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Theorem clwwlknclwwlkdifnum 29777
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 𝐡 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 = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)}
clwwlknclwwlkdif.b 𝐡 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)
clwwlknclwwlkdifnum.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
clwwlknclwwlkdifnum (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (β™―β€˜π΄) = ((𝐾↑𝑁) βˆ’ (β™―β€˜π΅)))
Distinct variable groups:   𝑀,𝐺   𝑀,𝑁   𝑀,𝑋   𝑀,𝐾   𝑀,𝑉
Allowed substitution hints:   𝐴(𝑀)   𝐡(𝑀)

Proof of Theorem clwwlknclwwlkdifnum
StepHypRef Expression
1 clwwlknclwwlkdif.a . . . . 5 𝐴 = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)}
2 clwwlknclwwlkdif.b . . . . 5 𝐡 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)
3 eqid 2727 . . . . 5 {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}
41, 2, 3clwwlknclwwlkdif 29776 . . . 4 𝐴 = ({𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} βˆ– 𝐡)
54fveq2i 6894 . . 3 (β™―β€˜π΄) = (β™―β€˜({𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} βˆ– 𝐡))
65a1i 11 . 2 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (β™―β€˜π΄) = (β™―β€˜({𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} βˆ– 𝐡)))
7 clwwlknclwwlkdifnum.v . . . . . . . 8 𝑉 = (Vtxβ€˜πΊ)
87eleq1i 2819 . . . . . . 7 (𝑉 ∈ Fin ↔ (Vtxβ€˜πΊ) ∈ Fin)
98biimpi 215 . . . . . 6 (𝑉 ∈ Fin β†’ (Vtxβ€˜πΊ) ∈ Fin)
109adantl 481 . . . . 5 ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) β†’ (Vtxβ€˜πΊ) ∈ Fin)
1110adantr 480 . . . 4 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (Vtxβ€˜πΊ) ∈ Fin)
12 wwlksnfi 29704 . . . 4 ((Vtxβ€˜πΊ) ∈ Fin β†’ (𝑁 WWalksN 𝐺) ∈ Fin)
13 rabfi 9285 . . . 4 ((𝑁 WWalksN 𝐺) ∈ Fin β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} ∈ Fin)
1411, 12, 133syl 18 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} ∈ Fin)
157iswwlksnon 29651 . . . . . . . 8 (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋)}
16 ancom 460 . . . . . . . . 9 (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋) ↔ ((π‘€β€˜π‘) = 𝑋 ∧ (π‘€β€˜0) = 𝑋))
1716rabbii 3433 . . . . . . . 8 {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋)} = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜π‘) = 𝑋 ∧ (π‘€β€˜0) = 𝑋)}
1815, 17eqtri 2755 . . . . . . 7 (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜π‘) = 𝑋 ∧ (π‘€β€˜0) = 𝑋)}
1918a1i 11 . . . . . 6 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜π‘) = 𝑋 ∧ (π‘€β€˜0) = 𝑋)})
202, 19eqtrid 2779 . . . . 5 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ 𝐡 = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜π‘) = 𝑋 ∧ (π‘€β€˜0) = 𝑋)})
21 simpr 484 . . . . . . 7 (((π‘€β€˜π‘) = 𝑋 ∧ (π‘€β€˜0) = 𝑋) β†’ (π‘€β€˜0) = 𝑋)
2221a1i 11 . . . . . 6 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ (((π‘€β€˜π‘) = 𝑋 ∧ (π‘€β€˜0) = 𝑋) β†’ (π‘€β€˜0) = 𝑋))
2322ss2rabi 4070 . . . . 5 {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜π‘) = 𝑋 ∧ (π‘€β€˜0) = 𝑋)} βŠ† {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}
2420, 23eqsstrdi 4032 . . . 4 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ 𝐡 βŠ† {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋})
2524adantl 481 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝐡 βŠ† {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋})
26 hashssdif 14395 . . 3 (({𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} ∈ Fin ∧ 𝐡 βŠ† {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}) β†’ (β™―β€˜({𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} βˆ– 𝐡)) = ((β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}) βˆ’ (β™―β€˜π΅)))
2714, 25, 26syl2anc 583 . 2 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (β™―β€˜({𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} βˆ– 𝐡)) = ((β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}) βˆ’ (β™―β€˜π΅)))
28 simpl 482 . . . . 5 ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) β†’ 𝐺 RegUSGraph 𝐾)
2928adantr 480 . . . 4 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝐺 RegUSGraph 𝐾)
30 simpr 484 . . . . 5 ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) β†’ 𝑉 ∈ Fin)
3130adantr 480 . . . 4 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝑉 ∈ Fin)
32 simpl 482 . . . . 5 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ 𝑋 ∈ 𝑉)
3332adantl 481 . . . 4 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝑋 ∈ 𝑉)
34 simpr 484 . . . . 5 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ 𝑁 ∈ β„•0)
3534adantl 481 . . . 4 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝑁 ∈ β„•0)
367rusgrnumwwlkg 29774 . . . 4 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}) = (𝐾↑𝑁))
3729, 31, 33, 35, 36syl13anc 1370 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}) = (𝐾↑𝑁))
3837oveq1d 7429 . 2 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ ((β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}) βˆ’ (β™―β€˜π΅)) = ((𝐾↑𝑁) βˆ’ (β™―β€˜π΅)))
396, 27, 383eqtrd 2771 1 (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (β™―β€˜π΄) = ((𝐾↑𝑁) βˆ’ (β™―β€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  {crab 3427   βˆ– cdif 3941   βŠ† wss 3944   class class class wbr 5142  β€˜cfv 6542  (class class class)co 7414  Fincfn 8955  0cc0 11130   βˆ’ cmin 11466  β„•0cn0 12494  β†‘cexp 14050  β™―chash 14313  lastSclsw 14536  Vtxcvtx 28796   RegUSGraph crusgr 29357   WWalksN cwwlksn 29624   WWalksNOn cwwlksnon 29625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-disj 5108  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-oi 9525  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-rp 12999  df-xadd 13117  df-fz 13509  df-fzo 13652  df-seq 13991  df-exp 14051  df-hash 14314  df-word 14489  df-lsw 14537  df-concat 14545  df-s1 14570  df-substr 14615  df-pfx 14645  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-sum 15657  df-vtx 28798  df-iedg 28799  df-edg 28848  df-uhgr 28858  df-ushgr 28859  df-upgr 28882  df-umgr 28883  df-uspgr 28950  df-usgr 28951  df-fusgr 29117  df-nbgr 29133  df-vtxdg 29267  df-rgr 29358  df-rusgr 29359  df-wwlks 29628  df-wwlksn 29629  df-wwlksnon 29630
This theorem is referenced by:  numclwwlkqhash  30172
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