MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  numclwwlkqhash Structured version   Visualization version   GIF version

Theorem numclwwlkqhash 30172
Description: In a 𝐾-regular graph, the size of the set of walks of length 𝑁 starting with a fixed vertex 𝑋 and ending not at this vertex is the difference between 𝐾 to the power of 𝑁 and the size of the set of closed walks of length 𝑁 on vertex 𝑋. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 30-May-2021.) (Revised by AV, 5-Mar-2022.) (Proof shortened by AV, 7-Jul-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)})
Assertion
Ref Expression
numclwwlkqhash (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ (β™―β€˜(𝑋𝑄𝑁)) = ((𝐾↑𝑁) βˆ’ (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))))
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)   𝐾(𝑣,𝑛)

Proof of Theorem numclwwlkqhash
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 numclwwlk.v . . . . 5 𝑉 = (Vtxβ€˜πΊ)
2 numclwwlk.q . . . . 5 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})
31, 2numclwwlkovq 30171 . . . 4 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑋𝑄𝑁) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)})
43adantl 481 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ (𝑋𝑄𝑁) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)})
54fveq2d 6895 . 2 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ (β™―β€˜(𝑋𝑄𝑁)) = (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)}))
6 nnnn0 12501 . . . 4 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„•0)
7 eqid 2727 . . . . 5 {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)} = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)}
8 eqid 2727 . . . . 5 (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)
97, 8, 1clwwlknclwwlkdifnum 29777 . . . 4 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)}) = ((𝐾↑𝑁) βˆ’ (β™―β€˜(𝑋(𝑁 WWalksNOn 𝐺)𝑋))))
106, 9sylanr2 682 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)}) = ((𝐾↑𝑁) βˆ’ (β™―β€˜(𝑋(𝑁 WWalksNOn 𝐺)𝑋))))
111iswwlksnon 29651 . . . . . . 7 (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋)}
12 wwlknlsw 29645 . . . . . . . . . . 11 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ (π‘€β€˜π‘) = (lastSβ€˜π‘€))
13 eqcom 2734 . . . . . . . . . . . 12 ((π‘€β€˜0) = 𝑋 ↔ 𝑋 = (π‘€β€˜0))
1413biimpi 215 . . . . . . . . . . 11 ((π‘€β€˜0) = 𝑋 β†’ 𝑋 = (π‘€β€˜0))
1512, 14eqeqan12d 2741 . . . . . . . . . 10 ((𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑋) β†’ ((π‘€β€˜π‘) = 𝑋 ↔ (lastSβ€˜π‘€) = (π‘€β€˜0)))
1615pm5.32da 578 . . . . . . . . 9 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋) ↔ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) = (π‘€β€˜0))))
1716biancomd 463 . . . . . . . 8 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋) ↔ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)))
1817rabbiia 3431 . . . . . . 7 {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋)} = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)}
1911, 18eqtri 2755 . . . . . 6 (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)}
2019fveq2i 6894 . . . . 5 (β™―β€˜(𝑋(𝑁 WWalksNOn 𝐺)𝑋)) = (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)})
2120a1i 11 . . . 4 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ (β™―β€˜(𝑋(𝑁 WWalksNOn 𝐺)𝑋)) = (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)}))
2221oveq2d 7430 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ ((𝐾↑𝑁) βˆ’ (β™―β€˜(𝑋(𝑁 WWalksNOn 𝐺)𝑋))) = ((𝐾↑𝑁) βˆ’ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)})))
2310, 22eqtrd 2767 . 2 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)}) = ((𝐾↑𝑁) βˆ’ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)})))
24 ovex 7447 . . . . 5 (𝑁 WWalksN 𝐺) ∈ V
2524rabex 5328 . . . 4 {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)} ∈ V
26 clwwlkvbij 29910 . . . . 5 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ βˆƒπ‘“ 𝑓:{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))
2726adantl 481 . . . 4 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ βˆƒπ‘“ 𝑓:{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))
28 hasheqf1oi 14334 . . . 4 ({𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)} ∈ V β†’ (βˆƒπ‘“ 𝑓:{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁) β†’ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)}) = (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))))
2925, 27, 28mpsyl 68 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)}) = (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)𝑁)))
3029oveq2d 7430 . 2 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ ((𝐾↑𝑁) βˆ’ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)})) = ((𝐾↑𝑁) βˆ’ (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))))
315, 23, 303eqtrd 2771 1 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ (β™―β€˜(𝑋𝑄𝑁)) = ((𝐾↑𝑁) βˆ’ (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534  βˆƒwex 1774   ∈ wcel 2099   β‰  wne 2935  {crab 3427  Vcvv 3469   class class class wbr 5142  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  Fincfn 8955  0cc0 11130   βˆ’ cmin 11466  β„•cn 12234  β„•0cn0 12494  β†‘cexp 14050  β™―chash 14313  lastSclsw 14536  Vtxcvtx 28796   RegUSGraph crusgr 29357   WWalksN cwwlksn 29624   WWalksNOn cwwlksnon 29625  ClWWalksNOncclwwlknon 29884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  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 5108  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-oi 9525  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-rp 12999  df-xadd 13117  df-fz 13509  df-fzo 13652  df-seq 13991  df-exp 14051  df-hash 14314  df-word 14489  df-lsw 14537  df-concat 14545  df-s1 14570  df-substr 14615  df-pfx 14645  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-sum 15657  df-vtx 28798  df-iedg 28799  df-edg 28848  df-uhgr 28858  df-ushgr 28859  df-upgr 28882  df-umgr 28883  df-uspgr 28950  df-usgr 28951  df-fusgr 29117  df-nbgr 29133  df-vtxdg 29267  df-rgr 29358  df-rusgr 29359  df-wwlks 29628  df-wwlksn 29629  df-wwlksnon 29630  df-clwwlk 29779  df-clwwlkn 29822  df-clwwlknon 29885
This theorem is referenced by:  numclwwlk2  30178
  Copyright terms: Public domain W3C validator