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Theorem numclwwlk1 30290

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 30277, 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 30298. If the general representation of (closed) walk is used, Huneke's statement can be proven even for n = 2, see numclwlk1 30300. 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 29492	. . . . 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 30289	. . . 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 14313	. . 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 2729	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
13	12	clwwlknonfin 30023	. . . . . 6 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
14	5	eleq1i 2819	. . . . . 6 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
15	7	eleq1i 2819	. . . . . 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 29493	. . . . . . 7 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
20	19	ancoms 458	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
21		fusgrfis 29257	. . . . . 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 2729	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
25	5, 24	nbusgrfi 29301	. . . 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 14399	. . 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 29511	. . . . . . . . 9 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑥 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾))
30		oveq2 7395	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
31	30	fveqeq2d 6866	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 ↔ (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
32	31	rspccv 3585	. . . . . . . . . 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 7403	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · 𝐾))
40		hashcl 14321	. . . . 5 ⊢ (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ₀)
41		nn0cn 12452	. . . . 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 4304	. . . . . . . 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 29499	. . . . . 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 12505	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝐾 ∈ ℂ)
51	42, 50	mulcomd 11195	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ((♯‘𝐹) · 𝐾) = (𝐾 · (♯‘𝐹)))
52	39, 51	eqtrd 2764	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = (𝐾 · (♯‘𝐹)))
53	11, 28, 52	3eqtrd 2768	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 2109 ≠ wne 2925 ∀wral 3044 {crab 3405 ∅c0 4296 class class class wbr 5107 × cxp 5636 ‘cfv 6511 (class class class)co 7387 ∈ cmpo 7389 ≈ cen 8915 Fincfn 8918 ℂcc 11066 · cmul 11073 − cmin 11405 2c2 12241 3c3 12242 ℕ₀cn0 12442 ℕ₀^*cxnn0 12515 ℤ_≥cuz 12793 ♯chash 14295 Vtxcvtx 28923 Edgcedg 28974 USGraphcusgr 29076 FinUSGraphcfusgr 29243 NeighbVtx cnbgr 29259 RegUSGraph crusgr 29484 ClWWalksNOncclwwlknon 30016

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-xnn0 12516 df-z 12530 df-uz 12794 df-rp 12952 df-xadd 13073 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-word 14479 df-lsw 14528 df-concat 14536 df-s1 14561 df-substr 14606 df-pfx 14636 df-s2 14814 df-vtx 28925 df-iedg 28926 df-edg 28975 df-uhgr 28985 df-ushgr 28986 df-upgr 29009 df-umgr 29010 df-uspgr 29077 df-usgr 29078 df-fusgr 29244 df-nbgr 29260 df-vtxdg 29394 df-rgr 29485 df-rusgr 29486 df-wwlks 29760 df-wwlksn 29761 df-clwwlk 29911 df-clwwlkn 29954 df-clwwlknon 30017

This theorem is referenced by: numclwlk1lem2 30299 numclwwlk3 30314

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