Theorem numclwwlk1 30693

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 30680, 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 30701. If the general representation of (closed) walk is used, Huneke's statement can be proven even for n = 2, see numclwlk1 30703. 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 29895	. . . . 5 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
2	1	ad2antlr 739	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐺 ∈ USGraph)
3		simprl 782	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝑋 ∈ 𝑉)
4		simprr 784	. . . 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 30692	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)))
9	2, 3, 4, 8	syl3anc 1398	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)))
10		hasheni 14386	. . 3 ⊢ ((𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)) → (♯‘(𝑋𝐶𝑁)) = (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))))
11	9, 10	syl 18	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘(𝑋𝐶𝑁)) = (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))))
12		eqid 2763	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
13	12	clwwlknonfin 30426	. . . . . 6 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
14	5	eleq1i 2854	. . . . . 6 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
15	7	eleq1i 2854	. . . . . 6 ⊢ (𝐹 ∈ Fin ↔ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
16	13, 14, 15	3imtr4i 295	. . . . 5 ⊢ (𝑉 ∈ Fin → 𝐹 ∈ Fin)
17	16	adantr 485	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐹 ∈ Fin)
18	17	adantr 485	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐹 ∈ Fin)
19	5	finrusgrfusgr 29896	. . . . . . 7 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
20	19	ancoms 463	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
21		fusgrfis 29661	. . . . . 6 ⊢ (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
22	20, 21	syl 18	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (Edg‘𝐺) ∈ Fin)
23	22	adantr 485	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (Edg‘𝐺) ∈ Fin)
24		eqid 2763	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
25	5, 24	nbusgrfi 29705	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ (Edg‘𝐺) ∈ Fin ∧ 𝑋 ∈ 𝑉) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
26	2, 23, 3, 25	syl3anc 1398	. . 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 595	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))))
29	5	rusgrpropnb 29914	. . . . . . . . 9 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑥 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾))
30		oveq2 7420	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
31	30	fveqeq2d 6891	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 ↔ (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
32	31	rspccv 3579	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 → (𝑋 ∈ 𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
33	32	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑥 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾) → (𝑋 ∈ 𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
34	29, 33	syl 18	. . . . . . . 8 ⊢ (𝐺 RegUSGraph 𝐾 → (𝑋 ∈ 𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
35	34	adantl 486	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝑋 ∈ 𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
36	35	com12 33	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
37	36	adantr 485	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
38	37	impcom 412	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾)
39	38	oveq2d 7428	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · 𝐾))
40		hashcl 14394	. . . . 5 ⊢ (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ0)
41		nn0cn 12515	. . . . 5 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℂ)
42	18, 40, 41	3syl 19	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘𝐹) ∈ ℂ)
43	20	adantr 485	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐺 ∈ FinUSGraph)
44		simplr 780	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐺 RegUSGraph 𝐾)
45		ne0i 4295	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑉 ≠ ∅)
46	45	adantr 485	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → 𝑉 ≠ ∅)
47	46	adantl 486	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝑉 ≠ ∅)
48	5	frusgrnn0 29902	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
49	43, 44, 47, 48	syl3anc 1398	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐾 ∈ ℕ0)
50	49	nn0cnd 12568	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐾 ∈ ℂ)
51	42, 50	mulcomd 11231	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → ((♯‘𝐹) · 𝐾) = (𝐾 · (♯‘𝐹)))
52	39, 51	eqtrd 2798	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = (𝐾 · (♯‘𝐹)))
53	11, 28, 52	3eqtrd 2802	1 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘(𝑋𝐶𝑁)) = (𝐾 · (♯‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  {crab 3416  ∅c0 4287   class class class wbr 5110   × cxp 5661  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414   ≈ cen 8941  Fincfn 8944  ℂcc 11099   · cmul 11106   − cmin 11442  2c2 12296  3c3 12297  ℕ₀cn0 12505  ℕ₀^*cxnn0 12578  ℤ_≥cuz 12863  ♯chash 14368  Vtxcvtx 29327  Edgcedg 29378  USGraphcusgr 29480  FinUSGraphcfusgr 29647   NeighbVtx cnbgr 29663   RegUSGraph crusgr 29887  ClWWalksNOncclwwlknon 30419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-oadd 8458  df-er 8695  df-map 8827  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-dju 9888  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-n0 12506  df-xnn0 12579  df-z 12593  df-uz 12864  df-rp 13018  df-xadd 13139  df-fz 13537  df-fzo 13685  df-seq 14040  df-exp 14100  df-hash 14369  df-word 14553  df-lsw 14602  df-concat 14610  df-s1 14636  df-substr 14681  df-pfx 14711  df-s2 14887  df-vtx 29329  df-iedg 29330  df-edg 29379  df-uhgr 29389  df-ushgr 29390  df-upgr 29413  df-umgr 29414  df-uspgr 29481  df-usgr 29482  df-fusgr 29648  df-nbgr 29664  df-vtxdg 29797  df-rgr 29888  df-rusgr 29889  df-wwlks 30160  df-wwlksn 30161  df-clwwlk 30314  df-clwwlkn 30357  df-clwwlknon 30420

This theorem is referenced by:  numclwlk1lem2  30702  numclwwlk3  30717