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

Theorem numclwwlkqhash 26627
 Description: In a k-regular graph, the size of the set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between k to the power of n and the size of the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Proof shortened by AV, 5-May-2021.)
Hypotheses
Ref Expression
numclwwlk.c 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
numclwwlk.f 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})
numclwwlk.g 𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
numclwwlk.q 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
Assertion
Ref Expression
numclwwlkqhash (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘(𝑋𝑄𝑁)) = ((𝐾𝑁) − (#‘(𝑋𝐹𝑁))))
Distinct variable groups:   𝑛,𝐸   𝑛,𝑁   𝑛,𝑉   𝑤,𝐶   𝑤,𝑁   𝐶,𝑛,𝑣,𝑤   𝑣,𝑁   𝑛,𝑋,𝑣,𝑤   𝑣,𝑉   𝑤,𝐸   𝑤,𝑉   𝑤,𝐹   𝑤,𝑄   𝑤,𝐾   𝑤,𝐺   𝑣,𝐸
Allowed substitution hints:   𝑄(𝑣,𝑛)   𝐹(𝑣,𝑛)   𝐺(𝑣,𝑛)   𝐾(𝑣,𝑛)

Proof of Theorem numclwwlkqhash
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 nnnn0 11176 . . . . . 6 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
21anim2i 591 . . . . 5 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑉𝑁 ∈ ℕ0))
32adantl 481 . . . 4 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (𝑋𝑉𝑁 ∈ ℕ0))
4 numclwwlk.c . . . . 5 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
5 numclwwlk.f . . . . 5 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})
6 numclwwlk.g . . . . 5 𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
7 numclwwlk.q . . . . 5 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
84, 5, 6, 7numclwwlkovq 26626 . . . 4 ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋𝑄𝑁) = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
93, 8syl 17 . . 3 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (𝑋𝑄𝑁) = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
109fveq2d 6107 . 2 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘(𝑋𝑄𝑁)) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}))
11 eqid 2610 . . 3 {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}
12 eqid 2610 . . 3 {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
1311, 12clwlknclwlkdifnum 26488 . 2 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}) = ((𝐾𝑁) − (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)})))
14 fvex 6113 . . . . . 6 ((𝑉 WWalksN 𝐸)‘𝑁) ∈ V
1514rabex 4740 . . . . 5 {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} ∈ V
16 rusisusgra 26458 . . . . . . . . 9 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 USGrph 𝐸)
17 usgrav 25867 . . . . . . . . 9 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
18 simpll 786 . . . . . . . . . . 11 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → 𝑉 ∈ V)
19 simpr 476 . . . . . . . . . . . 12 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝐸 ∈ V)
2019adantr 480 . . . . . . . . . . 11 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → 𝐸 ∈ V)
21 simpr 476 . . . . . . . . . . . 12 ((𝑋𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
2221adantl 481 . . . . . . . . . . 11 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → 𝑁 ∈ ℕ)
2318, 20, 223jca 1235 . . . . . . . . . 10 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ))
2423ex 449 . . . . . . . . 9 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑋𝑉𝑁 ∈ ℕ) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ)))
2516, 17, 243syl 18 . . . . . . . 8 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((𝑋𝑉𝑁 ∈ ℕ) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ)))
2625adantr 480 . . . . . . 7 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) → ((𝑋𝑉𝑁 ∈ ℕ) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ)))
2726imp 444 . . . . . 6 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ))
28 clwwlkvbij 26329 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋})
2927, 28syl 17 . . . . 5 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋})
30 hasheqf1oi 13002 . . . . 5 ({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} ∈ V → (∃𝑓 𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}) = (#‘{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋})))
3115, 29, 30mpsyl 66 . . . 4 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}) = (#‘{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}))
321adantl 481 . . . . . . . 8 ((𝑋𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
3332adantl 481 . . . . . . 7 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → 𝑁 ∈ ℕ0)
344numclwwlkfvc 26604 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝐶𝑁) = ((𝑉 ClWWalksN 𝐸)‘𝑁))
35 rabeq 3166 . . . . . . 7 ((𝐶𝑁) = ((𝑉 ClWWalksN 𝐸)‘𝑁) → {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋})
3633, 34, 353syl 18 . . . . . 6 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋})
3736eqcomd 2616 . . . . 5 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → {𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑋})
3837fveq2d 6107 . . . 4 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) = (#‘{𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑋}))
394, 5numclwwlkovf 26608 . . . . . . 7 ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑋})
403, 39syl 17 . . . . . 6 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑋})
4140eqcomd 2616 . . . . 5 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑋} = (𝑋𝐹𝑁))
4241fveq2d 6107 . . . 4 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘{𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑋}) = (#‘(𝑋𝐹𝑁)))
4331, 38, 423eqtrd 2648 . . 3 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}) = (#‘(𝑋𝐹𝑁)))
4443oveq2d 6565 . 2 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → ((𝐾𝑁) − (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)})) = ((𝐾𝑁) − (#‘(𝑋𝐹𝑁))))
4510, 13, 443eqtrd 2648 1 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘(𝑋𝑄𝑁)) = ((𝐾𝑁) − (#‘(𝑋𝐹𝑁))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Fincfn 7841  0cc0 9815   − cmin 10145  ℕcn 10897  2c2 10947  ℕ0cn0 11169  ℤ≥cuz 11563  ↑cexp 12722  #chash 12979   lastS clsw 13147   USGrph cusg 25859   WWalksN cwwlkn 26206   ClWWalksN cclwwlkn 26277   RegUSGrph crusgra 26450 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-usgra 25862  df-nbgra 25949  df-wlk 26036  df-wwlk 26207  df-wwlkn 26208  df-clwwlk 26279  df-clwwlkn 26280  df-vdgr 26421  df-rgra 26451  df-rusgra 26452 This theorem is referenced by:  numclwwlk2  26634
 Copyright terms: Public domain W3C validator