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Theorem clwwlknclwwlkdifnum 26946
Description: In a 𝐾-regular graph, the size of the set 𝐴 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 𝐵 of closed walks of length 𝑁 anchored at this vertex 𝑋. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.) (Revised by AV, 8-Mar-2022.) (Proof shortened by AV, 16-Mar-2022.)
Hypotheses
Ref Expression
clwwlknclwwlkdif.a 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}
clwwlknclwwlkdif.b 𝐵 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)
clwwlknclwwlkdifnum.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
clwwlknclwwlkdifnum (((𝐺RegUSGraph𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → (#‘𝐴) = ((𝐾𝑁) − (#‘𝐵)))
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑤,𝑋   𝑤,𝐾   𝑤,𝑉
Allowed substitution hints:   𝐴(𝑤)   𝐵(𝑤)

Proof of Theorem clwwlknclwwlkdifnum
StepHypRef Expression
1 clwwlknclwwlkdif.a . . . . 5 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}
2 clwwlknclwwlkdif.b . . . . 5 𝐵 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)
3 eqid 2651 . . . . 5 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}
41, 2, 3clwwlknclwwlkdif 26945 . . . 4 𝐴 = ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)
54fveq2i 6232 . . 3 (#‘𝐴) = (#‘({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵))
65a1i 11 . 2 (((𝐺RegUSGraph𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → (#‘𝐴) = (#‘({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)))
7 clwwlknclwwlkdifnum.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
87eleq1i 2721 . . . . . . 7 (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
98biimpi 206 . . . . . 6 (𝑉 ∈ Fin → (Vtx‘𝐺) ∈ Fin)
109adantl 481 . . . . 5 ((𝐺RegUSGraph𝐾𝑉 ∈ Fin) → (Vtx‘𝐺) ∈ Fin)
1110adantr 480 . . . 4 (((𝐺RegUSGraph𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → (Vtx‘𝐺) ∈ Fin)
12 wwlksnfi 26869 . . . 4 ((Vtx‘𝐺) ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)
13 rabfi 8226 . . . 4 ((𝑁 WWalksN 𝐺) ∈ Fin → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ Fin)
1411, 12, 133syl 18 . . 3 (((𝐺RegUSGraph𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ Fin)
157iswwlksnon 26802 . . . . . . . 8 (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤𝑁) = 𝑋)}
16 ancom 465 . . . . . . . . 9 (((𝑤‘0) = 𝑋 ∧ (𝑤𝑁) = 𝑋) ↔ ((𝑤𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋))
1716rabbii 3216 . . . . . . . 8 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤𝑁) = 𝑋)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋)}
1815, 17eqtri 2673 . . . . . . 7 (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋)}
1918a1i 11 . . . . . 6 ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋)})
202, 19syl5eq 2697 . . . . 5 ((𝑋𝑉𝑁 ∈ ℕ0) → 𝐵 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋)})
21 simpr 476 . . . . . . 7 (((𝑤𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋)
2221a1i 11 . . . . . 6 (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋))
2322ss2rabi 3717 . . . . 5 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋)} ⊆ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}
2420, 23syl6eqss 3688 . . . 4 ((𝑋𝑉𝑁 ∈ ℕ0) → 𝐵 ⊆ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
2524adantl 481 . . 3 (((𝐺RegUSGraph𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → 𝐵 ⊆ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
26 hashssdif 13238 . . 3 (({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ Fin ∧ 𝐵 ⊆ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) → (#‘({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)))
2714, 25, 26syl2anc 694 . 2 (((𝐺RegUSGraph𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → (#‘({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)))
28 simpl 472 . . . . 5 ((𝐺RegUSGraph𝐾𝑉 ∈ Fin) → 𝐺RegUSGraph𝐾)
2928adantr 480 . . . 4 (((𝐺RegUSGraph𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → 𝐺RegUSGraph𝐾)
30 simpr 476 . . . . 5 ((𝐺RegUSGraph𝐾𝑉 ∈ Fin) → 𝑉 ∈ Fin)
3130adantr 480 . . . 4 (((𝐺RegUSGraph𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → 𝑉 ∈ Fin)
32 simpl 472 . . . . 5 ((𝑋𝑉𝑁 ∈ ℕ0) → 𝑋𝑉)
3332adantl 481 . . . 4 (((𝐺RegUSGraph𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → 𝑋𝑉)
34 simpr 476 . . . . 5 ((𝑋𝑉𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
3534adantl 481 . . . 4 (((𝐺RegUSGraph𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → 𝑁 ∈ ℕ0)
367rusgrnumwwlkg 26943 . . . 4 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑋𝑉𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) = (𝐾𝑁))
3729, 31, 33, 35, 36syl13anc 1368 . . 3 (((𝐺RegUSGraph𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) = (𝐾𝑁))
3837oveq1d 6705 . 2 (((𝐺RegUSGraph𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)) = ((𝐾𝑁) − (#‘𝐵)))
396, 27, 383eqtrd 2689 1 (((𝐺RegUSGraph𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → (#‘𝐴) = ((𝐾𝑁) − (#‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823  {crab 2945  cdif 3604  wss 3607   class class class wbr 4685  cfv 5926  (class class class)co 6690  Fincfn 7997  0cc0 9974  cmin 10304  0cn0 11330  cexp 12900  #chash 13157   lastS clsw 13324  Vtxcvtx 25919  RegUSGraphcrusgr 26508   WWalksN cwwlksn 26774   WWalksNOn cwwlksnon 26775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-rp 11871  df-xadd 11985  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-word 13331  df-lsw 13332  df-concat 13333  df-s1 13334  df-substr 13335  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-vtx 25921  df-iedg 25922  df-edg 25985  df-uhgr 25998  df-ushgr 25999  df-upgr 26022  df-umgr 26023  df-uspgr 26090  df-usgr 26091  df-fusgr 26254  df-nbgr 26270  df-vtxdg 26418  df-rgr 26509  df-rusgr 26510  df-wwlks 26778  df-wwlksn 26779  df-wwlksnon 26780
This theorem is referenced by:  numclwwlkqhash  27355
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