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Theorem rusgrnumwwlk 27121
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 ((𝐺RegUSGraph𝐾 ∧ (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 6800 . . . . . . . 8 (𝑥 = 0 → (𝑃𝐿𝑥) = (𝑃𝐿0))
2 oveq2 6800 . . . . . . . 8 (𝑥 = 0 → (𝐾𝑥) = (𝐾↑0))
31, 2eqeq12d 2785 . . . . . . 7 (𝑥 = 0 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿0) = (𝐾↑0)))
43imbi2d 329 . . . . . 6 (𝑥 = 0 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿0) = (𝐾↑0))))
5 oveq2 6800 . . . . . . . 8 (𝑥 = 𝑦 → (𝑃𝐿𝑥) = (𝑃𝐿𝑦))
6 oveq2 6800 . . . . . . . 8 (𝑥 = 𝑦 → (𝐾𝑥) = (𝐾𝑦))
75, 6eqeq12d 2785 . . . . . . 7 (𝑥 = 𝑦 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿𝑦) = (𝐾𝑦)))
87imbi2d 329 . . . . . 6 (𝑥 = 𝑦 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑦) = (𝐾𝑦))))
9 oveq2 6800 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑃𝐿𝑥) = (𝑃𝐿(𝑦 + 1)))
10 oveq2 6800 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝐾𝑥) = (𝐾↑(𝑦 + 1)))
119, 10eqeq12d 2785 . . . . . . 7 (𝑥 = (𝑦 + 1) → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
1211imbi2d 329 . . . . . 6 (𝑥 = (𝑦 + 1) → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1)))))
13 oveq2 6800 . . . . . . . 8 (𝑥 = 𝑁 → (𝑃𝐿𝑥) = (𝑃𝐿𝑁))
14 oveq2 6800 . . . . . . . 8 (𝑥 = 𝑁 → (𝐾𝑥) = (𝐾𝑁))
1513, 14eqeq12d 2785 . . . . . . 7 (𝑥 = 𝑁 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿𝑁) = (𝐾𝑁)))
1615imbi2d 329 . . . . . 6 (𝑥 = 𝑁 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑁) = (𝐾𝑁))))
17 rusgrusgr 26694 . . . . . . . . 9 (𝐺RegUSGraph𝐾𝐺 ∈ USGraph)
18 usgruspgr 26294 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
1917, 18syl 17 . . . . . . . 8 (𝐺RegUSGraph𝐾𝐺 ∈ USPGraph)
20 simpr 471 . . . . . . . 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 27117 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑃𝐿0) = 1)
2419, 20, 23syl2anr 576 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿0) = 1)
25 simpl 468 . . . . . . . . . . 11 ((𝑉 ∈ Fin ∧ 𝑃𝑉) → 𝑉 ∈ Fin)
2625, 17anim12ci 593 . . . . . . . . . 10 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2721isfusgr 26432 . . . . . . . . . 10 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2826, 27sylibr 224 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → 𝐺 ∈ FinUSGraph)
29 simpr 471 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → 𝐺RegUSGraph𝐾)
30 ne0i 4067 . . . . . . . . . 10 (𝑃𝑉𝑉 ≠ ∅)
3130ad2antlr 698 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → 𝑉 ≠ ∅)
3221frusgrnn0 26701 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝐺RegUSGraph𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
3332nn0cnd 11554 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝐺RegUSGraph𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℂ)
3428, 29, 31, 33syl3anc 1475 . . . . . . . 8 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → 𝐾 ∈ ℂ)
3534exp0d 13208 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝐾↑0) = 1)
3624, 35eqtr4d 2807 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿0) = (𝐾↑0))
37 simpl 468 . . . . . . . . . . 11 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑉 ∈ Fin ∧ 𝑃𝑉))
3837anim1i 594 . . . . . . . . . 10 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝑦 ∈ ℕ0))
39 df-3an 1072 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0) ↔ ((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝑦 ∈ ℕ0))
4038, 39sylibr 224 . . . . . . . . 9 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) ∧ 𝑦 ∈ ℕ0) → (𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0))
4121, 22rusgrnumwwlks 27120 . . . . . . . . 9 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0)) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
4229, 40, 41syl2an2r 656 . . . . . . . 8 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
4342expcom 398 . . . . . . 7 (𝑦 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1)))))
4443a2d 29 . . . . . 6 (𝑦 ∈ ℕ0 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑦) = (𝐾𝑦)) → (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1)))))
454, 8, 12, 16, 36, 44nn0ind 11673 . . . . 5 (𝑁 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑁) = (𝐾𝑁)))
4645expd 400 . . . 4 (𝑁 ∈ ℕ0 → ((𝑉 ∈ Fin ∧ 𝑃𝑉) → (𝐺RegUSGraph𝐾 → (𝑃𝐿𝑁) = (𝐾𝑁))))
4746com12 32 . . 3 ((𝑉 ∈ Fin ∧ 𝑃𝑉) → (𝑁 ∈ ℕ0 → (𝐺RegUSGraph𝐾 → (𝑃𝐿𝑁) = (𝐾𝑁))))
48473impia 1108 . 2 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝐺RegUSGraph𝐾 → (𝑃𝐿𝑁) = (𝐾𝑁)))
4948impcom 394 1 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  {crab 3064  c0 4061   class class class wbr 4784  cfv 6031  (class class class)co 6792  cmpt2 6794  Fincfn 8108  cc 10135  0cc0 10137  1c1 10138   + caddc 10140  0cn0 11493  cexp 13066  chash 13320  Vtxcvtx 26094  USPGraphcuspgr 26264  USGraphcusgr 26265  FinUSGraphcfusgr 26430  RegUSGraphcrusgr 26686   WWalksN cwwlksn 26953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-disj 4753  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-sup 8503  df-oi 8570  df-card 8964  df-cda 9191  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-2 11280  df-3 11281  df-n0 11494  df-xnn0 11565  df-z 11579  df-uz 11888  df-rp 12035  df-xadd 12151  df-fz 12533  df-fzo 12673  df-seq 13008  df-exp 13067  df-hash 13321  df-word 13494  df-lsw 13495  df-concat 13496  df-s1 13497  df-substr 13498  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-clim 14426  df-sum 14624  df-vtx 26096  df-iedg 26097  df-edg 26160  df-uhgr 26173  df-ushgr 26174  df-upgr 26197  df-umgr 26198  df-uspgr 26266  df-usgr 26267  df-fusgr 26431  df-nbgr 26447  df-vtxdg 26596  df-rgr 26687  df-rusgr 26688  df-wwlks 26957  df-wwlksn 26958
This theorem is referenced by:  rusgrnumwwlkg  27122
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