Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frgrregorufrg Structured version   Visualization version   GIF version

Theorem frgrregorufrg 41505
 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 41490 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 2610 . . . . 5 (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
41, 2, 3frgrregorufr 41490 . . . 4 (𝐺 ∈ FriendGraph → (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
54adantr 480 . . 3 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
6 frgrusgr 41432 . . . . . . . 8 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
76adantr 480 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → 𝐺 ∈ USGraph )
87adantr 480 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → 𝐺 ∈ USGraph )
9 nn0xnn0 11244 . . . . . . . . 9 (𝑘 ∈ ℕ0𝑘 ∈ ℕ0*)
109adantl 481 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0*)
1110anim1i 590 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → (𝑘 ∈ ℕ0* ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘))
121, 3isrgr 40759 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (𝐺 RegGraph 𝑘 ↔ (𝑘 ∈ ℕ0* ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘)))
1312adantr 480 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → (𝐺 RegGraph 𝑘 ↔ (𝑘 ∈ ℕ0* ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘)))
1411, 13mpbird 246 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → 𝐺 RegGraph 𝑘)
15 isrusgr 40761 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (𝐺 RegUSGraph 𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝑘)))
1615adantr 480 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → (𝐺 RegUSGraph 𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝑘)))
178, 14, 16mpbir2and 959 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → 𝐺 RegUSGraph 𝑘)
1817ex 449 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘𝐺 RegUSGraph 𝑘))
1918orim1d 880 . . 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 46 . 2 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
2120ralrimiva 2949 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 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537  {csn 4125  {cpr 4127   class class class wbr 4583  ‘cfv 5804  ℕ0cn0 11169  ℕ0*cxnn0 11240  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379  VtxDegcvtxdg 40681   RegGraph crgr 40755   RegUSGraph crusgr 40756   FriendGraph cfrgr 41428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-uhgr 25724  df-ushgr 25725  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-nbgr 40554  df-vtxdg 40682  df-rgr 40757  df-rusgr 40758  df-frgr 41429 This theorem is referenced by:  av-friendshipgt3  41552
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