Theorem rusgrnumwwlk 30446

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 7421	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑃𝐿𝑥) = (𝑃𝐿0))
2		oveq2 7421	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾↑𝑥) = (𝐾↑0))
3	1, 2	eqeq12d 2776	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝑃𝐿𝑥) = (𝐾↑𝑥) ↔ (𝑃𝐿0) = (𝐾↑0)))
4	3	imbi2d 343	. . . . . 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 7421	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑃𝐿𝑥) = (𝑃𝐿𝑦))
6		oveq2 7421	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐾↑𝑥) = (𝐾↑𝑦))
7	5, 6	eqeq12d 2776	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑃𝐿𝑥) = (𝐾↑𝑥) ↔ (𝑃𝐿𝑦) = (𝐾↑𝑦)))
8	7	imbi2d 343	. . . . . 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 7421	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝑃𝐿𝑥) = (𝑃𝐿(𝑦 + 1)))
10		oveq2 7421	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝐾↑𝑥) = (𝐾↑(𝑦 + 1)))
11	9, 10	eqeq12d 2776	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝑃𝐿𝑥) = (𝐾↑𝑥) ↔ (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
12	11	imbi2d 343	. . . . . 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 7421	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑃𝐿𝑥) = (𝑃𝐿𝑁))
14		oveq2 7421	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐾↑𝑥) = (𝐾↑𝑁))
15	13, 14	eqeq12d 2776	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝑃𝐿𝑥) = (𝐾↑𝑥) ↔ (𝑃𝐿𝑁) = (𝐾↑𝑁)))
16	15	imbi2d 343	. . . . . 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 30024	. . . . . . . . 9 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
18		usgruspgr 29640	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
19	17, 18	syl 18	. . . . . . . 8 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USPGraph)
20		simpr 490	. . . . . . . 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 30442	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1)
24	19, 20, 23	syl2anr 609	. . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = 1)
25		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → 𝑉 ∈ Fin)
26	25, 17	anim12ci 626	. . . . . . . . . 10 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
27	21	isfusgr 29778	. . . . . . . . . 10 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
28	26, 27	sylibr 237	. . . . . . . . 9 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
29		simpr 490	. . . . . . . . 9 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾)
30		ne0i 4287	. . . . . . . . . 10 ⊢ (𝑃 ∈ 𝑉 → 𝑉 ≠ ∅)
31	30	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝑉 ≠ ∅)
32	21	frusgrnn0 30031	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
33	32	nn0cnd 12591	. . . . . . . . 9 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℂ)
34	28, 29, 31, 33	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 ∈ ℂ)
35	34	exp0d 14204	. . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝐾↑0) = 1)
36	24, 35	eqtr4d 2798	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = (𝐾↑0))
37		simpl 488	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉))
38	37	anim1i 627	. . . . . . . . . 10 ⊢ ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝑦 ∈ ℕ0))
39		df-3an 1105	. . . . . . . . . 10 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑦 ∈ ℕ0) ↔ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝑦 ∈ ℕ0))
40	38, 39	sylibr 237	. . . . . . . . 9 ⊢ ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑦 ∈ ℕ0))
41	21, 22	rusgrnumwwlks 30445	. . . . . . . . 9 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑦 ∈ ℕ0)) → ((𝑃𝐿𝑦) = (𝐾↑𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
42	29, 40, 41	syl2an2r 698	. . . . . . . 8 ⊢ ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → ((𝑃𝐿𝑦) = (𝐾↑𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
43	42	expcom 419	. . . . . . 7 ⊢ (𝑦 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → ((𝑃𝐿𝑦) = (𝐾↑𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1)))))
44	43	a2d 30	. . . . . 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 12716	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿𝑁) = (𝐾↑𝑁)))
46	45	expd 421	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → (𝐺 RegUSGraph 𝐾 → (𝑃𝐿𝑁) = (𝐾↑𝑁))))
47	46	com12 33	. . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → (𝑁 ∈ ℕ0 → (𝐺 RegUSGraph 𝐾 → (𝑃𝐿𝑁) = (𝐾↑𝑁))))
48	47	3impia 1135	. 2 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐺 RegUSGraph 𝐾 → (𝑃𝐿𝑁) = (𝐾↑𝑁)))
49	48	impcom 413	1 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾↑𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  {crab 3412  ∅c0 4279   class class class wbr 5103  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  Fincfn 8952  ℂcc 11122  0cc0 11124  1c1 11125   + caddc 11127  ℕ₀cn0 12528  ↑cexp 14125  ♯chash 14394  Vtxcvtx 29453  USPGraphcuspgr 29608  USGraphcusgr 29609  FinUSGraphcfusgr 29776   RegUSGraph crusgr 30016   WWalksN cwwlksn 30294

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-disj 5071  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-oadd 8459  df-er 8696  df-map 8828  df-pm 8829  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-oi 9482  df-dju 9906  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-n0 12529  df-xnn0 12602  df-z 12616  df-uz 12888  df-rp 13043  df-xadd 13164  df-fz 13562  df-fzo 13710  df-seq 14066  df-exp 14126  df-hash 14395  df-word 14579  df-lsw 14628  df-concat 14636  df-s1 14663  df-substr 14709  df-pfx 14741  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-clim 15575  df-sum 15774  df-vtx 29455  df-iedg 29456  df-edg 29505  df-uhgr 29515  df-ushgr 29516  df-upgr 29539  df-umgr 29540  df-uspgr 29610  df-usgr 29611  df-fusgr 29777  df-nbgr 29793  df-vtxdg 29926  df-rgr 30017  df-rusgr 30018  df-wwlks 30298  df-wwlksn 30299

This theorem is used by:  rusgrnumwwlkg  30447