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Theorem rusgrnumwwlk 26750
Description: In a 𝐾-regular graph, the number of walks of a fixed length 𝑁 from a fixed vertex is 𝐾 to the power of 𝑁. By definition, (𝑁 WWalksN 𝐺) is the set of walks (as words) with length 𝑁, and (𝑃𝐿𝑁) is the number of walks with length 𝑁 starting at the vertex 𝑃. Because of the 𝐾-regularity, a walk can be continued in 𝐾 different ways at the end vertex of the walk, and this repeated 𝑁 times.

This theorem even holds for 𝑁 = 0: in this case, the walk consists of only one vertex 𝑃, so the number of walks of length 𝑁 = 0 starting with 𝑃 is (𝐾↑0) = 1. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.)

Hypotheses
Ref Expression
rusgrnumwwlk.v 𝑉 = (Vtx‘𝐺)
rusgrnumwwlk.l 𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
Assertion
Ref Expression
rusgrnumwwlk ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾𝑁))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑃,𝑛,𝑣,𝑤   𝑛,𝑉,𝑣,𝑤   𝑤,𝐾
Allowed substitution hints:   𝐾(𝑣,𝑛)   𝐿(𝑤,𝑣,𝑛)

Proof of Theorem rusgrnumwwlk
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6618 . . . . . . . 8 (𝑥 = 0 → (𝑃𝐿𝑥) = (𝑃𝐿0))
2 oveq2 6618 . . . . . . . 8 (𝑥 = 0 → (𝐾𝑥) = (𝐾↑0))
31, 2eqeq12d 2636 . . . . . . 7 (𝑥 = 0 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿0) = (𝐾↑0)))
43imbi2d 330 . . . . . 6 (𝑥 = 0 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = (𝐾↑0))))
5 oveq2 6618 . . . . . . . 8 (𝑥 = 𝑦 → (𝑃𝐿𝑥) = (𝑃𝐿𝑦))
6 oveq2 6618 . . . . . . . 8 (𝑥 = 𝑦 → (𝐾𝑥) = (𝐾𝑦))
75, 6eqeq12d 2636 . . . . . . 7 (𝑥 = 𝑦 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿𝑦) = (𝐾𝑦)))
87imbi2d 330 . . . . . 6 (𝑥 = 𝑦 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑦) = (𝐾𝑦))))
9 oveq2 6618 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑃𝐿𝑥) = (𝑃𝐿(𝑦 + 1)))
10 oveq2 6618 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝐾𝑥) = (𝐾↑(𝑦 + 1)))
119, 10eqeq12d 2636 . . . . . . 7 (𝑥 = (𝑦 + 1) → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
1211imbi2d 330 . . . . . 6 (𝑥 = (𝑦 + 1) → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1)))))
13 oveq2 6618 . . . . . . . 8 (𝑥 = 𝑁 → (𝑃𝐿𝑥) = (𝑃𝐿𝑁))
14 oveq2 6618 . . . . . . . 8 (𝑥 = 𝑁 → (𝐾𝑥) = (𝐾𝑁))
1513, 14eqeq12d 2636 . . . . . . 7 (𝑥 = 𝑁 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿𝑁) = (𝐾𝑁)))
1615imbi2d 330 . . . . . 6 (𝑥 = 𝑁 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑁) = (𝐾𝑁))))
17 rusgrusgr 26343 . . . . . . . . 9 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph )
18 usgruspgr 25979 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
1917, 18syl 17 . . . . . . . 8 (𝐺 RegUSGraph 𝐾𝐺 ∈ USPGraph )
20 simpr 477 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝑃𝑉) → 𝑃𝑉)
21 rusgrnumwwlk.v . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
22 rusgrnumwwlk.l . . . . . . . . 9 𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
2321, 22rusgrnumwwlkb0 26746 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑃𝐿0) = 1)
2419, 20, 23syl2anr 495 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = 1)
25 simpl 473 . . . . . . . . . . 11 ((𝑉 ∈ Fin ∧ 𝑃𝑉) → 𝑉 ∈ Fin)
2625, 17anim12ci 590 . . . . . . . . . 10 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2721isfusgr 26111 . . . . . . . . . 10 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2826, 27sylibr 224 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph )
29 simpr 477 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾)
30 ne0i 3902 . . . . . . . . . 10 (𝑃𝑉𝑉 ≠ ∅)
3130ad2antlr 762 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝑉 ≠ ∅)
3221frusgrnn0 26350 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
3332nn0cnd 11304 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℂ)
3428, 29, 31, 33syl3anc 1323 . . . . . . . 8 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 ∈ ℂ)
3534exp0d 12949 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝐾↑0) = 1)
3624, 35eqtr4d 2658 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = (𝐾↑0))
37 simpl 473 . . . . . . . . . . 11 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑉 ∈ Fin ∧ 𝑃𝑉))
3837anim1i 591 . . . . . . . . . 10 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝑦 ∈ ℕ0))
39 df-3an 1038 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0) ↔ ((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝑦 ∈ ℕ0))
4038, 39sylibr 224 . . . . . . . . 9 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → (𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0))
4121, 22rusgrnumwwlks 26749 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0)) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
4229, 40, 41syl2an2r 875 . . . . . . . 8 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
4342expcom 451 . . . . . . 7 (𝑦 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1)))))
4443a2d 29 . . . . . 6 (𝑦 ∈ ℕ0 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑦) = (𝐾𝑦)) → (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1)))))
454, 8, 12, 16, 36, 44nn0ind 11423 . . . . 5 (𝑁 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑁) = (𝐾𝑁)))
4645expd 452 . . . 4 (𝑁 ∈ ℕ0 → ((𝑉 ∈ Fin ∧ 𝑃𝑉) → (𝐺 RegUSGraph 𝐾 → (𝑃𝐿𝑁) = (𝐾𝑁))))
4746com12 32 . . 3 ((𝑉 ∈ Fin ∧ 𝑃𝑉) → (𝑁 ∈ ℕ0 → (𝐺 RegUSGraph 𝐾 → (𝑃𝐿𝑁) = (𝐾𝑁))))
48473impia 1258 . 2 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝐺 RegUSGraph 𝐾 → (𝑃𝐿𝑁) = (𝐾𝑁)))
4948impcom 446 1 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  {crab 2911  c0 3896   class class class wbr 4618  cfv 5852  (class class class)co 6610  cmpt2 6612  Fincfn 7906  cc 9885  0cc0 9887  1c1 9888   + caddc 9890  0cn0 11243  cexp 12807  #chash 13064  Vtxcvtx 25787   USPGraph cuspgr 25949   USGraph cusgr 25950   FinUSGraph cfusgr 26109   RegUSGraph crusgr 26335   WWalksN cwwlksn 26600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-disj 4589  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-sup 8299  df-oi 8366  df-card 8716  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-n0 11244  df-xnn0 11315  df-z 11329  df-uz 11639  df-rp 11784  df-xadd 11898  df-fz 12276  df-fzo 12414  df-seq 12749  df-exp 12808  df-hash 13065  df-word 13245  df-lsw 13246  df-concat 13247  df-s1 13248  df-substr 13249  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-clim 14160  df-sum 14358  df-vtx 25789  df-iedg 25790  df-edg 25853  df-uhgr 25862  df-ushgr 25863  df-upgr 25886  df-umgr 25887  df-uspgr 25951  df-usgr 25952  df-fusgr 26110  df-nbgr 26128  df-vtxdg 26262  df-rgr 26336  df-rusgr 26337  df-wwlks 26604  df-wwlksn 26605
This theorem is referenced by:  rusgrnumwwlkg  26751
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