Theorem rusgrnumwwlk 29903

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 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))

Assertion

Ref	Expression
rusgrnumwwlk	⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾↑𝑁))

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.

Step	Hyp	Ref	Expression
1		oveq2 7411	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑃𝐿𝑥) = (𝑃𝐿0))
2		oveq2 7411	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾↑𝑥) = (𝐾↑0))
3	1, 2	eqeq12d 2751	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝑃𝐿𝑥) = (𝐾↑𝑥) ↔ (𝑃𝐿0) = (𝐾↑0)))
4	3	imbi2d 340	. . . . . 6 ⊢ (𝑥 = 0 → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑥) = (𝐾↑𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = (𝐾↑0))))
5		oveq2 7411	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑃𝐿𝑥) = (𝑃𝐿𝑦))
6		oveq2 7411	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐾↑𝑥) = (𝐾↑𝑦))
7	5, 6	eqeq12d 2751	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑃𝐿𝑥) = (𝐾↑𝑥) ↔ (𝑃𝐿𝑦) = (𝐾↑𝑦)))
8	7	imbi2d 340	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑥) = (𝐾↑𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑦) = (𝐾↑𝑦))))
9		oveq2 7411	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝑃𝐿𝑥) = (𝑃𝐿(𝑦 + 1)))
10		oveq2 7411	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝐾↑𝑥) = (𝐾↑(𝑦 + 1)))
11	9, 10	eqeq12d 2751	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝑃𝐿𝑥) = (𝐾↑𝑥) ↔ (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
12	11	imbi2d 340	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑥) = (𝐾↑𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1)))))
13		oveq2 7411	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑃𝐿𝑥) = (𝑃𝐿𝑁))
14		oveq2 7411	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐾↑𝑥) = (𝐾↑𝑁))
15	13, 14	eqeq12d 2751	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝑃𝐿𝑥) = (𝐾↑𝑥) ↔ (𝑃𝐿𝑁) = (𝐾↑𝑁)))
16	15	imbi2d 340	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑥) = (𝐾↑𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑁) = (𝐾↑𝑁))))
17		rusgrusgr 29490	. . . . . . . . 9 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
18		usgruspgr 29105	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
19	17, 18	syl 17	. . . . . . . 8 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USPGraph)
20		simpr 484	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → 𝑃 ∈ 𝑉)
21		rusgrnumwwlk.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
22		rusgrnumwwlk.l	. . . . . . . . 9 ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
23	21, 22	rusgrnumwwlkb0 29899	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1)
24	19, 20, 23	syl2anr 597	. . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = 1)
25		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → 𝑉 ∈ Fin)
26	25, 17	anim12ci 614	. . . . . . . . . 10 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
27	21	isfusgr 29243	. . . . . . . . . 10 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
28	26, 27	sylibr 234	. . . . . . . . 9 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
29		simpr 484	. . . . . . . . 9 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾)
30		ne0i 4316	. . . . . . . . . 10 ⊢ (𝑃 ∈ 𝑉 → 𝑉 ≠ ∅)
31	30	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝑉 ≠ ∅)
32	21	frusgrnn0 29497	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
33	32	nn0cnd 12562	. . . . . . . . 9 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℂ)
34	28, 29, 31, 33	syl3anc 1373	. . . . . . . 8 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 ∈ ℂ)
35	34	exp0d 14156	. . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝐾↑0) = 1)
36	24, 35	eqtr4d 2773	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = (𝐾↑0))
37		simpl 482	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉))
38	37	anim1i 615	. . . . . . . . . 10 ⊢ ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝑦 ∈ ℕ0))
39		df-3an 1088	. . . . . . . . . 10 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑦 ∈ ℕ0) ↔ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝑦 ∈ ℕ0))
40	38, 39	sylibr 234	. . . . . . . . 9 ⊢ ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑦 ∈ ℕ0))
41	21, 22	rusgrnumwwlks 29902	. . . . . . . . 9 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑦 ∈ ℕ0)) → ((𝑃𝐿𝑦) = (𝐾↑𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
42	29, 40, 41	syl2an2r 685	. . . . . . . 8 ⊢ ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑃𝐿𝑦) = (𝐾↑𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
43	42	expcom 413	. . . . . . 7 ⊢ (𝑦 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → ((𝑃𝐿𝑦) = (𝐾↑𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1)))))
44	43	a2d 29	. . . . . 6 ⊢ (𝑦 ∈ ℕ0 → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑦) = (𝐾↑𝑦)) → (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1)))))
45	4, 8, 12, 16, 36, 44	nn0ind 12686	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑁) = (𝐾↑𝑁)))
46	45	expd 415	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → (𝐺 RegUSGraph 𝐾 → (𝑃𝐿𝑁) = (𝐾↑𝑁))))
47	46	com12 32	. . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → (𝑁 ∈ ℕ0 → (𝐺 RegUSGraph 𝐾 → (𝑃𝐿𝑁) = (𝐾↑𝑁))))
48	47	3impia 1117	. 2 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐺 RegUSGraph 𝐾 → (𝑃𝐿𝑁) = (𝐾↑𝑁)))
49	48	impcom 407	1 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾↑𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  {crab 3415  ∅c0 4308   class class class wbr 5119  ‘cfv 6530  (class class class)co 7403   ∈ cmpo 7405  Fincfn 8957  ℂcc 11125  0cc0 11127  1c1 11128   + caddc 11130  ℕ₀cn0 12499  ↑cexp 14077  ♯chash 14346  Vtxcvtx 28921  USPGraphcuspgr 29073  USGraphcusgr 29074  FinUSGraphcfusgr 29241   RegUSGraph crusgr 29482   WWalksN cwwlksn 29754

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-inf2 9653  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-disj 5087  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-isom 6539  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-2o 8479  df-oadd 8482  df-er 8717  df-map 8840  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9452  df-oi 9522  df-dju 9913  df-card 9951  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-div 11893  df-nn 12239  df-2 12301  df-3 12302  df-n0 12500  df-xnn0 12573  df-z 12587  df-uz 12851  df-rp 13007  df-xadd 13127  df-fz 13523  df-fzo 13670  df-seq 14018  df-exp 14078  df-hash 14347  df-word 14530  df-lsw 14579  df-concat 14587  df-s1 14612  df-substr 14657  df-pfx 14687  df-cj 15116  df-re 15117  df-im 15118  df-sqrt 15252  df-abs 15253  df-clim 15502  df-sum 15701  df-vtx 28923  df-iedg 28924  df-edg 28973  df-uhgr 28983  df-ushgr 28984  df-upgr 29007  df-umgr 29008  df-uspgr 29075  df-usgr 29076  df-fusgr 29242  df-nbgr 29258  df-vtxdg 29392  df-rgr 29483  df-rusgr 29484  df-wwlks 29758  df-wwlksn 29759

This theorem is referenced by:  rusgrnumwwlkg  29904