Theorem numclwwlk1 27516

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 27501, 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 27526. If the general representation of (closed) walk is used, Huneke's statement can be proven even for n = 2, see numclwlk1 27528. 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.)

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 ∧ 𝐺RegUSGraph𝐾) ∧ (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 variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6837	. . 3 ⊢ (𝑋𝐶𝑁) ∈ V
2		rusgrusgr 26666	. . . . 5 ⊢ (𝐺RegUSGraph𝐾 → 𝐺 ∈ USGraph)
3	2	ad2antlr 765	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐺 ∈ USGraph)
4		simprl 811	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝑋 ∈ 𝑉)
5		simprr 813	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝑁 ∈ (ℤ≥‘3))
6		extwwlkfab.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
7		extwwlkfab.c	. . . . 5 ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
8		extwwlkfab.f	. . . . 5 ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))
9	6, 7, 8	numclwwlk1lem2 27515	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → ∃𝑓 𝑓:(𝑋𝐶𝑁)–1-1-onto→(𝐹 × (𝐺 NeighbVtx 𝑋)))
10	3, 4, 5, 9	syl3anc 1477	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → ∃𝑓 𝑓:(𝑋𝐶𝑁)–1-1-onto→(𝐹 × (𝐺 NeighbVtx 𝑋)))
11		hasheqf1oi 13330	. . 3 ⊢ ((𝑋𝐶𝑁) ∈ V → (∃𝑓 𝑓:(𝑋𝐶𝑁)–1-1-onto→(𝐹 × (𝐺 NeighbVtx 𝑋)) → (♯‘(𝑋𝐶𝑁)) = (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋)))))
12	1, 10, 11	mpsyl 68	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘(𝑋𝐶𝑁)) = (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))))
13		eqid 2756	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
14	13	clwwlknonfin 27237	. . . . . 6 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
15	6	eleq1i 2826	. . . . . 6 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
16	8	eleq1i 2826	. . . . . 6 ⊢ (𝐹 ∈ Fin ↔ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
17	14, 15, 16	3imtr4i 281	. . . . 5 ⊢ (𝑉 ∈ Fin → 𝐹 ∈ Fin)
18	17	adantr 472	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → 𝐹 ∈ Fin)
19	18	adantr 472	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐹 ∈ Fin)
20	6	finrusgrfusgr 26667	. . . . . . 7 ⊢ ((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
21	20	ancoms 468	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → 𝐺 ∈ FinUSGraph)
22		fusgrfis 26417	. . . . . 6 ⊢ (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
23	21, 22	syl 17	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (Edg‘𝐺) ∈ Fin)
24	23	adantr 472	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (Edg‘𝐺) ∈ Fin)
25		eqid 2756	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
26	6, 25	nbusgrfi 26470	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ (Edg‘𝐺) ∈ Fin ∧ 𝑋 ∈ 𝑉) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
27	3, 24, 4, 26	syl3anc 1477	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
28		hashxp 13409	. . 3 ⊢ ((𝐹 ∈ Fin ∧ (𝐺 NeighbVtx 𝑋) ∈ Fin) → (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))))
29	19, 27, 28	syl2anc 696	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))))
30	6	rusgrpropnb 26685	. . . . . . . . 9 ⊢ (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑥 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾))
31		oveq2 6817	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
32	31	fveq2d 6352	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (♯‘(𝐺 NeighbVtx 𝑥)) = (♯‘(𝐺 NeighbVtx 𝑋)))
33	32	eqeq1d 2758	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 ↔ (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
34	33	rspccv 3442	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 → (𝑋 ∈ 𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
35	34	3ad2ant3 1130	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑥 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾) → (𝑋 ∈ 𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
36	30, 35	syl 17	. . . . . . . 8 ⊢ (𝐺RegUSGraph𝐾 → (𝑋 ∈ 𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
37	36	adantl 473	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (𝑋 ∈ 𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
38	37	com12 32	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
39	38	adantr 472	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
40	39	impcom 445	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾)
41	40	oveq2d 6825	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · 𝐾))
42		hashcl 13335	. . . . 5 ⊢ (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ0)
43		nn0cn 11490	. . . . 5 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℂ)
44	19, 42, 43	3syl 18	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘𝐹) ∈ ℂ)
45	21	adantr 472	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐺 ∈ FinUSGraph)
46		simplr 809	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐺RegUSGraph𝐾)
47		ne0i 4060	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑉 ≠ ∅)
48	47	adantr 472	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → 𝑉 ≠ ∅)
49	48	adantl 473	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝑉 ≠ ∅)
50	6	frusgrnn0 26673	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺RegUSGraph𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
51	45, 46, 49, 50	syl3anc 1477	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐾 ∈ ℕ0)
52	51	nn0cnd 11541	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐾 ∈ ℂ)
53	44, 52	mulcomd 10249	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → ((♯‘𝐹) · 𝐾) = (𝐾 · (♯‘𝐹)))
54	41, 53	eqtrd 2790	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = (𝐾 · (♯‘𝐹)))
55	12, 29, 54	3eqtrd 2794	1 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘(𝑋𝐶𝑁)) = (𝐾 · (♯‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1628  ∃wex 1849   ∈ wcel 2135   ≠ wne 2928  ∀wral 3046  {crab 3050  Vcvv 3336  ∅c0 4054   class class class wbr 4800   × cxp 5260  –1-1-onto→wf1o 6044  ‘cfv 6045  (class class class)co 6809   ↦ cmpt2 6811  Fincfn 8117  ℂcc 10122   · cmul 10129   − cmin 10454  2c2 11258  3c3 11259  ℕ₀cn0 11480  ℕ₀^*cxnn0 11551  ℤ_≥cuz 11875  ♯chash 13307  Vtxcvtx 26069  Edgcedg 26134  USGraphcusgr 26239  FinUSGraphcfusgr 26403   NeighbVtx cnbgr 26419  RegUSGraphcrusgr 26658  ClWWalksNOncclwwlknon 27228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-cnex 10180  ax-resscn 10181  ax-1cn 10182  ax-icn 10183  ax-addcl 10184  ax-addrcl 10185  ax-mulcl 10186  ax-mulrcl 10187  ax-mulcom 10188  ax-addass 10189  ax-mulass 10190  ax-distr 10191  ax-i2m1 10192  ax-1ne0 10193  ax-1rid 10194  ax-rnegex 10195  ax-rrecex 10196  ax-cnre 10197  ax-pre-lttri 10198  ax-pre-lttrn 10199  ax-pre-ltadd 10200  ax-pre-mulgt0 10201

