df-pnf 11188  df-mnf 11189  df-xr 11190  df-ltxr 11191  df-le 11192  df-sub 11385  df-neg 11386  df-div 11814  df-nn 12165  df-2 12227  df-3 12228  df-4 12229  df-5 12230  df-6 12231  df-7 12232  df-8 12233  df-9 12234  df-n0 12421  df-xnn0 12494  df-z 12508  df-dec 12628  df-uz 12772  df-rp 12930  df-ico 13290  df-fz 13447  df-fzo 13594  df-fl 13732  df-ceil 13733  df-mod 13810  df-seq 13945  df-exp 14005  df-hash 14274  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-dvds 16200  df-struct 17094  df-slot 17129  df-ndx 17141  df-base 17157  df-edgf 28970  df-vtx 28979  df-iedg 28980  df-edg 29029  df-uhgr 29039  df-ushgr 29040  df-upgr 29063  df-umgr 29064  df-uspgr 29131  df-usgr 29132  df-subgr 29249  df-nbgr 29314  df-clnbgr 47814  df-isubgr 47855  df-grim 47872  df-gric 47875  df-stgr 47945  df-grlim 47971  df-grlic 47974  df-gpg 48026

This theorem is referenced by:  lgricngricex  48113