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Theorem numclwwlk1 30380

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 30367, 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 30388. If the general representation of (closed) walk is used, Huneke's statement can be proven even for n = 2, see numclwlk1 30390. 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 29582	. . . . 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 771	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝑋 ∈ 𝑉)
4		simprr 773	. . . 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 30379	. . . 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 14387	. . 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 2737	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
13	12	clwwlknonfin 30113	. . . . . 6 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
14	5	eleq1i 2832	. . . . . 6 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
15	7	eleq1i 2832	. . . . . 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 29583	. . . . . . 7 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
20	19	ancoms 458	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
21		fusgrfis 29347	. . . . . 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 2737	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
25	5, 24	nbusgrfi 29391	. . . 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 14473	. . 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 29601	. . . . . . . . 9 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑥 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾))
30		oveq2 7439	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
31	30	fveqeq2d 6914	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 ↔ (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
32	31	rspccv 3619	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 → (𝑋 ∈ 𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
33	32	3ad2ant3 1136	. . . . . . . . 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 7447	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · 𝐾))
40		hashcl 14395	. . . . 5 ⊢ (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ₀)
41		nn0cn 12536	. . . . 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 769	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝐺 RegUSGraph 𝐾)
45		ne0i 4341	. . . . . . . 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 29589	. . . . . 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 12589	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝐾 ∈ ℂ)
51	42, 50	mulcomd 11282	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ((♯‘𝐹) · 𝐾) = (𝐾 · (♯‘𝐹)))
52	39, 51	eqtrd 2777	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = (𝐾 · (♯‘𝐹)))
53	11, 28, 52	3eqtrd 2781	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 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 {crab 3436 ∅c0 4333 class class class wbr 5143 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ≈ cen 8982 Fincfn 8985 ℂcc 11153 · cmul 11160 − cmin 11492 2c2 12321 3c3 12322 ℕ₀cn0 12526 ℕ₀^*cxnn0 12599 ℤ_≥cuz 12878 ♯chash 14369 Vtxcvtx 29013 Edgcedg 29064 USGraphcusgr 29166 FinUSGraphcfusgr 29333 NeighbVtx cnbgr 29349 RegUSGraph crusgr 29574 ClWWalksNOncclwwlknon 30106

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-rp 13035 df-xadd 13155 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-s2 14887 df-vtx 29015 df-iedg 29016 df-edg 29065 df-uhgr 29075 df-ushgr 29076 df-upgr 29099 df-umgr 29100 df-uspgr 29167 df-usgr 29168 df-fusgr 29334 df-nbgr 29350 df-vtxdg 29484 df-rgr 29575 df-rusgr 29576 df-wwlks 29850 df-wwlksn 29851 df-clwwlk 30001 df-clwwlkn 30044 df-clwwlknon 30107

This theorem is referenced by: numclwlk1lem2 30389 numclwwlk3 30404

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