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Theorem rusgrnumwwlk 29496
Description: In a 𝐟-regular graph, the number of walks of a fixed length 𝑁 from a fixed vertex is 𝐟 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 𝐟-regularity, a walk can be continued in 𝐟 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 (𝐟↑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 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↩ (♯‘{𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑣}))
Assertion
Ref Expression
rusgrnumwwlk ((𝐺 RegUSGraph 𝐟 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐟↑𝑁))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑃,𝑛,𝑣,𝑀   𝑛,𝑉,𝑣,𝑀   𝑀,𝐟
Allowed substitution hints:   𝐟(𝑣,𝑛)   𝐿(𝑀,𝑣,𝑛)

Proof of Theorem rusgrnumwwlk
Dummy variables 𝑥 𝑊 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7419 . . . . . . . 8 (𝑥 = 0 → (𝑃𝐿𝑥) = (𝑃𝐿0))
2 oveq2 7419 . . . . . . . 8 (𝑥 = 0 → (𝐟↑𝑥) = (𝐟↑0))
31, 2eqeq12d 2746 . . . . . . 7 (𝑥 = 0 → ((𝑃𝐿𝑥) = (𝐟↑𝑥) ↔ (𝑃𝐿0) = (𝐟↑0)))
43imbi2d 339 . . . . . 6 (𝑥 = 0 → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑥) = (𝐟↑𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿0) = (𝐟↑0))))
5 oveq2 7419 . . . . . . . 8 (𝑥 = 𝑊 → (𝑃𝐿𝑥) = (𝑃𝐿𝑊))
6 oveq2 7419 . . . . . . . 8 (𝑥 = 𝑊 → (𝐟↑𝑥) = (𝐟↑𝑊))
75, 6eqeq12d 2746 . . . . . . 7 (𝑥 = 𝑊 → ((𝑃𝐿𝑥) = (𝐟↑𝑥) ↔ (𝑃𝐿𝑊) = (𝐟↑𝑊)))
87imbi2d 339 . . . . . 6 (𝑥 = 𝑊 → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑥) = (𝐟↑𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑊) = (𝐟↑𝑊))))
9 oveq2 7419 . . . . . . . 8 (𝑥 = (𝑊 + 1) → (𝑃𝐿𝑥) = (𝑃𝐿(𝑊 + 1)))
10 oveq2 7419 . . . . . . . 8 (𝑥 = (𝑊 + 1) → (𝐟↑𝑥) = (𝐟↑(𝑊 + 1)))
119, 10eqeq12d 2746 . . . . . . 7 (𝑥 = (𝑊 + 1) → ((𝑃𝐿𝑥) = (𝐟↑𝑥) ↔ (𝑃𝐿(𝑊 + 1)) = (𝐟↑(𝑊 + 1))))
1211imbi2d 339 . . . . . 6 (𝑥 = (𝑊 + 1) → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑥) = (𝐟↑𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿(𝑊 + 1)) = (𝐟↑(𝑊 + 1)))))
13 oveq2 7419 . . . . . . . 8 (𝑥 = 𝑁 → (𝑃𝐿𝑥) = (𝑃𝐿𝑁))
14 oveq2 7419 . . . . . . . 8 (𝑥 = 𝑁 → (𝐟↑𝑥) = (𝐟↑𝑁))
1513, 14eqeq12d 2746 . . . . . . 7 (𝑥 = 𝑁 → ((𝑃𝐿𝑥) = (𝐟↑𝑥) ↔ (𝑃𝐿𝑁) = (𝐟↑𝑁)))
1615imbi2d 339 . . . . . 6 (𝑥 = 𝑁 → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑥) = (𝐟↑𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑁) = (𝐟↑𝑁))))
17 rusgrusgr 29088 . . . . . . . . 9 (𝐺 RegUSGraph 𝐟 → 𝐺 ∈ USGraph)
18 usgruspgr 28705 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
1917, 18syl 17 . . . . . . . 8 (𝐺 RegUSGraph 𝐟 → 𝐺 ∈ USPGraph)
20 simpr 483 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → 𝑃 ∈ 𝑉)
21 rusgrnumwwlk.v . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
22 rusgrnumwwlk.l . . . . . . . . 9 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↩ (♯‘{𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑣}))
2321, 22rusgrnumwwlkb0 29492 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1)
2419, 20, 23syl2anr 595 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿0) = 1)
25 simpl 481 . . . . . . . . . . 11 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → 𝑉 ∈ Fin)
2625, 17anim12ci 612 . . . . . . . . . 10 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2721isfusgr 28842 . . . . . . . . . 10 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2826, 27sylibr 233 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → 𝐺 ∈ FinUSGraph)
29 simpr 483 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → 𝐺 RegUSGraph 𝐟)
30 ne0i 4333 . . . . . . . . . 10 (𝑃 ∈ 𝑉 → 𝑉 ≠ ∅)
