Theorem rusgrnumwwlk 29662

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 7420	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑃𝐿𝑥) = (𝑃𝐿0))
2		oveq2 7420	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾↑𝑥) = (𝐾↑0))
3	1, 2	eqeq12d 2747	. . . . . . 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 7420	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑃𝐿𝑥) = (𝑃𝐿𝑦))
6		oveq2 7420	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐾↑𝑥) = (𝐾↑𝑦))
7	5, 6	eqeq12d 2747	. . . . . . 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 7420	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝑃𝐿𝑥) = (𝑃𝐿(𝑦 + 1)))
10		oveq2 7420	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝐾↑𝑥) = (𝐾↑(𝑦 + 1)))
11	9, 10	eqeq12d 2747	. . . . . . 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 7420	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑃𝐿𝑥) = (𝑃𝐿𝑁))
14		oveq2 7420	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐾↑𝑥) = (𝐾↑𝑁))
15	13, 14	eqeq12d 2747	. . . . . . 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 29254	. . . . . . . . 9 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
18		usgruspgr 28871	. . . . . . . . 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 29658	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1)
24	19, 20, 23	syl2anr 596	. . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝑃𝐿0) = 1)
25		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → 𝑉 ∈ Fin)
26	25, 17	anim12ci 613	. . . . . . . . . 10 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
27	21	isfusgr 29008	. . . . . . . . . 10 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
28	26, 27	sylibr 233	. . . . . . . . 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 4334	. . . . . . . . . 10 ⊢ (𝑃 ∈ 𝑉 → 𝑉 ≠ ∅)
31	30	ad2antlr 724	. . . . . . . . 9 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝑉 ≠ ∅)
32	21	frusgrnn0 29261	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
33	32	nn0cnd 12541	. . . . . . . . 9 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℂ)
34	28, 29, 31, 33	syl3anc 1370	. . . . . . . 8 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 ∈ ℂ)
35	34	exp0d 14112	. . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) → (𝐾↑0) = 1)
36	24, 35	eqtr4d 2774	. . . . . 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 614	. . . . . . . . . 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 233	. . . . . . . . 9 ⊢ ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐾) ∧ 𝑦 ∈ ℕ0) → (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑦 ∈ ℕ0))
41	21, 22	rusgrnumwwlks 29661	. . . . . . . . 9 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑦 ∈ ℕ0)) → ((𝑃𝐿𝑦) = (𝐾↑𝑦) → (𝑃𝐿(𝑦 + 1)) = (𝐾↑(𝑦 + 1))))
42	29, 40, 41	syl2an2r 682	. . . . . . . 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 12664	. . . . 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 1116	. 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 2105   ≠ wne 2939  {crab 3431  ∅c0 4322   class class class wbr 5148  ‘cfv 6543  (class class class)co 7412   ∈ cmpo 7414  Fincfn 8945  ℂcc 11114  0cc0 11116  1c1 11117   + caddc 11119  ℕ₀cn0 12479  ↑cexp 14034  ♯chash 14297  Vtxcvtx 28689  USPGraphcuspgr 28841  USGraphcusgr 28842  FinUSGraphcfusgr 29006   RegUSGraph crusgr 29246   WWalksN cwwlksn 29513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-oadd 8476  df-er 8709  df-map 8828  df-pm 8829  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-oi 9511  df-dju 9902  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-xnn0 12552  df-z 12566  df-uz 12830  df-rp 12982  df-xadd 13100  df-fz 13492  df-fzo 13635  df-seq 13974  df-exp 14035  df-hash 14298  df-word 14472  df-lsw 14520  df-concat 14528  df-s1 14553  df-substr 14598  df-pfx 14628  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15439  df-sum 15640  df-vtx 28691  df-iedg 28692  df-edg 28741  df-uhgr 28751  df-ushgr 28752  df-upgr 28775  df-umgr 28776  df-uspgr 28843  df-usgr 28844  df-fusgr 29007  df-nbgr 29023  df-vtxdg 29156  df-rgr 29247  df-rusgr 29248  df-wwlks 29517  df-wwlksn 29518

This theorem is referenced by:  rusgrnumwwlkg  29663