Theorem gpg5grlic 48997

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 12935	. . . 4 ⊢ 5 ∈ (ℤ≥‘3)
2		3z 12654	. . . . . . 7 ⊢ 3 ∈ ℤ
3		1lt3 12443	. . . . . . 7 ⊢ 1 < 3
4		eluz2b1 12971	. . . . . . 7 ⊢ (3 ∈ (ℤ≥‘2) ↔ (3 ∈ ℤ ∧ 1 < 3))
5	2, 3, 4	mpbir2an 724	. . . . . 6 ⊢ 3 ∈ (ℤ≥‘2)
6		fzo1lb 13771	. . . . . 6 ⊢ (1 ∈ (1..^3) ↔ 3 ∈ (ℤ≥‘2))
7	5, 6	mpbir 234	. . . . 5 ⊢ 1 ∈ (1..^3)
8		ceil5half3 48221	. . . . . . 7 ⊢ (⌈‘(5 / 2)) = 3
9	8	eqcomi 2771	. . . . . 6 ⊢ 3 = (⌈‘(5 / 2))
10	9	oveq2i 7427	. . . . 5 ⊢ (1..^3) = (1..^(⌈‘(5 / 2)))
11	7, 10	eleqtri 2860	. . . 4 ⊢ 1 ∈ (1..^(⌈‘(5 / 2)))
12		gpgusgra 48960	. . . 4 ⊢ ((5 ∈ (ℤ≥‘3) ∧ 1 ∈ (1..^(⌈‘(5 / 2)))) → (5 gPetersenGr 1) ∈ USGraph)
13	1, 11, 12	mp2an 705	. . 3 ⊢ (5 gPetersenGr 1) ∈ USGraph
14		pglem 48994	. . . 4 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
15		gpgusgra 48960	. . . 4 ⊢ ((5 ∈ (ℤ≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) → (5 gPetersenGr 2) ∈ USGraph)
16	1, 14, 15	mp2an 705	. . 3 ⊢ (5 gPetersenGr 2) ∈ USGraph
17		2eluzge1 12934	. . . . . 6 ⊢ 2 ∈ (ℤ≥‘1)
18		eluzfz1 13587	. . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → 1 ∈ (1...2))
19	17, 18	ax-mp 5	. . . . 5 ⊢ 1 ∈ (1...2)
20		gpg5order 48963	. . . . 5 ⊢ (1 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 1))) = ;10)
21	19, 20	ax-mp 5	. . . 4 ⊢ (♯‘(Vtx‘(5 gPetersenGr 1))) = ;10
22		eluzfz2 13588	. . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → 2 ∈ (1...2))
23	17, 22	ax-mp 5	. . . . 5 ⊢ 2 ∈ (1...2)
24		gpg5order 48963	. . . . 5 ⊢ (2 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10)
25	23, 24	ax-mp 5	. . . 4 ⊢ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10
26		eqtr3 2784	. . . . 5 ⊢ (((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 ∧ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10) → (♯‘(Vtx‘(5 gPetersenGr 1))) = (♯‘(Vtx‘(5 gPetersenGr 2))))
27		fvex 6895	. . . . . . 7 ⊢ (Vtx‘(5 gPetersenGr 1)) ∈ V
28		10nn0 12761	. . . . . . 7 ⊢ ;10 ∈ ℕ0
29		hashvnfin 14426	. . . . . . 7 ⊢ (((Vtx‘(5 gPetersenGr 1)) ∈ V ∧ ;10 ∈ ℕ0) → ((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 → (Vtx‘(5 gPetersenGr 1)) ∈ Fin))
30	27, 28, 29	mp2an 705	. . . . . 6 ⊢ ((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 → (Vtx‘(5 gPetersenGr 1)) ∈ Fin)
31		fvex 6895	. . . . . . 7 ⊢ (Vtx‘(5 gPetersenGr 2)) ∈ V
32		hashvnfin 14426	. . . . . . 7 ⊢ (((Vtx‘(5 gPetersenGr 2)) ∈ V ∧ ;10 ∈ ℕ0) → ((♯‘(Vtx‘(5 gPetersenGr 2))) = ;10 → (Vtx‘(5 gPetersenGr 2)) ∈ Fin))
33	31, 28, 32	mp2an 705	. . . . . 6 ⊢ ((♯‘(Vtx‘(5 gPetersenGr 2))) = ;10 → (Vtx‘(5 gPetersenGr 2)) ∈ Fin)