3130ad2antlr 723 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → 𝑉 ≠ ∅)
3221frusgrnn0 29095 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐟 ∧ 𝑉 ≠ ∅) → 𝐟 ∈ ℕ0)
3332nn0cnd 12538 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐟 ∧ 𝑉 ≠ ∅) → 𝐟 ∈ ℂ)
3428, 29, 31, 33syl3anc 1369 . . . . . . . 8 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → 𝐟 ∈ ℂ)
3534exp0d 14109 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝐟↑0) = 1)
3624, 35eqtr4d 2773 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿0) = (𝐟↑0))
37 simpl 481 . . . . . . . . . . 11 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉))
3837anim1i 613 . . . . . . . . . 10 ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) ∧ 𝑊 ∈ ℕ0) → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝑊 ∈ ℕ0))
39 df-3an 1087 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑊 ∈ ℕ0) ↔ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝑊 ∈ ℕ0))
4038, 39sylibr 233 . . . . . . . . 9 ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) ∧ 𝑊 ∈ ℕ0) → (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑊 ∈ ℕ0))
4121, 22rusgrnumwwlks 29495 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐟 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑊 ∈ ℕ0)) → ((𝑃𝐿𝑊) = (𝐟↑𝑊) → (𝑃𝐿(𝑊 + 1)) = (𝐟↑(𝑊 + 1))))
4229, 40, 41syl2an2r 681 . . . . . . . 8 ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) ∧ 𝑊 ∈ ℕ0) → ((𝑃𝐿𝑊) = (𝐟↑𝑊) → (𝑃𝐿(𝑊 + 1)) = (𝐟↑(𝑊 + 1))))
4342expcom 412 . . . . . . 7 (𝑊 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → ((𝑃𝐿𝑊) = (𝐟↑𝑊) → (𝑃𝐿(𝑊 + 1)) = (𝐟↑(𝑊 + 1)))))
4443a2d 29 . . . . . 6 (𝑊 ∈ ℕ0 → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑊) = (𝐟↑𝑊)) → (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿(𝑊 + 1)) = (𝐟↑(𝑊 + 1)))))
454, 8, 12, 16, 36, 44nn0ind 12661 . . . . 5 (𝑁 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑁) = (𝐟↑𝑁)))
4645expd 414 . . . 4 (𝑁 ∈ ℕ0 → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → (𝐺 RegUSGraph 𝐟 → (𝑃𝐿𝑁) = (𝐟↑𝑁))))
4746com12 32 . . 3 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → (𝑁 ∈ ℕ0 → (𝐺 RegUSGraph 𝐟 → (𝑃𝐿𝑁) = (𝐟↑𝑁))))
48473impia 1115 . 2 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐺 RegUSGraph 𝐟 → (𝑃𝐿𝑁) = (𝐟↑𝑁)))
4948impcom 406 1 ((𝐺 RegUSGraph 𝐟 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐟↑𝑁))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  {crab 3430  âˆ…c0 4321   class class class wbr 5147  â€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  Fincfn 8941  â„‚cc 11110  0cc0 11112  1c1 11113   + caddc 11115  â„•0cn0 12476  â†‘cexp 14031  â™¯chash 14294  Vtxcvtx 28523  USPGraphcuspgr 28675  USGraphcusgr 28676  FinUSGraphcfusgr 28840   RegUSGraph crusgr 29080   WWalksN cwwlksn 29347
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-disj 5113  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-oi 9507  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-rp 12979  df-xadd 13097  df-fz 13489  df-fzo 13632  df-seq 13971  df-exp 14032  df-hash 14295  df-word 14469  df-lsw 14517  df-concat 14525  df-s1 14550  df-substr 14595  df-pfx 14625  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-sum 15637  df-vtx 28525  df-iedg 28526  df-edg 28575  df-uhgr 28585  df-ushgr 28586  df-upgr 28609  df-umgr 28610  df-uspgr 28677  df-usgr 28678  df-fusgr 28841  df-nbgr 28857  df-vtxdg 28990  df-rgr 29081  df-rusgr 29082  df-wwlks 29351  df-wwlksn 29352
This theorem is referenced by:  rusgrnumwwlkg  29497
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