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Theorem numclwwlk2 30143
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 29740, 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 12838 . . . . . . . 8 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑁 ∈ β„‚)
2 2cnd 12294 . . . . . . . 8 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 2 ∈ β„‚)
31, 2npcand 11579 . . . . . . 7 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ ((𝑁 βˆ’ 2) + 2) = 𝑁)
43eqcomd 2732 . . . . . 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 481 . . . 4 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝑁 = ((𝑁 βˆ’ 2) + 2))
76oveq2d 7421 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (𝑋𝐻𝑁) = (𝑋𝐻((𝑁 βˆ’ 2) + 2)))
87fveq2d 6889 . 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 766 . . 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 12879 . . . . 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 481 . . 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 30142 . . 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 482 . . . 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 612 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin))
2211anim2i 616 . . . . 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 481 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (𝑋 ∈ 𝑉 ∧ (𝑁 βˆ’ 2) ∈ β„•))
2514, 15numclwwlkqhash 30137 . . 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 583 . 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 2772 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  {crab 3426   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  Fincfn 8941  0cc0 11112   + caddc 11115   βˆ’ cmin 11448  β„•cn 12216  2c2 12271  3c3 12272  β„€β‰₯cuz 12826  β†‘cexp 14032  β™―chash 14295  lastSclsw 14518  Vtxcvtx 28764   RegUSGraph crusgr 29322   WWalksN cwwlksn 29589  ClWWalksNOncclwwlknon 29849   FriendGraph cfrgr 30020
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-disj 5107  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-2o 8468  df-oadd 8471  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-oi 9507  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-rp 12981  df-xadd 13099  df-fz 13491  df-fzo 13634  df-seq 13973  df-exp 14033  df-hash 14296  df-word 14471  df-lsw 14519  df-concat 14527  df-s1 14552  df-substr 14597  df-pfx 14627  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15438  df-sum 15639  df-vtx 28766  df-iedg 28767  df-edg 28816  df-uhgr 28826  df-ushgr 28827  df-upgr 28850  df-umgr 28851  df-uspgr 28918  df-usgr 28919  df-fusgr 29082  df-nbgr 29098  df-vtxdg 29232  df-rgr 29323  df-rusgr 29324  df-wwlks 29593  df-wwlksn 29594  df-wwlksnon 29595  df-clwwlk 29744  df-clwwlkn 29787  df-clwwlknon 29850  df-frgr 30021
This theorem is referenced by:  numclwwlk3  30147
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