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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 29590, using the definition RegUSGraph (df-rusgr 28812). (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 28817 | . . . . . . 7 β’ (πΊ RegUSGraph πΎ β πΊ RegGraph πΎ) | |
2 | frrusgrord0.v | . . . . . . . 8 β’ π = (VtxβπΊ) | |
3 | eqid 2732 | . . . . . . . 8 β’ (VtxDegβπΊ) = (VtxDegβπΊ) | |
4 | 2, 3 | rgrprop 28814 | . . . . . . 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 496 | . . . . 5 β’ (πΊ RegUSGraph πΎ β βπ£ β π ((VtxDegβπΊ)βπ£) = πΎ) |
7 | 2 | frrusgrord0 29590 | . . . . 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 1120 | . . 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 414 | . 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 411 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 βwral 3061 β c0 4322 class class class wbr 5148 βcfv 6543 (class class class)co 7408 Fincfn 8938 1c1 11110 + caddc 11112 Β· cmul 11114 β cmin 11443 β0*cxnn0 12543 β―chash 14289 Vtxcvtx 28253 VtxDegcvtxdg 28719 RegGraph crgr 28809 RegUSGraph crusgr 28810 FriendGraph cfrgr 29508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-ac2 10457 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-oadd 8469 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-oi 9504 df-dju 9895 df-card 9933 df-ac 10110 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-xnn0 12544 df-z 12558 df-uz 12822 df-rp 12974 df-xadd 13092 df-fz 13484 df-fzo 13627 df-seq 13966 df-exp 14027 df-hash 14290 df-word 14464 df-concat 14520 df-s1 14545 df-s2 14798 df-s3 14799 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-clim 15431 df-sum 15632 df-vtx 28255 df-iedg 28256 df-edg 28305 df-uhgr 28315 df-ushgr 28316 df-upgr 28339 df-umgr 28340 df-uspgr 28407 df-usgr 28408 df-fusgr 28571 df-nbgr 28587 df-vtxdg 28720 df-rgr 28811 df-rusgr 28812 df-wlks 28853 df-wlkson 28854 df-trls 28946 df-trlson 28947 df-pths 28970 df-spths 28971 df-pthson 28972 df-spthson 28973 df-wwlks 29081 df-wwlksn 29082 df-wwlksnon 29083 df-wspthsn 29084 df-wspthsnon 29085 df-frgr 29509 |
This theorem is referenced by: numclwwlk7 29641 frgrregord013 29645 |
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