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Mirrors > Home > MPE Home > Th. List > frrusgrord | Structured version Visualization version GIF version |
Description: If a nonempty finite friendship graph is k-regular, its order is k(k-1)+1. This corresponds to claim 3 in [Huneke] p. 2: "Next we claim that the number n of vertices in G is exactly k(k-1)+1.". Variant of frrusgrord0 30192, using the definition RegUSGraph (df-rusgr 29414). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.) |
Ref | Expression |
---|---|
frrusgrord0.v | β’ π = (VtxβπΊ) |
Ref | Expression |
---|---|
frrusgrord | β’ ((π β Fin β§ π β β ) β ((πΊ β FriendGraph β§ πΊ RegUSGraph πΎ) β (β―βπ) = ((πΎ Β· (πΎ β 1)) + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rusgrrgr 29419 | . . . . . . 7 β’ (πΊ RegUSGraph πΎ β πΊ RegGraph πΎ) | |
2 | frrusgrord0.v | . . . . . . . 8 β’ π = (VtxβπΊ) | |
3 | eqid 2725 | . . . . . . . 8 β’ (VtxDegβπΊ) = (VtxDegβπΊ) | |
4 | 2, 3 | rgrprop 29416 | . . . . . . 7 β’ (πΊ RegGraph πΎ β (πΎ β β0* β§ βπ£ β π ((VtxDegβπΊ)βπ£) = πΎ)) |
5 | 1, 4 | syl 17 | . . . . . 6 β’ (πΊ RegUSGraph πΎ β (πΎ β β0* β§ βπ£ β π ((VtxDegβπΊ)βπ£) = πΎ)) |
6 | 5 | simprd 494 | . . . . 5 β’ (πΊ RegUSGraph πΎ β βπ£ β π ((VtxDegβπΊ)βπ£) = πΎ) |
7 | 2 | frrusgrord0 30192 | . . . . 5 β’ ((πΊ β FriendGraph β§ π β Fin β§ π β β ) β (βπ£ β π ((VtxDegβπΊ)βπ£) = πΎ β (β―βπ) = ((πΎ Β· (πΎ β 1)) + 1))) |
8 | 6, 7 | syl5 34 | . . . 4 β’ ((πΊ β FriendGraph β§ π β Fin β§ π β β ) β (πΊ RegUSGraph πΎ β (β―βπ) = ((πΎ Β· (πΎ β 1)) + 1))) |
9 | 8 | 3expb 1117 | . . 3 β’ ((πΊ β FriendGraph β§ (π β Fin β§ π β β )) β (πΊ RegUSGraph πΎ β (β―βπ) = ((πΎ Β· (πΎ β 1)) + 1))) |
10 | 9 | expcom 412 | . 2 β’ ((π β Fin β§ π β β ) β (πΊ β FriendGraph β (πΊ RegUSGraph πΎ β (β―βπ) = ((πΎ Β· (πΎ β 1)) + 1)))) |
11 | 10 | impd 409 | 1 β’ ((π β Fin β§ π β β ) β ((πΊ β FriendGraph β§ πΊ RegUSGraph πΎ) β (β―βπ) = ((πΎ Β· (πΎ β 1)) + 1))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 βwral 3051 β c0 4318 class class class wbr 5143 βcfv 6542 (class class class)co 7415 Fincfn 8960 1c1 11137 + caddc 11139 Β· cmul 11141 β cmin 11472 β0*cxnn0 12572 β―chash 14319 Vtxcvtx 28851 VtxDegcvtxdg 29321 RegGraph crgr 29411 RegUSGraph crusgr 29412 FriendGraph cfrgr 30110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-ac2 10484 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-disj 5109 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-oadd 8487 df-er 8721 df-map 8843 df-pm 8844 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-oi 9531 df-dju 9922 df-card 9960 df-ac 10137 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-xnn0 12573 df-z 12587 df-uz 12851 df-rp 13005 df-xadd 13123 df-fz 13515 df-fzo 13658 df-seq 13997 df-exp 14057 df-hash 14320 df-word 14495 df-concat 14551 df-s1 14576 df-s2 14829 df-s3 14830 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-clim 15462 df-sum 15663 df-vtx 28853 df-iedg 28854 df-edg 28903 df-uhgr 28913 df-ushgr 28914 df-upgr 28937 df-umgr 28938 df-uspgr 29005 df-usgr 29006 df-fusgr 29172 df-nbgr 29188 df-vtxdg 29322 df-rgr 29413 df-rusgr 29414 df-wlks 29455 df-wlkson 29456 df-trls 29548 df-trlson 29549 df-pths 29572 df-spths 29573 df-pthson 29574 df-spthson 29575 df-wwlks 29683 df-wwlksn 29684 df-wwlksnon 29685 df-wspthsn 29686 df-wspthsnon 29687 df-frgr 30111 |
This theorem is referenced by: numclwwlk7 30243 frgrregord013 30247 |
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