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Theorem rusgrnumwwlk 29995
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 7439 . . . . . . . 8 (𝑥 = 0 → (𝑃𝐿𝑥) = (𝑃𝐿0))
2 oveq2 7439 . . . . . . . 8 (𝑥 = 0 → (𝐾𝑥) = (𝐾↑0))
31, 2eqeq12d 2753 . . . . . . 7 (𝑥 = 0 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿0) = (𝐾↑0)))
43imbi2d 340 . . . . . 6 (𝑥 = 0 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = (𝐾↑0))))
5 oveq2 7439 . . . . . . . 8 (𝑥 = 𝑦 → (𝑃𝐿𝑥) = (𝑃𝐿𝑦))
6 oveq2 7439 . . . . . . . 8 (𝑥 = 𝑦 → (𝐾𝑥) = (𝐾𝑦))
75, 6eqeq12d 2753 . . . . . . 7 (𝑥 = 𝑦 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿𝑦) = (𝐾𝑦)))
87imbi2d 340 . . . . . 6 (𝑥 = 𝑦 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑦) = (𝐾𝑦))))
9 oveq2 7439 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑃𝐿𝑥) = (𝑃𝐿(𝑦 + 1)))
10 oveq2 7439 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝐾𝑥) = (𝐾↑(𝑦 + 1)))
119, 10eqeq12d 2753 . . . . . . 7 (𝑥 = (𝑦 + 1) → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
1211imbi2d 340 . . . . . 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 7439 . . . . . . . 8 (𝑥 = 𝑁 → (𝑃𝐿𝑥) = (𝑃𝐿𝑁))
14 oveq2 7439 . . . . . . . 8 (𝑥 = 𝑁 → (𝐾𝑥) = (𝐾𝑁))
1513, 14eqeq12d 2753 . . . . . . 7 (𝑥 = 𝑁 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿𝑁) = (𝐾𝑁)))
1615imbi2d 340 . . . . . 6 (𝑥 = 𝑁 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑁) = (𝐾𝑁))))
17 rusgrusgr 29582 . . . . . . . . 9 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
18 usgruspgr 29197 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
1917, 18syl 17 . . . . . . . 8 (𝐺 RegUSGraph 𝐾𝐺 ∈ USPGraph)
20 simpr 484 . . . . . . . 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 29991 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑃𝐿0) = 1)
2419, 20, 23syl2anr 597 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = 1)
25 simpl 482 . . . . . . . . . . 11 ((𝑉 ∈ Fin ∧ 𝑃𝑉) → 𝑉 ∈ Fin)
2625, 17anim12ci 614 . . . . . . . . . 10 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2721isfusgr 29335 . . . . . . . . . 10 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2826, 27sylibr 234 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
29 simpr 484 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾)
30 ne0i 4341 . . . . . . . . . 10 (𝑃𝑉𝑉 ≠ ∅)
3130ad2antlr 727 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝑉 ≠ ∅)
3221frusgrnn0 29589 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
3332nn0cnd 12589 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℂ)
3428, 29, 31, 33syl3anc 1373 . . . . . . . 8 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 ∈ ℂ)
3534exp0d 14180 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝐾↑0) = 1)
3624, 35eqtr4d 2780 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = (𝐾↑0))
37 simpl 482 . . . . . . . . . . 11 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑉 ∈ Fin ∧ 𝑃𝑉))
3837anim1i 615 . . . . . . . . . 10 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝑦 ∈ ℕ0))
39 df-3an 1089 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0) ↔ ((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝑦 ∈ ℕ0))
4038, 39sylibr 234 . . . . . . . . 9 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → (𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0))
4121, 22rusgrnumwwlks 29994 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0)) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
4229, 40, 41syl2an2r 685 . . . . . . . 8 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
4342expcom 413 . . . . . . 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 12713 . . . . 5 (𝑁 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑁) = (𝐾𝑁)))
4645expd 415 . . . 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 1118 . 2 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝐺 RegUSGraph 𝐾 → (𝑃𝐿𝑁) = (𝐾𝑁)))
4948impcom 407 1 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  {crab 3436  c0 4333   class class class wbr 5143  cfv 6561  (class class class)co 7431  cmpo 7433  Fincfn 8985  cc 11153  0cc0 11155  1c1 11156   + caddc 11158  0cn0 12526  cexp 14102  chash 14369  Vtxcvtx 29013  USPGraphcuspgr 29165  USGraphcusgr 29166  FinUSGraphcfusgr 29333   RegUSGraph crusgr 29574   WWalksN cwwlksn 29846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-rp 13035  df-xadd 13155  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-word 14553  df-lsw 14601  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-vtx 29015  df-iedg 29016  df-edg 29065  df-uhgr 29075  df-ushgr 29076  df-upgr 29099  df-umgr 29100  df-uspgr 29167  df-usgr 29168  df-fusgr 29334  df-nbgr 29350  df-vtxdg 29484  df-rgr 29575  df-rusgr 29576  df-wwlks 29850  df-wwlksn 29851
This theorem is referenced by:  rusgrnumwwlkg  29996
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