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Mirrors > Home > MPE Home > Th. List > frgrregorufr | Structured version Visualization version GIF version |
Description: If there is a vertex having degree πΎ for each (nonnegative integer) πΎ in a friendship graph, then either all vertices have degree πΎ or 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". (Contributed by Alexander van der Vekens, 1-Jan-2018.) |
Ref | Expression |
---|---|
frgrregorufr0.v | β’ π = (VtxβπΊ) |
frgrregorufr0.e | β’ πΈ = (EdgβπΊ) |
frgrregorufr0.d | β’ π· = (VtxDegβπΊ) |
Ref | Expression |
---|---|
frgrregorufr | β’ (πΊ β FriendGraph β (βπ β π (π·βπ) = πΎ β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrregorufr0.v | . . 3 β’ π = (VtxβπΊ) | |
2 | frgrregorufr0.e | . . 3 β’ πΈ = (EdgβπΊ) | |
3 | frgrregorufr0.d | . . 3 β’ π· = (VtxDegβπΊ) | |
4 | 1, 2, 3 | frgrregorufr0 29310 | . 2 β’ (πΊ β FriendGraph β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ)) |
5 | orc 866 | . . . 4 β’ (βπ£ β π (π·βπ£) = πΎ β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ)) | |
6 | 5 | a1d 25 | . . 3 β’ (βπ£ β π (π·βπ£) = πΎ β (βπ β π (π·βπ) = πΎ β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
7 | fveq2 6847 | . . . . . . . 8 β’ (π£ = π β (π·βπ£) = (π·βπ)) | |
8 | 7 | neeq1d 3004 | . . . . . . 7 β’ (π£ = π β ((π·βπ£) β πΎ β (π·βπ) β πΎ)) |
9 | 8 | rspcva 3582 | . . . . . 6 β’ ((π β π β§ βπ£ β π (π·βπ£) β πΎ) β (π·βπ) β πΎ) |
10 | df-ne 2945 | . . . . . . 7 β’ ((π·βπ) β πΎ β Β¬ (π·βπ) = πΎ) | |
11 | pm2.21 123 | . . . . . . 7 β’ (Β¬ (π·βπ) = πΎ β ((π·βπ) = πΎ β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) | |
12 | 10, 11 | sylbi 216 | . . . . . 6 β’ ((π·βπ) β πΎ β ((π·βπ) = πΎ β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
13 | 9, 12 | syl 17 | . . . . 5 β’ ((π β π β§ βπ£ β π (π·βπ£) β πΎ) β ((π·βπ) = πΎ β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
14 | 13 | ancoms 460 | . . . 4 β’ ((βπ£ β π (π·βπ£) β πΎ β§ π β π) β ((π·βπ) = πΎ β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
15 | 14 | rexlimdva 3153 | . . 3 β’ (βπ£ β π (π·βπ£) β πΎ β (βπ β π (π·βπ) = πΎ β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
16 | olc 867 | . . . 4 β’ (βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ)) | |
17 | 16 | a1d 25 | . . 3 β’ (βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ β (βπ β π (π·βπ) = πΎ β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
18 | 6, 15, 17 | 3jaoi 1428 | . 2 β’ ((βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ) β (βπ β π (π·βπ) = πΎ β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
19 | 4, 18 | syl 17 | 1 β’ (πΊ β FriendGraph β (βπ β π (π·βπ) = πΎ β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β¨ wo 846 β¨ w3o 1087 = wceq 1542 β wcel 2107 β wne 2944 βwral 3065 βwrex 3074 β cdif 3912 {csn 4591 {cpr 4593 βcfv 6501 Vtxcvtx 27989 Edgcedg 28040 VtxDegcvtxdg 28455 FriendGraph cfrgr 29244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-xadd 13041 df-fz 13432 df-hash 14238 df-edg 28041 df-uhgr 28051 df-ushgr 28052 df-upgr 28075 df-umgr 28076 df-uspgr 28143 df-usgr 28144 df-nbgr 28323 df-vtxdg 28456 df-frgr 29245 |
This theorem is referenced by: frgrregorufrg 29312 |
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