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Theorem numclwwlk1 30342

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 not for n=2, since 𝐹 = ∅, but (𝑋𝐶2), the set of closed walks with length 2 on 𝑋, see 2clwwlk2 30329, needs not be ∅ in this case. This is because of the special definition of 𝐹 and the usage of words to represent (closed) walks, and does not contradict Huneke's statement, which would read "the number of closed 2-walks v(0) v(1) v(2) from v = v(0) = v(2) ... is kf(0)", where f(0)=1 is the number of empty closed walks on v, see numclwlk1lem1 30350. If the general representation of (closed) walk is used, Huneke's statement can be proven even for n = 2, see numclwlk1 30352. This case, however, is not required to prove the friendship theorem. (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 6-Mar-2022.) (Proof shortened by AV, 31-Jul-2022.)

**Hypotheses**
Ref	Expression
extwwlkfab.v	⊢ 𝑉 = (Vtx‘𝐺)
extwwlkfab.c	⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
extwwlkfab.f	⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))

**Assertion**
Ref	Expression
numclwwlk1	⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝑋𝐶𝑁)) = (𝐾 · (♯‘𝐹)))

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 variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rusgrusgr 29544	. . . . 5 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
2	1	ad2antlr 727	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝐺 ∈ USGraph)
3		simprl 770	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝑋 ∈ 𝑉)
4		simprr 772	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝑁 ∈ (ℤ_≥‘3))
5		extwwlkfab.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
6		extwwlkfab.c	. . . . 5 ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
7		extwwlkfab.f	. . . . 5 ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))
8	5, 6, 7	numclwwlk1lem2 30341	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)))
9	2, 3, 4, 8	syl3anc 1373	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)))
10		hasheni 14366	. . 3 ⊢ ((𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)) → (♯‘(𝑋𝐶𝑁)) = (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))))
11	9, 10	syl 17	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝑋𝐶𝑁)) = (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))))
12		eqid 2735	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
13	12	clwwlknonfin 30075	. . . . . 6 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
14	5	eleq1i 2825	. . . . . 6 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
15	7	eleq1i 2825	. . . . . 6 ⊢ (𝐹 ∈ Fin ↔ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
16	13, 14, 15	3imtr4i 292	. . . . 5 ⊢ (𝑉 ∈ Fin → 𝐹 ∈ Fin)
17	16	adantr 480	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐹 ∈ Fin)
18	17	adantr 480	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝐹 ∈ Fin)
19	5	finrusgrfusgr 29545	. . . . . . 7 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
20	19	ancoms 458	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
21		fusgrfis 29309	. . . . . 6 ⊢ (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
22	20, 21	syl 17	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (Edg‘𝐺) ∈ Fin)
23	22	adantr 480	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (Edg‘𝐺) ∈ Fin)
24		eqid 2735	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
25	5, 24	nbusgrfi 29353	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ (Edg‘𝐺) ∈ Fin ∧ 𝑋 ∈ 𝑉) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
26	2, 23, 3, 25	syl3anc 1373	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
27		hashxp 14452	. . 3 ⊢ ((𝐹 ∈ Fin ∧ (𝐺 NeighbVtx 𝑋) ∈ Fin) → (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))))
28	18, 26, 27	syl2anc 584	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))))
29	5	rusgrpropnb 29563	. . . . . . . . 9 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑥 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾))
30		oveq2 7413	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
31	30	fveqeq2d 6884	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 ↔ (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
32	31	rspccv 3598	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 → (𝑋 ∈ 𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
33	32	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑥 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾) → (𝑋 ∈ 𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
34	29, 33	syl 17	. . . . . . . 8 ⊢ (𝐺 RegUSGraph 𝐾 → (𝑋 ∈ 𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
35	34	adantl 481	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝑋 ∈ 𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
36	35	com12 32	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
37	36	adantr 480	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
38	37	impcom 407	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾)
39	38	oveq2d 7421	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · 𝐾))
40		hashcl 14374	. . . . 5 ⊢ (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ₀)
41		nn0cn 12511	. . . . 5 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ ℂ)
42	18, 40, 41	3syl 18	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘𝐹) ∈ ℂ)
43	20	adantr 480	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝐺 ∈ FinUSGraph)
44		simplr 768	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝐺 RegUSGraph 𝐾)
45		ne0i 4316	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑉 ≠ ∅)
46	45	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑉 ≠ ∅)
47	46	adantl 481	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝑉 ≠ ∅)
48	5	frusgrnn0 29551	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ₀)
49	43, 44, 47, 48	syl3anc 1373	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝐾 ∈ ℕ₀)
50	49	nn0cnd 12564	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝐾 ∈ ℂ)
51	42, 50	mulcomd 11256	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ((♯‘𝐹) · 𝐾) = (𝐾 · (♯‘𝐹)))
52	39, 51	eqtrd 2770	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = (𝐾 · (♯‘𝐹)))
53	11, 28, 52	3eqtrd 2774	1 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝑋𝐶𝑁)) = (𝐾 · (♯‘𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 {crab 3415 ∅c0 4308 class class class wbr 5119 × cxp 5652 ‘cfv 6531 (class class class)co 7405 ∈ cmpo 7407 ≈ cen 8956 Fincfn 8959 ℂcc 11127 · cmul 11134 − cmin 11466 2c2 12295 3c3 12296 ℕ₀cn0 12501 ℕ₀^*cxnn0 12574 ℤ_≥cuz 12852 ♯chash 14348 Vtxcvtx 28975 Edgcedg 29026 USGraphcusgr 29128 FinUSGraphcfusgr 29295 NeighbVtx cnbgr 29311 RegUSGraph crusgr 29536 ClWWalksNOncclwwlknon 30068

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 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

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-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-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 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8719 df-map 8842 df-pm 8843 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-dju 9915 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-n0 12502 df-xnn0 12575 df-z 12589 df-uz 12853 df-rp 13009 df-xadd 13129 df-fz 13525 df-fzo 13672 df-seq 14020 df-exp 14080 df-hash 14349 df-word 14532 df-lsw 14581 df-concat 14589 df-s1 14614 df-substr 14659 df-pfx 14689 df-s2 14867 df-vtx 28977 df-iedg 28978 df-edg 29027 df-uhgr 29037 df-ushgr 29038 df-upgr 29061 df-umgr 29062 df-uspgr 29129 df-usgr 29130 df-fusgr 29296 df-nbgr 29312 df-vtxdg 29446 df-rgr 29537 df-rusgr 29538 df-wwlks 29812 df-wwlksn 29813 df-clwwlk 29963 df-clwwlkn 30006 df-clwwlknon 30069

This theorem is referenced by: numclwlk1lem2 30351 numclwwlk3 30366

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