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Theorem numclwwlk3 29902

Description: Statement 12 in [Huneke] p. 2: "Thus f(n) = (k - 1)f(n - 2) + k^(n-2)." - the number of the closed walks v(0) ... v(n-2) v(n-1) v(n) is the sum of the number of the closed walks v(0) ... v(n-2) v(n-1) v(n) with v(n-2) = v(n) (see numclwwlk1 29878) and with v(n-2) =/= v(n) (see numclwwlk2 29898): f(n) = kf(n-2) + k^(n-2) - f(n-2) = (k-1)f(n-2) + k^(n-2). (Contributed by Alexander van der Vekens, 26-Aug-2018.) (Revised by AV, 6-Mar-2022.)

**Hypothesis**
Ref	Expression
numclwwlk3.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
numclwwlk3	⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁)) = (((𝐾 − 1) · (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))) + (𝐾↑(𝑁 − 2))))

Proof of Theorem numclwwlk3 Dummy variables 𝑛 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) → 𝐺 RegUSGraph 𝐾)
2		simp1 1135	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑉 ∈ Fin)
3		numclwwlk3.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	finrusgrfusgr 29086	. . . 4 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
5	1, 2, 4	syl2an 595	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝐺 ∈ FinUSGraph)
6		simpr2 1194	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝑋 ∈ 𝑉)
7		uzuzle23 12878	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ (ℤ_≥‘2))
8	7	3ad2ant3 1134	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑁 ∈ (ℤ_≥‘2))
9	8	adantl 481	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝑁 ∈ (ℤ_≥‘2))
10		eqid 2731	. . . 4 ⊢ (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
11		eqid 2731	. . . 4 ⊢ (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
12	10, 11	numclwwlk3lem2 29901	. . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁)) = ((♯‘(𝑋(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})𝑁)) + (♯‘(𝑋(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})𝑁))))
13	5, 6, 9, 12	syl21anc 835	. 2 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁)) = ((♯‘(𝑋(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})𝑁)) + (♯‘(𝑋(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})𝑁))))
14		eqid 2731	. . . 4 ⊢ (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
15	3, 14, 11	numclwwlk2 29898	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝑋(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})𝑁)) = ((𝐾↑(𝑁 − 2)) − (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))))
16	1, 2	anim12ci 613	. . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾))
17		3simpc 1149	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)))
18	17	adantl 481	. . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)))
19		eqid 2731	. . . . 5 ⊢ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))
20	3, 10, 19	numclwwlk1 29878	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝑋(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})𝑁)) = (𝐾 · (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))))
21	16, 18, 20	syl2anc 583	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝑋(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})𝑁)) = (𝐾 · (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))))
22	15, 21	oveq12d 7430	. 2 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ((♯‘(𝑋(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})𝑁)) + (♯‘(𝑋(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})𝑁))) = (((𝐾↑(𝑁 − 2)) − (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))) + (𝐾 · (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))))))
23		simpll 764	. . . . 5 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝐺 RegUSGraph 𝐾)
24		ne0i 4335	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑉 ≠ ∅)
25	24	3ad2ant2 1133	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑉 ≠ ∅)
26	25	adantl 481	. . . . 5 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝑉 ≠ ∅)
27	3	frusgrnn0 29092	. . . . 5 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ₀)
28	5, 23, 26, 27	syl3anc 1370	. . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝐾 ∈ ℕ₀)
29	28	nn0cnd 12539	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → 𝐾 ∈ ℂ)
30		uz3m2nn 12880	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (𝑁 − 2) ∈ ℕ)
31	30	3anim3i 1153	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ (𝑁 − 2) ∈ ℕ))
32	31	adantl 481	. . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ (𝑁 − 2) ∈ ℕ))
33	3	clwwlknonfin 29611	. . . . 5 ⊢ (𝑉 ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
34	33	3ad2ant1 1132	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ (𝑁 − 2) ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
35		hashcl 14321	. . . . 5 ⊢ ((𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin → (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) ∈ ℕ₀)
36	35	nn0cnd 12539	. . . 4 ⊢ ((𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin → (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) ∈ ℂ)
37	32, 34, 36	3syl 18	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) ∈ ℂ)
38		numclwwlk3lem1 29899	. . 3 ⊢ ((𝐾 ∈ ℂ ∧ (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) ∈ ℂ ∧ 𝑁 ∈ (ℤ_≥‘2)) → (((𝐾↑(𝑁 − 2)) − (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))) + (𝐾 · (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))))) = (((𝐾 − 1) · (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))) + (𝐾↑(𝑁 − 2))))
39	29, 37, 9, 38	syl3anc 1370	. 2 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (((𝐾↑(𝑁 − 2)) − (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))) + (𝐾 · (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))))) = (((𝐾 − 1) · (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))) + (𝐾↑(𝑁 − 2))))
40	13, 22, 39	3eqtrd 2775	1 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁)) = (((𝐾 − 1) · (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))) + (𝐾↑(𝑁 − 2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 {crab 3431 ∅c0 4323 class class class wbr 5149 ‘cfv 6544 (class class class)co 7412 ∈ cmpo 7414 Fincfn 8942 ℂcc 11111 0cc0 11113 1c1 11114 + caddc 11116 · cmul 11118 − cmin 11449 ℕcn 12217 2c2 12272 3c3 12273 ℕ₀cn0 12477 ℤ_≥cuz 12827 ↑cexp 14032 ♯chash 14295 lastSclsw 14517 Vtxcvtx 28520 FinUSGraphcfusgr 28837 RegUSGraph crusgr 29077 WWalksN cwwlksn 29344 ClWWalksNOncclwwlknon 29604 FriendGraph cfrgr 29775

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-oadd 8473 df-er 8706 df-map 8825 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9440 df-oi 9508 df-dju 9899 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-xnn0 12550 df-z 12564 df-uz 12828 df-rp 12980 df-xadd 13098 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-word 14470 df-lsw 14518 df-concat 14526 df-s1 14551 df-substr 14596 df-pfx 14626 df-s2 14804 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 df-vtx 28522 df-iedg 28523 df-edg 28572 df-uhgr 28582 df-ushgr 28583 df-upgr 28606 df-umgr 28607 df-uspgr 28674 df-usgr 28675 df-fusgr 28838 df-nbgr 28854 df-vtxdg 28987 df-rgr 29078 df-rusgr 29079 df-wwlks 29348 df-wwlksn 29349 df-wwlksnon 29350 df-clwwlk 29499 df-clwwlkn 29542 df-clwwlknon 29605 df-frgr 29776

This theorem is referenced by: numclwwlk5 29905

W3C validator