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Theorem rusgrnumwwlk 27312
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 6918 . . . . . . . 8 (𝑥 = 0 → (𝑃𝐿𝑥) = (𝑃𝐿0))
2 oveq2 6918 . . . . . . . 8 (𝑥 = 0 → (𝐾𝑥) = (𝐾↑0))
31, 2eqeq12d 2840 . . . . . . 7 (𝑥 = 0 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿0) = (𝐾↑0)))
43imbi2d 332 . . . . . 6 (𝑥 = 0 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿0) = (𝐾↑0))))
5 oveq2 6918 . . . . . . . 8 (𝑥 = 𝑦 → (𝑃𝐿𝑥) = (𝑃𝐿𝑦))
6 oveq2 6918 . . . . . . . 8 (𝑥 = 𝑦 → (𝐾𝑥) = (𝐾𝑦))
75, 6eqeq12d 2840 . . . . . . 7 (𝑥 = 𝑦 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿𝑦) = (𝐾𝑦)))
87imbi2d 332 . . . . . 6 (𝑥 = 𝑦 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑦) = (𝐾𝑦))))
9 oveq2 6918 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑃𝐿𝑥) = (𝑃𝐿(𝑦 + 1)))
10 oveq2 6918 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝐾𝑥) = (𝐾↑(𝑦 + 1)))
119, 10eqeq12d 2840 . . . . . . 7 (𝑥 = (𝑦 + 1) → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
1211imbi2d 332 . . . . . 6 (𝑥 = (𝑦 + 1) → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1)))))
13 oveq2 6918 . . . . . . . 8 (𝑥 = 𝑁 → (𝑃𝐿𝑥) = (𝑃𝐿𝑁))
14 oveq2 6918 . . . . . . . 8 (𝑥 = 𝑁 → (𝐾𝑥) = (𝐾𝑁))
1513, 14eqeq12d 2840 . . . . . . 7 (𝑥 = 𝑁 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿𝑁) = (𝐾𝑁)))
1615imbi2d 332 . . . . . 6 (𝑥 = 𝑁 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑁) = (𝐾𝑁))))
17 rusgrusgr 26869 . . . . . . . . 9 (𝐺RegUSGraph𝐾𝐺 ∈ USGraph)
18 usgruspgr 26484 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
1917, 18syl 17 . . . . . . . 8 (𝐺RegUSGraph𝐾𝐺 ∈ USPGraph)
20 simpr 479 . . . . . . . 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 27307 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑃𝐿0) = 1)
2419, 20, 23syl2anr 590 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿0) = 1)
25 simpl 476 . . . . . . . . . . 11 ((𝑉 ∈ Fin ∧ 𝑃𝑉) → 𝑉 ∈ Fin)
2625, 17anim12ci 607 . . . . . . . . . 10 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2721isfusgr 26622 . . . . . . . . . 10 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2826, 27sylibr 226 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → 𝐺 ∈ FinUSGraph)
29 simpr 479 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → 𝐺RegUSGraph𝐾)
30 ne0i 4152 . . . . . . . . . 10 (𝑃𝑉𝑉 ≠ ∅)
3130ad2antlr 718 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → 𝑉 ≠ ∅)
3221frusgrnn0 26876 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝐺RegUSGraph𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
3332nn0cnd 11687 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝐺RegUSGraph𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℂ)
3428, 29, 31, 33syl3anc 1494 . . . . . . . 8 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → 𝐾 ∈ ℂ)
3534exp0d 13303 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝐾↑0) = 1)
3624, 35eqtr4d 2864 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿0) = (𝐾↑0))
37 simpl 476 . . . . . . . . . . 11 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑉 ∈ Fin ∧ 𝑃𝑉))
3837anim1i 608 . . . . . . . . . 10 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝑦 ∈ ℕ0))
39 df-3an 1113 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0) ↔ ((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝑦 ∈ ℕ0))
4038, 39sylibr 226 . . . . . . . . 9 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) ∧ 𝑦 ∈ ℕ0) → (𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0))
4121, 22rusgrnumwwlks 27310 . . . . . . . . 9 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0)) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
4229, 40, 41syl2an2r 675 . . . . . . . 8 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
4342expcom 404 . . . . . . 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 11807 . . . . 5 (𝑁 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺RegUSGraph𝐾) → (𝑃𝐿𝑁) = (𝐾𝑁)))
4645expd 406 . . . 4 (𝑁 ∈ ℕ0 → ((𝑉 ∈ Fin ∧ 𝑃𝑉) → (𝐺RegUSGraph𝐾 → (𝑃𝐿𝑁) = (𝐾𝑁))))
4746com12 32 . . 3 ((𝑉 ∈ Fin ∧ 𝑃𝑉) → (𝑁 ∈ ℕ0 → (𝐺RegUSGraph𝐾 → (𝑃𝐿𝑁) = (𝐾𝑁))))
48473impia 1149 . 2 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝐺RegUSGraph𝐾 → (𝑃𝐿𝑁) = (𝐾𝑁)))
4948impcom 398 1 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999  {crab 3121  c0 4146   class class class wbr 4875  cfv 6127  (class class class)co 6910  cmpt2 6912  Fincfn 8228  cc 10257  0cc0 10259  1c1 10260   + caddc 10262  0cn0 11625  cexp 13161  chash 13417  Vtxcvtx 26301  USPGraphcuspgr 26454  USGraphcusgr 26455  FinUSGraphcfusgr 26620  RegUSGraphcrusgr 26861   WWalksN cwwlksn 27132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-inf2 8822  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336  ax-pre-sup 10337
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-disj 4844  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-map 8129  df-pm 8130  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-sup 8623  df-oi 8691  df-card 9085  df-cda 9312  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-div 11017  df-nn 11358  df-2 11421  df-3 11422  df-n0 11626  df-xnn0 11698  df-z 11712  df-uz 11976  df-rp 12120  df-xadd 12240  df-fz 12627  df-fzo 12768  df-seq 13103  df-exp 13162  df-hash 13418  df-word 13582  df-lsw 13630  df-concat 13638  df-s1 13663  df-substr 13708  df-pfx 13757  df-cj 14223  df-re 14224  df-im 14225  df-sqrt 14359  df-abs 14360  df-clim 14603  df-sum 14801  df-vtx 26303  df-iedg 26304  df-edg 26353  df-uhgr 26363  df-ushgr 26364  df-upgr 26387  df-umgr 26388  df-uspgr 26456  df-usgr 26457  df-fusgr 26621  df-nbgr 26637  df-vtxdg 26771  df-rgr 26862  df-rusgr 26863  df-wwlks 27136  df-wwlksn 27137
This theorem is referenced by:  rusgrnumwwlkg  27313
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