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Theorem frgrregorufrg 41503
Description: If there is a vertex having degree 𝑘 for each nonnegative integer 𝑘 in a friendship graph, then there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "Suppose there is a vertex of degree k > 1. ... all vertices have degree k, unless there is a universal friend. ... It follows that G is k-regular, i.e., the degree of every vertex is k". Variant of frgrregorufr 41488 with generalization. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Revised by AV, 26-May-2021.)
Hypotheses
Ref Expression
frrusgrord0.v 𝑉 = (Vtx‘𝐺)
frgrregorufrg.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
frgrregorufrg (𝐺 ∈ FriendGraph → ∀𝑘 ∈ ℕ0 (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
Distinct variable groups:   𝑣,𝐺   𝑣,𝑉   𝐺,𝑎,𝑘,𝑤,𝑣   𝐸,𝑎,𝑣   𝑉,𝑎,𝑤   𝑘,𝑎,𝑣,𝑤
Allowed substitution hints:   𝐸(𝑤,𝑘)   𝑉(𝑘)

Proof of Theorem frgrregorufrg
StepHypRef Expression
1 frrusgrord0.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 frgrregorufrg.e . . . . 5 𝐸 = (Edg‘𝐺)
3 eqid 2605 . . . . 5 (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
41, 2, 3frgrregorufr 41488 . . . 4 (𝐺 ∈ FriendGraph → (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
54adantr 479 . . 3 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
6 frgrusgr 41430 . . . . . . . 8 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
76adantr 479 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → 𝐺 ∈ USGraph )
87adantr 479 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → 𝐺 ∈ USGraph )
9 nn0xnn0 40199 . . . . . . . . 9 (𝑘 ∈ ℕ0𝑘 ∈ ℕ0*)
109adantl 480 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0*)
1110anim1i 589 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → (𝑘 ∈ ℕ0* ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘))
121, 3isrgr 40757 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (𝐺 RegGraph 𝑘 ↔ (𝑘 ∈ ℕ0* ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘)))
1312adantr 479 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → (𝐺 RegGraph 𝑘 ↔ (𝑘 ∈ ℕ0* ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘)))
1411, 13mpbird 245 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → 𝐺 RegGraph 𝑘)
15 isrusgr 40759 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (𝐺 RegUSGraph 𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝑘)))
1615adantr 479 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → (𝐺 RegUSGraph 𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝑘)))
178, 14, 16mpbir2and 958 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → 𝐺 RegUSGraph 𝑘)
1817ex 448 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘𝐺 RegUSGraph 𝑘))
1918orim1d 879 . . 3 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → ((∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
205, 19syld 45 . 2 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
2120ralrimiva 2944 1 (𝐺 ∈ FriendGraph → ∀𝑘 ∈ ℕ0 (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1975  wral 2891  wrex 2892  cdif 3532  {csn 4120  {cpr 4122   class class class wbr 4573  cfv 5786  0cn0 11135  0*cxnn0 40195  Vtxcvtx 40227  Edgcedga 40349   USGraph cusgr 40377  VtxDegcvtxdg 40679   RegGraph crgr 40753   RegUSGraph crusgr 40754   FriendGraph cfrgr 41426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-card 8621  df-cda 8846  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-n0 11136  df-z 11207  df-uz 11516  df-xadd 11775  df-fz 12149  df-hash 12931  df-xnn0 40196  df-uhgr 40278  df-ushgr 40279  df-upgr 40306  df-umgr 40307  df-edga 40350  df-uspgr 40378  df-usgr 40379  df-nbgr 40552  df-vtxdg 40680  df-rgr 40755  df-rusgr 40756  df-frgr 41427
This theorem is referenced by:  av-friendshipgt3  41550
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