Theorem rusgrnumwwlk 30576

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 7432	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑃𝐿𝑥) = (𝑃𝐿0))
2		oveq2 7432	. . . . . . . 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 7432	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑃𝐿𝑥) = (𝑃𝐿𝑦))
6		oveq2 7432	. . . . . . . 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 7432	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝑃𝐿𝑥) = (𝑃𝐿(𝑦 + 1)))
10		oveq2 7432	. . . . . . . 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 7432	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑃𝐿𝑥) = (𝑃𝐿𝑁))
14		oveq2 7432	. . . . . . . 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 30154	. . . . . . . . 9 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
18		usgruspgr 29770	. . . . . . . . 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 30572	. . . . . . . 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 29908	. . . . . . . . . 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 4286	. . . . . . . . . 10 ⊢ (𝑃 ∈ 𝑉 → 𝑉 ≠ ∅)
31	30	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝑉 ≠ ∅)
32	21	frusgrnn0 30161	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
33	32	nn0cnd 12676	. . . . . . . . 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 14290	. . . . . . 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 30575	. . . . . . . . 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 12801	. . . . 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 1569   ∈ wcel 2144   ≠ wne 2955  {crab 3412  ∅c0 4277   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424   ∈ cmpo 7426  Fincfn 8980  ℂcc 11205  0cc0 11207  1c1 11208   + caddc 11210  ℕ₀cn0 12613  ↑cexp 14211  ♯chash 14481  Vtxcvtx 29583  USPGraphcuspgr 29738  USGraphcusgr 29739  FinUSGraphcfusgr 29906   RegUSGraph crusgr 30146   WWalksN cwwlksn 30424

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284  ax-pre-sup 11285

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-disj 5070  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-2o 8484  df-oadd 8487  df-er 8724  df-map 8856  df-pm 8857  df-en 8981  df-dom 8982  df-sdom 8983  df-fin 8984  df-sup 9441  df-oi 9511  df-dju 9989  df-card 10027  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-div 11981  df-nn 12343  df-2 12412  df-3 12413  df-n0 12614  df-xnn0 12687  df-z 12701  df-uz 12973  df-rp 13128  df-xadd 13249  df-fz 13647  df-fzo 13796  df-seq 14152  df-exp 14212  df-hash 14482  df-word 14666  df-lsw 14715  df-concat 14723  df-s1 14750  df-substr 14797  df-pfx 14829  df-cj 15275  df-re 15276  df-im 15277  df-sqrt 15411  df-abs 15412  df-clim 15664  df-sum 15863  df-vtx 29585  df-iedg 29586  df-edg 29635  df-uhgr 29645  df-ushgr 29646  df-upgr 29669  df-umgr 29670  df-uspgr 29740  df-usgr 29741  df-fusgr 29907  df-nbgr 29923  df-vtxdg 30056  df-rgr 30147  df-rusgr 30148  df-wwlks 30428  df-wwlksn 30429

This theorem is used by:  rusgrnumwwlkg  30577