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Theorem numclwlk1 30133
Description: Statement 9 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 kf(n-2)". Since 𝐺 is k-regular, the vertex v(n-2) = v has k neighbors v(n-1), so there are k walks from v(n-2) = v to v(n) = v (via each of v's neighbors) completing each of the f(n-2) walks from v=v(0) to v(n-2)=v. This theorem holds even for k=0. (Contributed by AV, 23-May-2022.)
Hypotheses
Ref Expression
numclwlk1.v 𝑉 = (Vtxβ€˜πΊ)
numclwlk1.c 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)}
numclwlk1.f 𝐹 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = (𝑁 βˆ’ 2) ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}
Assertion
Ref Expression
numclwlk1 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2))) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)))
Distinct variable groups:   𝑀,𝐺   𝑀,𝐾   𝑀,𝑁   𝑀,𝑉   𝑀,𝑋   𝑀,𝐢   𝑀,𝐹

Proof of Theorem numclwlk1
StepHypRef Expression
1 uzp1 12867 . . . 4 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ (𝑁 = 2 ∨ 𝑁 ∈ (β„€β‰₯β€˜(2 + 1))))
2 numclwlk1.v . . . . . . . 8 𝑉 = (Vtxβ€˜πΊ)
3 numclwlk1.c . . . . . . . 8 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)}
4 numclwlk1.f . . . . . . . 8 𝐹 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = (𝑁 βˆ’ 2) ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}
52, 3, 4numclwlk1lem1 30131 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)))
65expcom 413 . . . . . 6 ((𝑋 ∈ 𝑉 ∧ 𝑁 = 2) β†’ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ))))
76expcom 413 . . . . 5 (𝑁 = 2 β†’ (𝑋 ∈ 𝑉 β†’ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)))))
82, 3, 4numclwlk1lem2 30132 . . . . . . . 8 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)))
98expcom 413 . . . . . . 7 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ))))
109expcom 413 . . . . . 6 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (𝑋 ∈ 𝑉 β†’ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)))))
11 2p1e3 12358 . . . . . . 7 (2 + 1) = 3
1211fveq2i 6888 . . . . . 6 (β„€β‰₯β€˜(2 + 1)) = (β„€β‰₯β€˜3)
1310, 12eleq2s 2845 . . . . 5 (𝑁 ∈ (β„€β‰₯β€˜(2 + 1)) β†’ (𝑋 ∈ 𝑉 β†’ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)))))
147, 13jaoi 854 . . . 4 ((𝑁 = 2 ∨ 𝑁 ∈ (β„€β‰₯β€˜(2 + 1))) β†’ (𝑋 ∈ 𝑉 β†’ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)))))
151, 14syl 17 . . 3 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ (𝑋 ∈ 𝑉 β†’ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)))))
1615impcom 407 . 2 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ))))
1716impcom 407 1 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2))) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3426   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  1st c1st 7972  2nd c2nd 7973  Fincfn 8941  0cc0 11112  1c1 11113   + caddc 11115   Β· cmul 11117   βˆ’ cmin 11448  2c2 12271  3c3 12272  β„€β‰₯cuz 12826  β™―chash 14295  Vtxcvtx 28764   RegUSGraph crusgr 29322  ClWalkscclwlks 29536
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-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
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1060  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-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-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-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-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-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-s2 14805  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-wlks 29365  df-clwlks 29537  df-wwlks 29593  df-wwlksn 29594  df-clwwlk 29744  df-clwwlkn 29787  df-clwwlknon 29850
This theorem is referenced by: (None)
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