Theorem frgrregorufrg 30274

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 30273 with generalization. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.)

Hypotheses

Ref	Expression
frgrregorufrg.v	⊢ 𝑉 = (Vtx‘𝐺)
frgrregorufrg.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
frgrregorufrg	⊢ (𝐺 ∈ FriendGraph → ∀𝑘 ∈ ℕ0 (∃𝑎 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))

Distinct variable groups:   𝐺,𝑎,𝑘,𝑣,𝑤   𝐸,𝑎,𝑣   𝑉,𝑎,𝑣,𝑤   𝑘,𝑎,𝑣,𝑤

Allowed substitution hints:   𝐸(𝑤,𝑘)   𝑉(𝑘)

Proof of Theorem frgrregorufrg

Step	Hyp	Ref	Expression
1		frgrregorufrg.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		frgrregorufrg.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
3		eqid 2729	. . . . 5 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
4	1, 2, 3	frgrregorufr 30273	. . . 4 ⊢ (𝐺 ∈ FriendGraph → (∃𝑎 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
5	4	adantr 480	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∃𝑎 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
6		frgrusgr 30209	. . . . 5 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
7		nn0xnn0 12461	. . . . 5 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℕ0*)
8	1, 3	usgreqdrusgr 29518	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → 𝐺 RegUSGraph 𝑘)
9	8	3expia 1121	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑘 ∈ ℕ0*) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 → 𝐺 RegUSGraph 𝑘))
10	6, 7, 9	syl2an 596	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 → 𝐺 RegUSGraph 𝑘))
11	10	orim1d 967	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → ((∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
12	5, 11	syld 47	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∃𝑎 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
13	12	ralrimiva 3121	1 ⊢ (𝐺 ∈ FriendGraph → ∀𝑘 ∈ ℕ0 (∃𝑎 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∖ cdif 3900  {csn 4577  {cpr 4579   class class class wbr 5092  ‘cfv 6482  ℕ₀cn0 12384  ℕ₀^*cxnn0 12457  Vtxcvtx 28945  Edgcedg 28996  USGraphcusgr 29098  VtxDegcvtxdg 29415   RegUSGraph crusgr 29506   FriendGraph cfrgr 30206

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-n0 12385  df-xnn0 12458  df-z 12472  df-uz 12736  df-xadd 13015  df-fz 13411  df-hash 14238  df-edg 28997  df-uhgr 29007  df-ushgr 29008  df-upgr 29031  df-umgr 29032  df-uspgr 29099  df-usgr 29100  df-nbgr 29282  df-vtxdg 29416  df-rgr 29507  df-rusgr 29508  df-frgr 30207

This theorem is referenced by:  friendshipgt3  30346