Theorem numclwwlk1 27231

Description: Statement 9 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v is kf(n-2)". Since 𝐺 is k-regular, the vertex v(n-2) = v has k neighbors v(n-1), so there are k walks from v(n-2) = v to v(n) = v (via each of v's neighbors) completing each of the f(n-2) walks from v=v(0) to v(n-2)=v. This theorem holds even for k=0, but only for finite graphs! (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 29-May-2021.)

Hypotheses

Ref	Expression
extwwlkfab.v	⊢ 𝑉 = (Vtx‘𝐺)
extwwlkfab.f	⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
extwwlkfab.c	⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})

Assertion

Ref	Expression
numclwwlk1	⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (#‘(𝑋𝐶𝑁)) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))

Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑛,𝑉,𝑣,𝑤   𝑛,𝑋,𝑣,𝑤   𝑤,𝐹

Allowed substitution hints:   𝐶(𝑤,𝑣,𝑛)   𝐹(𝑣,𝑛)   𝐾(𝑤,𝑣,𝑛)

Proof of Theorem numclwwlk1

Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . 3 ⊢ (𝑋𝐶𝑁) ∈ V
2		rusgrusgr 26460	. . . . 5 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph )
3	2	ad2antlr 763	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐺 ∈ USGraph )
4		simprl 794	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝑋 ∈ 𝑉)
5		simprr 796	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝑁 ∈ (ℤ≥‘3))
6		extwwlkfab.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
7		extwwlkfab.f	. . . . 5 ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
8		extwwlkfab.c	. . . . 5 ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
9	6, 7, 8	numclwlk1lem2 27230	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → ∃𝑓 𝑓:(𝑋𝐶𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋)))
10	3, 4, 5, 9	syl3anc 1326	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → ∃𝑓 𝑓:(𝑋𝐶𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋)))
11		hasheqf1oi 13140	. . 3 ⊢ ((𝑋𝐶𝑁) ∈ V → (∃𝑓 𝑓:(𝑋𝐶𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋)) → (#‘(𝑋𝐶𝑁)) = (#‘((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋)))))
12	1, 10, 11	mpsyl 68	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (#‘(𝑋𝐶𝑁)) = (#‘((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋))))
13		simpll 790	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝑉 ∈ Fin)
14		uz3m2nn 11731	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ)
15	14	adantl 482	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → (𝑁 − 2) ∈ ℕ)
16	15	adantl 482	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (𝑁 − 2) ∈ ℕ)
17	7, 6	numclwwlkffin 27214	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ (𝑁 − 2) ∈ ℕ) → (𝑋𝐹(𝑁 − 2)) ∈ Fin)
18	13, 4, 16, 17	syl3anc 1326	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (𝑋𝐹(𝑁 − 2)) ∈ Fin)
19	6	finrusgrfusgr 26461	. . . . . . 7 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph )
20	19	ancoms 469	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph )
21		fusgrfis 26222	. . . . . 6 ⊢ (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
22	20, 21	syl 17	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (Edg‘𝐺) ∈ Fin)
23	22	adantr 481	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (Edg‘𝐺) ∈ Fin)
24		eqid 2622	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
25	6, 24	nbusgrfi 26276	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ (Edg‘𝐺) ∈ Fin ∧ 𝑋 ∈ 𝑉) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
26	3, 23, 4, 25	syl3anc 1326	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
27		hashxp 13221	. . 3 ⊢ (((𝑋𝐹(𝑁 − 2)) ∈ Fin ∧ (𝐺 NeighbVtx 𝑋) ∈ Fin) → (#‘((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋))) = ((#‘(𝑋𝐹(𝑁 − 2))) · (#‘(𝐺 NeighbVtx 𝑋))))
