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Mirrors > Home > MPE Home > Th. List > frgrregorufrg | Structured version Visualization version GIF version |
Description: If there is a vertex having degree π for each nonnegative integer π 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 30109 with generalization. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.) |
Ref | Expression |
---|---|
frgrregorufrg.v | β’ π = (VtxβπΊ) |
frgrregorufrg.e | β’ πΈ = (EdgβπΊ) |
Ref | Expression |
---|---|
frgrregorufrg | β’ (πΊ β FriendGraph β βπ β β0 (βπ β π ((VtxDegβπΊ)βπ) = π β (πΊ RegUSGraph π β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrregorufrg.v | . . . . 5 β’ π = (VtxβπΊ) | |
2 | frgrregorufrg.e | . . . . 5 β’ πΈ = (EdgβπΊ) | |
3 | eqid 2727 | . . . . 5 β’ (VtxDegβπΊ) = (VtxDegβπΊ) | |
4 | 1, 2, 3 | frgrregorufr 30109 | . . . 4 β’ (πΊ β FriendGraph β (βπ β π ((VtxDegβπΊ)βπ) = π β (βπ£ β π ((VtxDegβπΊ)βπ£) = π β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
5 | 4 | adantr 480 | . . 3 β’ ((πΊ β FriendGraph β§ π β β0) β (βπ β π ((VtxDegβπΊ)βπ) = π β (βπ£ β π ((VtxDegβπΊ)βπ£) = π β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
6 | frgrusgr 30045 | . . . . 5 β’ (πΊ β FriendGraph β πΊ β USGraph) | |
7 | nn0xnn0 12564 | . . . . 5 β’ (π β β0 β π β β0*) | |
8 | 1, 3 | usgreqdrusgr 29356 | . . . . . 6 β’ ((πΊ β USGraph β§ π β β0* β§ βπ£ β π ((VtxDegβπΊ)βπ£) = π) β πΊ RegUSGraph π) |
9 | 8 | 3expia 1119 | . . . . 5 β’ ((πΊ β USGraph β§ π β β0*) β (βπ£ β π ((VtxDegβπΊ)βπ£) = π β πΊ RegUSGraph π)) |
10 | 6, 7, 9 | syl2an 595 | . . . 4 β’ ((πΊ β FriendGraph β§ π β β0) β (βπ£ β π ((VtxDegβπΊ)βπ£) = π β πΊ RegUSGraph π)) |
11 | 10 | orim1d 964 | . . 3 β’ ((πΊ β FriendGraph β§ π β β0) β ((βπ£ β π ((VtxDegβπΊ)βπ£) = π β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ) β (πΊ RegUSGraph π β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
12 | 5, 11 | syld 47 | . 2 β’ ((πΊ β FriendGraph β§ π β β0) β (βπ β π ((VtxDegβπΊ)βπ) = π β (πΊ RegUSGraph π β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
13 | 12 | ralrimiva 3141 | 1 β’ (πΊ β FriendGraph β βπ β β0 (βπ β π ((VtxDegβπΊ)βπ) = π β (πΊ RegUSGraph π β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β¨ wo 846 = wceq 1534 β wcel 2099 βwral 3056 βwrex 3065 β cdif 3941 {csn 4624 {cpr 4626 class class class wbr 5142 βcfv 6542 β0cn0 12488 β0*cxnn0 12560 Vtxcvtx 28783 Edgcedg 28834 USGraphcusgr 28936 VtxDegcvtxdg 29253 RegUSGraph crusgr 29344 FriendGraph cfrgr 30042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-dju 9910 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-n0 12489 df-xnn0 12561 df-z 12575 df-uz 12839 df-xadd 13111 df-fz 13503 df-hash 14308 df-edg 28835 df-uhgr 28845 df-ushgr 28846 df-upgr 28869 df-umgr 28870 df-uspgr 28937 df-usgr 28938 df-nbgr 29120 df-vtxdg 29254 df-rgr 29345 df-rusgr 29346 df-frgr 30043 |
This theorem is referenced by: friendshipgt3 30182 |
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