Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rusgranumwlk Structured version   Visualization version   GIF version

Theorem rusgranumwlk 26484
 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. We denote with (𝑊‘𝑛) the set of walks with length n (in a given undirected simple graph) and with (𝑣𝐿𝑛) the number of walks with length n starting at the vertex v. This theorem corresponds to statement 11 in [Huneke] p. 2: "The total number of walks v(0) v(1) ... v(n-2) from a fixed vertex v = v(0) is k^(n-2) as G is k-regular.". Because of the k-regularity, the walk can be continued in k different ways at each vertex in the walk, therefore n times. This theorem even holds for n=0: then the walk consists only of one vertex v(0), so the number of walks of length n=0 starting with v=v(0) is 1=k^0. (Contributed by Alexander van der Vekens, 24-Aug-2018.)
Hypotheses
Ref Expression
rusgranumwlk.w 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})
rusgranumwlk.l 𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))
Assertion
Ref Expression
rusgranumwlk ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾𝑁))
Distinct variable groups:   𝐸,𝑐,𝑛   𝑁,𝑐,𝑛   𝑉,𝑐,𝑛   𝑣,𝑁,𝑤   𝑃,𝑛,𝑣,𝑤   𝑣,𝑉   𝑛,𝑊,𝑣,𝑤   𝑤,𝑉,𝑐   𝑣,𝐸,𝑤   𝑤,𝐾
Allowed substitution hints:   𝑃(𝑐)   𝐾(𝑣,𝑛,𝑐)   𝐿(𝑤,𝑣,𝑛,𝑐)   𝑊(𝑐)

Proof of Theorem rusgranumwlk
Dummy variables 𝑖 𝑦 𝑥 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) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 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) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 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) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 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) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑃𝐿𝑥) = (𝐾𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑃𝐿𝑁) = (𝐾𝑁))))
17 rusisusgra 26458 . . . . . . . 8 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 USGrph 𝐸)
18 simpr 476 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝑃𝑉) → 𝑃𝑉)
19 rusgranumwlk.w . . . . . . . . 9 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})
20 rusgranumwlk.l . . . . . . . . 9 𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))
2119, 20rusgranumwlkb0 26480 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑃𝑉) → (𝑃𝐿0) = 1)
2217, 18, 21syl2anr 494 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑃𝐿0) = 1)
23 rusgraprop 26456 . . . . . . . . . 10 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑖𝑉 ((𝑉 VDeg 𝐸)‘𝑖) = 𝐾))
24 nn0cn 11179 . . . . . . . . . . 11 (𝐾 ∈ ℕ0𝐾 ∈ ℂ)
25243ad2ant2 1076 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑖𝑉 ((𝑉 VDeg 𝐸)‘𝑖) = 𝐾) → 𝐾 ∈ ℂ)
26 exp0 12726 . . . . . . . . . 10 (𝐾 ∈ ℂ → (𝐾↑0) = 1)
2723, 25, 263syl 18 . . . . . . . . 9 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝐾↑0) = 1)
2827eqcomd 2616 . . . . . . . 8 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → 1 = (𝐾↑0))
2928adantl 481 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → 1 = (𝐾↑0))
3022, 29eqtrd 2644 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑃𝐿0) = (𝐾↑0))
31 simplr 788 . . . . . . . . 9 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ 𝑦 ∈ ℕ0) → ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾)
32 simpl 472 . . . . . . . . . . 11 (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑉 ∈ Fin ∧ 𝑃𝑉))
3332anim1i 590 . . . . . . . . . 10 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝑦 ∈ ℕ0))
34 df-3an 1033 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0) ↔ ((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ 𝑦 ∈ ℕ0))
3533, 34sylibr 223 . . . . . . . . 9 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ 𝑦 ∈ ℕ0) → (𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0))
3619, 20rusgranumwlks 26483 . . . . . . . . 9 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑦 ∈ ℕ0)) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
3731, 35, 36syl2anc 691 . . . . . . . 8 ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
3837expcom 450 . . . . . . 7 (𝑦 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((𝑃𝐿𝑦) = (𝐾𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1)))))
3938a2d 29 . . . . . 6 (𝑦 ∈ ℕ0 → ((((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑃𝐿𝑦) = (𝐾𝑦)) → (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1)))))
404, 8, 12, 16, 30, 39nn0ind 11348 . . . . 5 (𝑁 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃𝑉) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑃𝐿𝑁) = (𝐾𝑁)))
4140expd 451 . . . 4 (𝑁 ∈ ℕ0 → ((𝑉 ∈ Fin ∧ 𝑃𝑉) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑃𝐿𝑁) = (𝐾𝑁))))
4241com12 32 . . 3 ((𝑉 ∈ Fin ∧ 𝑃𝑉) → (𝑁 ∈ ℕ0 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑃𝐿𝑁) = (𝐾𝑁))))
43423impia 1253 . 2 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑃𝐿𝑁) = (𝐾𝑁)))
4443impcom 445 1 ((⟨𝑉, 𝐸⟩ RegUSGrph 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  ∀wral 2896  {crab 2900  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Fincfn 7841  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818  ℕ0cn0 11169  ↑cexp 12722  #chash 12979   USGrph cusg 25859   Walks cwalk 26026   VDeg cvdg 26420   RegUSGrph crusgra 26450 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-usgra 25862  df-nbgra 25949  df-wlk 26036  df-wwlk 26207  df-wwlkn 26208  df-vdgr 26421  df-rgra 26451  df-rusgra 26452 This theorem is referenced by:  rusgranumwlkg  26485
 Copyright terms: Public domain W3C validator