28	18, 26, 27	syl2anc 693	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (#‘((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋))) = ((#‘(𝑋𝐹(𝑁 − 2))) · (#‘(𝐺 NeighbVtx 𝑋))))
29	6	rusgrpropnb 26479	. . . . . . . . 9 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑥 ∈ 𝑉 (#‘(𝐺 NeighbVtx 𝑥)) = 𝐾))
30		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
31	30	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (#‘(𝐺 NeighbVtx 𝑥)) = (#‘(𝐺 NeighbVtx 𝑋)))
32	31	eqeq1d 2624	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((#‘(𝐺 NeighbVtx 𝑥)) = 𝐾 ↔ (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
33	32	rspccv 3306	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑉 (#‘(𝐺 NeighbVtx 𝑥)) = 𝐾 → (𝑋 ∈ 𝑉 → (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
34	33	3ad2ant3 1084	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑥 ∈ 𝑉 (#‘(𝐺 NeighbVtx 𝑥)) = 𝐾) → (𝑋 ∈ 𝑉 → (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
35	29, 34	syl 17	. . . . . . . 8 ⊢ (𝐺 RegUSGraph 𝐾 → (𝑋 ∈ 𝑉 → (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
36	35	adantl 482	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝑋 ∈ 𝑉 → (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
37	36	com12 32	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
38	37	adantr 481	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
39	38	impcom 446	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾)
40	39	oveq2d 6666	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → ((#‘(𝑋𝐹(𝑁 − 2))) · (#‘(𝐺 NeighbVtx 𝑋))) = ((#‘(𝑋𝐹(𝑁 − 2))) · 𝐾))
41		hashcl 13147	. . . . 5 ⊢ ((𝑋𝐹(𝑁 − 2)) ∈ Fin → (#‘(𝑋𝐹(𝑁 − 2))) ∈ ℕ0)
42		nn0cn 11302	. . . . 5 ⊢ ((#‘(𝑋𝐹(𝑁 − 2))) ∈ ℕ0 → (#‘(𝑋𝐹(𝑁 − 2))) ∈ ℂ)
43	18, 41, 42	3syl 18	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (#‘(𝑋𝐹(𝑁 − 2))) ∈ ℂ)
44	20	adantr 481	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐺 ∈ FinUSGraph )
45		simplr 792	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐺 RegUSGraph 𝐾)
46		ne0i 3921	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑉 ≠ ∅)
47	46	adantr 481	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → 𝑉 ≠ ∅)
48	47	adantl 482	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝑉 ≠ ∅)
49	6	frusgrnn0 26467	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
50	44, 45, 48, 49	syl3anc 1326	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐾 ∈ ℕ0)
51	50	nn0cnd 11353	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐾 ∈ ℂ)
52	43, 51	mulcomd 10061	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → ((#‘(𝑋𝐹(𝑁 − 2))) · 𝐾) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))
53	40, 52	eqtrd 2656	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → ((#‘(𝑋𝐹(𝑁 − 2))) · (#‘(𝐺 NeighbVtx 𝑋))) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))
54	12, 28, 53	3eqtrd 2660	1 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (#‘(𝑋𝐶𝑁)) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  {crab 2916  Vcvv 3200  ∅c0 3915   class class class wbr 4653   × cxp 5112  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Fincfn 7955  ℂcc 9934  0cc0 9936   · cmul 9941   − cmin 10266  ℕcn 11020  2c2 11070  3c3 11071  ℕ₀cn0 11292  ℕ₀^*cxnn0 11363  ℤ_≥cuz 11687  #chash 13117  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   FinUSGraph cfusgr 26208   NeighbVtx cnbgr 26224   RegUSGraph crusgr 26452   ClWWalksN cclwwlksn 26876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-s2 13593  df-vtx 25876  df-iedg 25877  df-edg 25940  df-uhgr 25953  df-ushgr 25954  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-fusgr 26209  df-nbgr 26228  df-vtxdg 26362  df-rgr 26453  df-rusgr 26454  df-wwlks 26722  df-wwlksn 26723  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  numclwwlk3OLD  27242  numclwwlk3  27243