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-fal 1634  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-nel 3032  df-ral 3051  df-rex 3052  df-reu 3053  df-rmo 3054  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-riota 6770  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-2o 7726  df-oadd 7729  df-er 7907  df-map 8021  df-pm 8022  df-en 8118  df-dom 8119  df-sdom 8120  df-fin 8121  df-card 8951  df-cda 9178  df-pnf 10264  df-mnf 10265  df-xr 10266  df-ltxr 10267  df-le 10268  df-sub 10456  df-neg 10457  df-nn 11209  df-2 11267  df-3 11268  df-n0 11481  df-xnn0 11552  df-z 11566  df-uz 11876  df-rp 12022  df-xadd 12136  df-fz 12516  df-fzo 12656  df-seq 12992  df-exp 13051  df-hash 13308  df-word 13481  df-lsw 13482  df-concat 13483  df-s1 13484  df-substr 13485  df-s2 13789  df-vtx 26071  df-iedg 26072  df-edg 26135  df-uhgr 26148  df-ushgr 26149  df-upgr 26172  df-umgr 26173  df-uspgr 26240  df-usgr 26241  df-fusgr 26404  df-nbgr 26420  df-vtxdg 26568  df-rgr 26659  df-rusgr 26660  df-wwlks 26929  df-wwlksn 26930  df-clwwlk 27101  df-clwwlkn 27145  df-clwwlknon 27229

This theorem is referenced by:  numclwlk1lem2  27527  numclwwlk3  27549