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Theorem rusgrnumwwlk 41178
 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 6557 . . . . . . . 8 (𝑥 = 0 → (𝑃𝐿𝑥) = (𝑃𝐿0))
2 oveq2 6557 . . . . . . . 8 (𝑥 = 0 → (𝐾𝑥) = (𝐾↑0))
31, 2eqeq12d 2625 . . . . . . 7 (𝑥 = 0 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿0) = (𝐾↑0)))
43imbi2d 329 . . . . . 6 (𝑥 = 0 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = (𝐾↑0))))
5 oveq2 6557 . . . . . . . 8 (𝑥 = 𝑦 → (𝑃𝐿𝑥) = (𝑃𝐿𝑦))
6 oveq2 6557 . . . . . . . 8 (𝑥 = 𝑦 → (𝐾𝑥) = (𝐾𝑦))
75, 6eqeq12d 2625 . . . . . . 7 (𝑥 = 𝑦 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿𝑦) = (𝐾𝑦)))
87imbi2d 329 . . . . . 6 (𝑥 = 𝑦 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑦) = (𝐾𝑦))))
9 oveq2 6557 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑃𝐿𝑥) = (𝑃𝐿(𝑦 + 1)))
10 oveq2 6557 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝐾𝑥) = (𝐾↑(𝑦 + 1)))
119, 10eqeq12d 2625 . . . . . . 7 (𝑥 = (𝑦 + 1) → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
1211imbi2d 329 . . . . . 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 6557 . . . . . . . 8 (𝑥 = 𝑁 → (𝑃𝐿𝑥) = (𝑃𝐿𝑁))
14 oveq2 6557 . . . . . . . 8 (𝑥 = 𝑁 → (𝐾𝑥) = (𝐾𝑁))
1513, 14eqeq12d 2625 . . . . . . 7 (𝑥 = 𝑁 → ((𝑃𝐿𝑥) = (𝐾𝑥) ↔ (𝑃𝐿𝑁) = (𝐾𝑁)))
1615imbi2d 329 . . . . . 6 (𝑥 = 𝑁 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑁) = (𝐾𝑁))))
17 rusgrusgr 40764 . . . . . . . . 9 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph )
18 usgruspgr 40408 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
1917, 18syl 17 . . . . . . . 8 (𝐺 RegUSGraph 𝐾𝐺 ∈ USPGraph )
20 simpr 476 . . . . . . . 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 41174 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑃𝐿0) = 1)
2419, 20, 23syl2anr 494 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = 1)
25 simpl 472 . . . . . . . . . . 11 ((𝑉 ∈ Fin ∧ 𝑃𝑉) → 𝑉 ∈ Fin)
2625, 17anim12ci 589 . . . . . . . . . 10 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2721isfusgr 40537 . . . . . . . . . 10 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2826, 27sylibr 223 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph )
29 simpr 476 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾)
30 ne0i 3880 . . . . . . . . . 10 (𝑃𝑉𝑉 ≠ ∅)
3130ad2antlr 759 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝑉 ≠ ∅)
3221frusgrnn0 40771 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
3332nn0cnd 11230 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℂ)
3428, 29, 31, 33syl3anc 1318 . . . . . . . 8 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 ∈ ℂ)
3534exp0d 12864 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝐾↑0) = 1)
3624, 35eqtr4d 2647 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = (𝐾↑0))
37 simpl 472 . . . . . . . . . . 11 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑉 ∈ Fin ∧ 𝑃𝑉))
3837anim1i 590 . . . . . . . . . 10 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝑦 ∈ ℕ0))
39 df-3an 1033 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0) ↔ ((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝑦 ∈ ℕ0))
4038, 39sylibr 223 . . . . . . . . 9 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → (𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0))
4121, 22rusgrnumwwlks 41177 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0)) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
4229, 40, 41syl2an2r 872 . . . . . . . 8 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
4342expcom 450 . . . . . . 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 11348 . . . . 5 (𝑁 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑁) = (𝐾𝑁)))
4645expd 451 . . . 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 1253 . 2 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝐺 RegUSGraph 𝐾 → (𝑃𝐿𝑁) = (𝐾𝑁)))
4948impcom 445 1 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900  ∅c0 3874   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Fincfn 7841  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818  ℕ0cn0 11169  ↑cexp 12722  #chash 12979  Vtxcvtx 25673   USPGraph cuspgr 40378   USGraph cusgr 40379   FinUSGraph cfusgr 40535   RegUSGraph crusgr 40756   WWalkSN cwwlksn 41029 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-xadd 11823  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-vtx 25675  df-iedg 25676  df-uhgr 25724  df-ushgr 25725  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-fusgr 40536  df-nbgr 40554  df-vtxdg 40682  df-rgr 40757  df-rusgr 40758  df-wwlks 41033  df-wwlksn 41034 This theorem is referenced by:  rusgrnumwwlkg  41179
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