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Theorem numclwwlk2 30233
Description: Statement 10 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v is k^(n-2) - f(n-2)." According to rusgrnumwlkg 29830, we have k^(n-2) different walks of length (n-2): v(0) ... v(n-2). From this number, the number of closed walks of length (n-2), which is f(n-2) per definition, must be subtracted, because for these walks v(n-2) =/= v(0) = v would hold. Because of the friendship condition, there is exactly one vertex v(n-1) which is a neighbor of v(n-2) as well as of v(n)=v=v(0), because v(n-2) and v(n)=v are different, so the number of walks v(0) ... v(n-2) is identical with the number of walks v(0) ... v(n), that means each (not closed) walk v(0) ... v(n-2) can be extended by two edges to a closed walk v(0) ... v(n)=v=v(0) in exactly one way. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.) (Revised by AV, 1-May-2022.)
Hypotheses
Ref Expression
numclwwlk.v 𝑉 = (Vtxβ€˜πΊ)
numclwwlk.q 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})
numclwwlk.h 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})
Assertion
Ref Expression
numclwwlk2 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜(𝑋𝐻𝑁)) = ((𝐾↑(𝑁 βˆ’ 2)) βˆ’ (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)))))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑀   𝑀,𝐾   𝑀,𝑉
Allowed substitution hints:   𝑄(𝑀,𝑣,𝑛)   𝐻(𝑀,𝑣,𝑛)   𝐾(𝑣,𝑛)

Proof of Theorem numclwwlk2
StepHypRef Expression
1 eluzelcn 12862 . . . . . . . 8 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑁 ∈ β„‚)
2 2cnd 12318 . . . . . . . 8 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 2 ∈ β„‚)
31, 2npcand 11603 . . . . . . 7 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ ((𝑁 βˆ’ 2) + 2) = 𝑁)
43eqcomd 2731 . . . . . 6 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑁 = ((𝑁 βˆ’ 2) + 2))
543ad2ant3 1132 . . . . 5 ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑁 = ((𝑁 βˆ’ 2) + 2))
65adantl 480 . . . 4 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝑁 = ((𝑁 βˆ’ 2) + 2))
76oveq2d 7431 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (𝑋𝐻𝑁) = (𝑋𝐻((𝑁 βˆ’ 2) + 2)))
87fveq2d 6895 . 2 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜(𝑋𝐻𝑁)) = (β™―β€˜(𝑋𝐻((𝑁 βˆ’ 2) + 2))))
9 simplr 767 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝐺 ∈ FriendGraph )
10 simpr2 1192 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝑋 ∈ 𝑉)
11 uz3m2nn 12903 . . . . 5 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (𝑁 βˆ’ 2) ∈ β„•)
12113ad2ant3 1132 . . . 4 ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑁 βˆ’ 2) ∈ β„•)
1312adantl 480 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (𝑁 βˆ’ 2) ∈ β„•)
14 numclwwlk.v . . . 4 𝑉 = (Vtxβ€˜πΊ)
15 numclwwlk.q . . . 4 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})
16 numclwwlk.h . . . 4 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})
1714, 15, 16numclwwlk2lem3 30232 . . 3 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ (𝑁 βˆ’ 2) ∈ β„•) β†’ (β™―β€˜(𝑋𝑄(𝑁 βˆ’ 2))) = (β™―β€˜(𝑋𝐻((𝑁 βˆ’ 2) + 2))))
189, 10, 13, 17syl3anc 1368 . 2 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜(𝑋𝑄(𝑁 βˆ’ 2))) = (β™―β€˜(𝑋𝐻((𝑁 βˆ’ 2) + 2))))
19 simpl 481 . . . 4 ((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) β†’ 𝐺 RegUSGraph 𝐾)
20 simp1 1133 . . . 4 ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑉 ∈ Fin)
2119, 20anim12i 611 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin))
2211anim2i 615 . . . . 5 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑋 ∈ 𝑉 ∧ (𝑁 βˆ’ 2) ∈ β„•))
23223adant1 1127 . . . 4 ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑋 ∈ 𝑉 ∧ (𝑁 βˆ’ 2) ∈ β„•))
2423adantl 480 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (𝑋 ∈ 𝑉 ∧ (𝑁 βˆ’ 2) ∈ β„•))
2514, 15numclwwlkqhash 30227 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ (𝑁 βˆ’ 2) ∈ β„•)) β†’ (β™―β€˜(𝑋𝑄(𝑁 βˆ’ 2))) = ((𝐾↑(𝑁 βˆ’ 2)) βˆ’ (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)))))
2621, 24, 25syl2anc 582 . 2 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜(𝑋𝑄(𝑁 βˆ’ 2))) = ((𝐾↑(𝑁 βˆ’ 2)) βˆ’ (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)))))
278, 18, 263eqtr2d 2771 1 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜(𝑋𝐻𝑁)) = ((𝐾↑(𝑁 βˆ’ 2)) βˆ’ (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  {crab 3419   class class class wbr 5143  β€˜cfv 6542  (class class class)co 7415   ∈ cmpo 7417  Fincfn 8960  0cc0 11136   + caddc 11139   βˆ’ cmin 11472  β„•cn 12240  2c2 12295  3c3 12296  β„€β‰₯cuz 12850  β†‘cexp 14056  β™―chash 14319  lastSclsw 14542  Vtxcvtx 28851   RegUSGraph crusgr 29412   WWalksN cwwlksn 29679  ClWWalksNOncclwwlknon 29939   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-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-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-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-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-lsw 14543  df-concat 14551  df-s1 14576  df-substr 14621  df-pfx 14651  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-wwlks 29683  df-wwlksn 29684  df-wwlksnon 29685  df-clwwlk 29834  df-clwwlkn 29877  df-clwwlknon 29940  df-frgr 30111
This theorem is referenced by:  numclwwlk3  30237
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