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

Theorem numclwwlkqhash 29148
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 = (Vtx‘𝐺)
numclwwlk.q 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
Assertion
Ref Expression
numclwwlkqhash (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (♯‘(𝑋𝑄𝑁)) = ((𝐾𝑁) − (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁))))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑤   𝑤,𝐾   𝑤,𝑉
Allowed substitution hints:   𝑄(𝑤,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 = (Vtx‘𝐺)
2 numclwwlk.q . . . . 5 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
31, 2numclwwlkovq 29147 . . . 4 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
43adantl 483 . . 3 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
54fveq2d 6844 . 2 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (♯‘(𝑋𝑄𝑁)) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}))
6 nnnn0 12379 . . . 4 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
7 eqid 2738 . . . . 5 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}
8 eqid 2738 . . . . 5 (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)
97, 8, 1clwwlknclwwlkdifnum 28753 . . . 4 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}) = ((𝐾𝑁) − (♯‘(𝑋(𝑁 WWalksNOn 𝐺)𝑋))))
106, 9sylanr2 682 . . 3 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}) = ((𝐾𝑁) − (♯‘(𝑋(𝑁 WWalksNOn 𝐺)𝑋))))
111iswwlksnon 28627 . . . . . . 7 (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤𝑁) = 𝑋)}
12 wwlknlsw 28621 . . . . . . . . . . 11 (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑤𝑁) = (lastS‘𝑤))
13 eqcom 2745 . . . . . . . . . . . 12 ((𝑤‘0) = 𝑋𝑋 = (𝑤‘0))
1413biimpi 215 . . . . . . . . . . 11 ((𝑤‘0) = 𝑋𝑋 = (𝑤‘0))
1512, 14eqeqan12d 2752 . . . . . . . . . 10 ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) → ((𝑤𝑁) = 𝑋 ↔ (lastS‘𝑤) = (𝑤‘0)))
1615pm5.32da 580 . . . . . . . . 9 (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝑋 ∧ (𝑤𝑁) = 𝑋) ↔ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) = (𝑤‘0))))
1716biancomd 465 . . . . . . . 8 (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝑋 ∧ (𝑤𝑁) = 𝑋) ↔ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)))
1817rabbiia 3410 . . . . . . 7 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤𝑁) = 𝑋)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
1911, 18eqtri 2766 . . . . . 6 (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
2019fveq2i 6843 . . . . 5 (♯‘(𝑋(𝑁 WWalksNOn 𝐺)𝑋)) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)})
2120a1i 11 . . . 4 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (♯‘(𝑋(𝑁 WWalksNOn 𝐺)𝑋)) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}))
2221oveq2d 7368 . . 3 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → ((𝐾𝑁) − (♯‘(𝑋(𝑁 WWalksNOn 𝐺)𝑋))) = ((𝐾𝑁) − (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)})))
2310, 22eqtrd 2778 . 2 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}) = ((𝐾𝑁) − (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)})))
24 ovex 7385 . . . . 5 (𝑁 WWalksN 𝐺) ∈ V
2524rabex 5288 . . . 4 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} ∈ V
26 clwwlkvbij 28886 . . . . 5 ((𝑋𝑉𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
2726adantl 483 . . . 4 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
28 hasheqf1oi 14205 . . . 4 ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} ∈ V → (∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁))))
2925, 27, 28mpsyl 68 . . 3 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁)))
3029oveq2d 7368 . 2 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → ((𝐾𝑁) − (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)})) = ((𝐾𝑁) − (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁))))
315, 23, 303eqtrd 2782 1 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (♯‘(𝑋𝑄𝑁)) = ((𝐾𝑁) − (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2942  {crab 3406  Vcvv 3444   class class class wbr 5104  1-1-ontowf1o 6493  cfv 6494  (class class class)co 7352  cmpo 7354  Fincfn 8842  0cc0 11010  cmin 11344  cn 12112  0cn0 12372  cexp 13922  chash 14184  lastSclsw 14404  Vtxcvtx 27776   RegUSGraph crusgr 28333   WWalksN cwwlksn 28600   WWalksNOn cwwlksnon 28601  ClWWalksNOncclwwlknon 28860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665  ax-inf2 9536  ax-cnex 11066  ax-resscn 11067  ax-1cn 11068  ax-icn 11069  ax-addcl 11070  ax-addrcl 11071  ax-mulcl 11072  ax-mulrcl 11073  ax-mulcom 11074  ax-addass 11075  ax-mulass 11076  ax-distr 11077  ax-i2m1 11078  ax-1ne0 11079  ax-1rid 11080  ax-rnegex 11081  ax-rrecex 11082  ax-cnre 11083  ax-pre-lttri 11084  ax-pre-lttrn 11085  ax-pre-ltadd 11086  ax-pre-mulgt0 11087  ax-pre-sup 11088
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-iun 4955  df-disj 5070  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5530  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5587  df-se 5588  df-we 5589  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6252  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7308  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7796  df-1st 7914  df-2nd 7915  df-frecs 8205  df-wrecs 8236  df-recs 8310  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8607  df-map 8726  df-pm 8727  df-en 8843  df-dom 8844  df-sdom 8845  df-fin 8846  df-sup 9337  df-oi 9405  df-dju 9796  df-card 9834  df-pnf 11150  df-mnf 11151  df-xr 11152  df-ltxr 11153  df-le 11154  df-sub 11346  df-neg 11347  df-div 11772  df-nn 12113  df-2 12175  df-3 12176  df-n0 12373  df-xnn0 12445  df-z 12459  df-uz 12723  df-rp 12871  df-xadd 12989  df-fz 13380  df-fzo 13523  df-seq 13862  df-exp 13923  df-hash 14185  df-word 14357  df-lsw 14405  df-concat 14413  df-s1 14438  df-substr 14487  df-pfx 14517  df-cj 14944  df-re 14945  df-im 14946  df-sqrt 15080  df-abs 15081  df-clim 15330  df-sum 15531  df-vtx 27778  df-iedg 27779  df-edg 27828  df-uhgr 27838  df-ushgr 27839  df-upgr 27862  df-umgr 27863  df-uspgr 27930  df-usgr 27931  df-fusgr 28094  df-nbgr 28110  df-vtxdg 28243  df-rgr 28334  df-rusgr 28335  df-wwlks 28604  df-wwlksn 28605  df-wwlksnon 28606  df-clwwlk 28755  df-clwwlkn 28798  df-clwwlknon 28861
This theorem is referenced by:  numclwwlk2  29154
  Copyright terms: Public domain W3C validator