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Theorem clwlknclwlkdifnum 26254
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.)
Hypotheses
Ref Expression
clwlknclwlkdif.a 𝐴 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}
clwlknclwlkdif.b 𝐵 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
Assertion
Ref Expression
clwlknclwlkdifnum (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘𝐴) = ((𝐾𝑁) − (#‘𝐵)))
Distinct variable groups:   𝑤,𝐸   𝑤,𝑁   𝑤,𝑉   𝑤,𝑋
Allowed substitution hints:   𝐴(𝑤)   𝐵(𝑤)   𝐾(𝑤)

Proof of Theorem clwlknclwlkdifnum
StepHypRef Expression
1 clwlknclwlkdif.a . . . 4 𝐴 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}
2 clwlknclwlkdif.b . . . 4 𝐵 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
31, 2clwlknclwlkdifs 26253 . . 3 𝐴 = ({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)
43fveq2i 6091 . 2 (#‘𝐴) = (#‘({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵))
5 simpr 475 . . . . . 6 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) → 𝑉 ∈ Fin)
65adantr 479 . . . . 5 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → 𝑉 ∈ Fin)
7 wwlknfi 26032 . . . . 5 (𝑉 ∈ Fin → ((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin)
8 rabfi 8047 . . . . 5 (((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∈ Fin)
96, 7, 83syl 18 . . . 4 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∈ Fin)
10 simpr 475 . . . . . . 7 ((( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋)
1110a1i 11 . . . . . 6 (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋))
1211ss2rabi 3646 . . . . 5 {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} ⊆ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}
132, 12eqsstri 3597 . . . 4 𝐵 ⊆ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}
14 hashssdif 13013 . . . 4 (({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∈ Fin ∧ 𝐵 ⊆ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) → (#‘({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)))
159, 13, 14sylancl 692 . . 3 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)))
16 simpl 471 . . . . . 6 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) → ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾)
1716adantr 479 . . . . 5 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾)
18 simpl 471 . . . . . 6 ((𝑋𝑉𝑁 ∈ ℕ) → 𝑋𝑉)
1918adantl 480 . . . . 5 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → 𝑋𝑉)
20 nnnn0 11146 . . . . . . 7 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
2120adantl 480 . . . . . 6 ((𝑋𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
2221adantl 480 . . . . 5 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → 𝑁 ∈ ℕ0)
23 rusgranumwwlkg 26252 . . . . 5 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑋𝑉𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) = (𝐾𝑁))
2417, 6, 19, 22, 23syl13anc 1319 . . . 4 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) = (𝐾𝑁))
2524oveq1d 6542 . . 3 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)) = ((𝐾𝑁) − (#‘𝐵)))
2615, 25eqtrd 2643 . 2 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((𝐾𝑁) − (#‘𝐵)))
274, 26syl5eq 2655 1 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘𝐴) = ((𝐾𝑁) − (#‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wne 2779  {crab 2899  cdif 3536  wss 3539  cop 4130   class class class wbr 4577  cfv 5790  (class class class)co 6527  Fincfn 7818  0cc0 9792  cmin 10117  cn 10867  0cn0 11139  cexp 12677  #chash 12934   lastS clsw 13093   WWalksN cwwlkn 25972   RegUSGrph crusgra 26216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-disj 4548  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-oi 8275  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-rp 11665  df-xadd 11779  df-fz 12153  df-fzo 12290  df-seq 12619  df-exp 12678  df-hash 12935  df-word 13100  df-lsw 13101  df-concat 13102  df-s1 13103  df-substr 13104  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-clim 14013  df-sum 14211  df-usgra 25628  df-nbgra 25715  df-wlk 25802  df-wwlk 25973  df-wwlkn 25974  df-vdgr 26187  df-rgra 26217  df-rusgra 26218
This theorem is referenced by:  numclwwlkqhash  26393
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