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Theorem numclwwlkqhash 28164
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 28163 . . . 4 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
43adantl 485 . . 3 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
54fveq2d 6653 . 2 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (♯‘(𝑋𝑄𝑁)) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}))
6 nnnn0 11896 . . . 4 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
7 eqid 2801 . . . . 5 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}
8 eqid 2801 . . . . 5 (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)
97, 8, 1clwwlknclwwlkdifnum 27769 . . . 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 27643 . . . . . . 7 (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤𝑁) = 𝑋)}
12 wwlknlsw 27637 . . . . . . . . . . 11 (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑤𝑁) = (lastS‘𝑤))
13 eqcom 2808 . . . . . . . . . . . 12 ((𝑤‘0) = 𝑋𝑋 = (𝑤‘0))
1413biimpi 219 . . . . . . . . . . 11 ((𝑤‘0) = 𝑋𝑋 = (𝑤‘0))
1512, 14eqeqan12d 2818 . . . . . . . . . 10 ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) → ((𝑤𝑁) = 𝑋 ↔ (lastS‘𝑤) = (𝑤‘0)))
1615pm5.32da 582 . . . . . . . . 9 (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝑋 ∧ (𝑤𝑁) = 𝑋) ↔ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) = (𝑤‘0))))
1716biancomd 467 . . . . . . . 8 (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝑋 ∧ (𝑤𝑁) = 𝑋) ↔ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)))
1817rabbiia 3422 . . . . . . 7 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤𝑁) = 𝑋)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
1911, 18eqtri 2824 . . . . . 6 (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
2019fveq2i 6652 . . . . 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 7155 . . 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 2836 . 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 7172 . . . . 5 (𝑁 WWalksN 𝐺) ∈ V
2524rabex 5202 . . . 4 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} ∈ V
26 clwwlkvbij 27902 . . . . 5 ((𝑋𝑉𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
2726adantl 485 . . . 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 13712 . . . 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 7155 . 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 2840 1 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (♯‘(𝑋𝑄𝑁)) = ((𝐾𝑁) − (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2112  wne 2990  {crab 3113  Vcvv 3444   class class class wbr 5033  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  cmpo 7141  Fincfn 8496  0cc0 10530  cmin 10863  cn 11629  0cn0 11889  cexp 13429  chash 13690  lastSclsw 13909  Vtxcvtx 26793   RegUSGraph crusgr 27350   WWalksN cwwlksn 27616   WWalksNOn cwwlksnon 27617  ClWWalksNOncclwwlknon 27876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-disj 4999  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-oi 8962  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-rp 12382  df-xadd 12500  df-fz 12890  df-fzo 13033  df-seq 13369  df-exp 13430  df-hash 13691  df-word 13862  df-lsw 13910  df-concat 13918  df-s1 13945  df-substr 13998  df-pfx 14028  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-clim 14841  df-sum 15039  df-vtx 26795  df-iedg 26796  df-edg 26845  df-uhgr 26855  df-ushgr 26856  df-upgr 26879  df-umgr 26880  df-uspgr 26947  df-usgr 26948  df-fusgr 27111  df-nbgr 27127  df-vtxdg 27260  df-rgr 27351  df-rusgr 27352  df-wwlks 27620  df-wwlksn 27621  df-wwlksnon 27622  df-clwwlk 27771  df-clwwlkn 27814  df-clwwlknon 27877
This theorem is referenced by:  numclwwlk2  28170
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