Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  av-numclwwlkqhash Structured version   Visualization version   GIF version

Theorem av-numclwwlkqhash 41530
 Description: In a 𝐾-regular graph, the size of the set of walks of length 𝑛 starting with a fixed vertex 𝑣 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 starting and ending at this vertex 𝑣. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 30-May-2021.)
Hypotheses
Ref Expression
av-numclwwlk.v 𝑉 = (Vtx‘𝐺)
av-numclwwlk.q 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
av-numclwwlk.f 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑣})
Assertion
Ref Expression
av-numclwwlkqhash (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘(𝑋𝑄𝑁)) = ((𝐾𝑁) − (#‘(𝑋𝐹𝑁))))
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)   𝐹(𝑤,𝑣,𝑛)   𝐾(𝑣,𝑛)

Proof of Theorem av-numclwwlkqhash
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 av-numclwwlk.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 av-numclwwlk.q . . . . 5 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
31, 2av-numclwwlkovq 41529 . . . 4 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
43adantl 481 . . 3 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
54fveq2d 6107 . 2 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘(𝑋𝑄𝑁)) = (#‘{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}))
6 eqid 2610 . . 3 {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)} = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}
7 eqid 2610 . . 3 {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
86, 7, 1clwwlknclwwlkdifnum 41182 . 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)})))
9 ovex 6577 . . . . . 6 (𝑁 WWalkSN 𝐺) ∈ V
109rabex 4740 . . . . 5 {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} ∈ V
11 clwwlksvbij 41229 . . . . . 6 (𝑁 ∈ ℕ → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})
1211ad2antll 761 . . . . 5 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})
13 hasheqf1oi 13002 . . . . 5 ({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→{𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} → (#‘{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}) = (#‘{𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})))
1410, 12, 13mpsyl 66 . . . 4 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}) = (#‘{𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋}))
15 av-numclwwlk.f . . . . . . . 8 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑣})
1615av-numclwwlkovf 41511 . . . . . . 7 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})
1716adantl 481 . . . . . 6 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})
1817eqcomd 2616 . . . . 5 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} = (𝑋𝐹𝑁))
1918fveq2d 6107 . . . 4 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘{𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋}) = (#‘(𝑋𝐹𝑁)))
2014, 19eqtrd 2644 . . 3 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}) = (#‘(𝑋𝐹𝑁)))
2120oveq2d 6565 . 2 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → ((𝐾𝑁) − (#‘{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)})) = ((𝐾𝑁) − (#‘(𝑋𝐹𝑁))))
225, 8, 213eqtrd 2648 1 (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘(𝑋𝑄𝑁)) = ((𝐾𝑁) − (#‘(𝑋𝐹𝑁))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173   class class class wbr 4583  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Fincfn 7841  0cc0 9815   − cmin 10145  ℕcn 10897  ↑cexp 12722  #chash 12979   lastS clsw 13147  Vtxcvtx 25673   RegUSGraph crusgr 40756   WWalkSN cwwlksn 41029   ClWWalkSN cclwwlksn 41184 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-xadd 11823  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-vtx 25675  df-iedg 25676  df-uhgr 25724  df-ushgr 25725  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-fusgr 40536  df-nbgr 40554  df-vtxdg 40682  df-rgr 40757  df-rusgr 40758  df-wwlks 41033  df-wwlksn 41034  df-clwwlks 41185  df-clwwlksn 41186 This theorem is referenced by:  av-numclwwlk2  41537
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