34		hashen 14413	. . . . . 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 608	. . . . 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 235	. . . 4 ⊢ (((♯‘(Vtx‘(5 gPetersenGr 1))) = ;10 ∧ (♯‘(Vtx‘(5 gPetersenGr 2))) = ;10) → (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2)))
37	21, 25, 36	mp2an 705	. . 3 ⊢ (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2))
38	13, 16, 37	3pm3.2i 1358	. 2 ⊢ ((5 gPetersenGr 1) ∈ USGraph ∧ (5 gPetersenGr 2) ∈ USGraph ∧ (Vtx‘(5 gPetersenGr 1)) ≈ (Vtx‘(5 gPetersenGr 2)))
39		eqid 2762	. . . . 5 ⊢ (5 gPetersenGr 1) = (5 gPetersenGr 1)
40	39	gpg5gricstgr3 48993	. . . 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 703	. . 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 2762	. . . . 5 ⊢ (5 gPetersenGr 2) = (5 gPetersenGr 2)
44	43	gpg5gricstgr3 48993	. . . 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 703	. . 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 12549	. . 3 ⊢ 3 ∈ ℕ0
48		eqid 2762	. . 3 ⊢ (Vtx‘(5 gPetersenGr 1)) = (Vtx‘(5 gPetersenGr 1))
49		eqid 2762	. . 3 ⊢ (Vtx‘(5 gPetersenGr 2)) = (Vtx‘(5 gPetersenGr 2))
50	47, 48, 49	clnbgr3stgrgrlic 48923	. 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 1490	1 ⊢ (5 gPetersenGr 1) ≃𝑙𝑔𝑟 (5 gPetersenGr 2)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   class class class wbr 5107  ‘cfv 6537  (class class class)co 7416   ≈ cen 8952  Fincfn 8955  0cc0 11127  1c1 11128   < clt 11270   / cdiv 11898  2c2 12322  3c3 12323  5c5 12325  ℕ₀cn0 12531  ℤcz 12618  ;cdc 12739  ℤ_≥cuz 12890  ...cfz 13563  ..^cfzo 13711  ⌈cceil 13854  ♯chash 14396  Vtxcvtx 29439  USGraphcusgr 29595   ClNeighbVtx cclnbgr 48721   ISubGr cisubgr 48763   ≃_𝑔𝑟 cgric 48779  StarGrcstgr 48854   ≃_𝑙𝑔𝑟 cgrlic 48880   gPetersenGr cgpg 48943

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  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 proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-oadd 8462  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9415  df-inf 9416  df-dju 9909  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-xnn0 12605  df-z 12619  df-dec 12740  df-uz 12891  df-rp 13045  df-ico 13406  df-fz 13564  df-fzo 13712  df-fl 13855  df-ceil 13856  df-mod 13933  df-seq 14068  df-exp 14128  df-hash 14397  df-cj 15188  df-re 15189  df-im 15190  df-sqrt 15324  df-abs 15325  df-dvds 16347  df-struct 17243  df-slot 17278  df-ndx 17290  df-base 17306  df-edgf 29432  df-vtx 29441  df-iedg 29442  df-edg 29491  df-uhgr 29501  df-ushgr 29502  df-upgr 29525  df-umgr 29526  df-uspgr 29596  df-usgr 29597  df-subgr 29714  df-nbgr 29779  df-clnbgr 48722  df-isubgr 48764  df-grim 48781  df-gric 48784  df-stgr 48855  df-grlim 48881  df-grlic 48884  df-gpg 48944

This theorem is used by:  lgricngricex  49032