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Theorem numclwlk1lem1 29886

Description: Lemma 1 for numclwlk1 29888 (Statement 9 in [Huneke] p. 2 for n=2): "the number of closed 2-walks v(0) v(1) v(2) from v = v(0) = v(2) ... is kf(0)". (Contributed by AV, 23-May-2022.)

**Hypotheses**
Ref	Expression
numclwlk1.v	⊢ 𝑉 = (Vtx‘𝐺)
numclwlk1.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
numclwlk1.f	⊢ 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = (𝑁 − 2) ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}

**Assertion**
Ref	Expression
numclwlk1lem1	⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))

Distinct variable groups: 𝑤,𝐺 𝑤,𝐾 𝑤,𝑁 𝑤,𝑉 𝑤,𝑋 Allowed substitution hints: 𝐶(𝑤) 𝐹(𝑤)

**Proof of Theorem numclwlk1lem1**
Step	Hyp	Ref	Expression
1		3anass 1094	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘(1^st ‘𝑤)) = 2 ∧ (((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)))
2		anidm 564	. . . . . . . 8 ⊢ ((((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) ↔ ((2^nd ‘𝑤)‘0) = 𝑋)
3	2	anbi2i 622	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝑤)) = 2 ∧ (((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) ↔ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋))
4	1, 3	bitri 274	. . . . . 6 ⊢ (((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋))
5	4	rabbii 3437	. . . . 5 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}
6	5	fveq2i 6895	. . . 4 ⊢ (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)})
7		simpl 482	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝑉 ∈ Fin)
8		simpr 484	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾)
9		simpl 482	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 2) → 𝑋 ∈ 𝑉)
10		numclwlk1.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
11	10	clwlknon2num 29885	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) = 𝐾)
12	7, 8, 9, 11	syl2an3an 1421	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) = 𝐾)
13	6, 12	eqtrid 2783	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) = 𝐾)
14		rusgrusgr 29085	. . . . . . . . 9 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
15	14	anim2i 616	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
16	15	ancomd 461	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
17	10	isfusgr 28839	. . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
18	16, 17	sylibr 233	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
19		ne0i 4335	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑉 ≠ ∅)
20	19	adantr 480	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 2) → 𝑉 ≠ ∅)
21	10	frusgrnn0 29092	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ₀)
22	18, 8, 20, 21	syl2an3an 1421	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → 𝐾 ∈ ℕ₀)
23	22	nn0red 12538	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → 𝐾 ∈ ℝ)
24		ax-1rid 11183	. . . 4 ⊢ (𝐾 ∈ ℝ → (𝐾 · 1) = 𝐾)
25	23, 24	syl 17	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (𝐾 · 1) = 𝐾)
26	10	wlkl0 29884	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
27	26	ad2antrl 725	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
28	27	fveq2d 6896	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) = (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}))
29		opex 5465	. . . . . 6 ⊢ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ V
30		hashsng 14334	. . . . . 6 ⊢ (⟨∅, {⟨0, 𝑋⟩}⟩ ∈ V → (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}) = 1)
31	29, 30	ax-mp 5	. . . . 5 ⊢ (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}) = 1
32	28, 31	eqtr2di 2788	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → 1 = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}))
33	32	oveq2d 7428	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (𝐾 · 1) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)})))
34	13, 25, 33	3eqtr2d 2777	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)})))
35		numclwlk1.c	. . . . . 6 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
36		eqeq2 2743	. . . . . . . 8 ⊢ (𝑁 = 2 → ((♯‘(1^st ‘𝑤)) = 𝑁 ↔ (♯‘(1^st ‘𝑤)) = 2))
37		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑁 = 2 → (𝑁 − 2) = (2 − 2))
38		2cn 12292	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
39	38	subidi 11536	. . . . . . . . . 10 ⊢ (2 − 2) = 0
40	37, 39	eqtrdi 2787	. . . . . . . . 9 ⊢ (𝑁 = 2 → (𝑁 − 2) = 0)
41	40	fveqeq2d 6900	. . . . . . . 8 ⊢ (𝑁 = 2 → (((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2^nd ‘𝑤)‘0) = 𝑋))
42	36, 41	3anbi13d 1437	. . . . . . 7 ⊢ (𝑁 = 2 → (((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋) ↔ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)))
43	42	rabbidv 3439	. . . . . 6 ⊢ (𝑁 = 2 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)})
44	35, 43	eqtrid 2783	. . . . 5 ⊢ (𝑁 = 2 → 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)})
45	44	fveq2d 6896	. . . 4 ⊢ (𝑁 = 2 → (♯‘𝐶) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}))
46		numclwlk1.f	. . . . . . 7 ⊢ 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = (𝑁 − 2) ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}
47	40	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑁 = 2 → ((♯‘(1^st ‘𝑤)) = (𝑁 − 2) ↔ (♯‘(1^st ‘𝑤)) = 0))
48	47	anbi1d 629	. . . . . . . 8 ⊢ (𝑁 = 2 → (((♯‘(1^st ‘𝑤)) = (𝑁 − 2) ∧ ((2^nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)))
49	48	rabbidv 3439	. . . . . . 7 ⊢ (𝑁 = 2 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = (𝑁 − 2) ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)})
50	46, 49	eqtrid 2783	. . . . . 6 ⊢ (𝑁 = 2 → 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)})
51	50	fveq2d 6896	. . . . 5 ⊢ (𝑁 = 2 → (♯‘𝐹) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}))
52	51	oveq2d 7428	. . . 4 ⊢ (𝑁 = 2 → (𝐾 · (♯‘𝐹)) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)})))
53	45, 52	eqeq12d 2747	. . 3 ⊢ (𝑁 = 2 → ((♯‘𝐶) = (𝐾 · (♯‘𝐹)) ↔ (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}))))
54	53	ad2antll 726	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → ((♯‘𝐶) = (𝐾 · (♯‘𝐹)) ↔ (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}))))
55	34, 54	mpbird 256	1 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 {crab 3431 Vcvv 3473 ∅c0 4323 {csn 4629 ⟨cop 4635 class class class wbr 5149 ‘cfv 6544 (class class class)co 7412 1^st c1st 7976 2^nd c2nd 7977 Fincfn 8942 ℝcr 11112 0cc0 11113 1c1 11114 · cmul 11118 − cmin 11449 2c2 12272 ℕ₀cn0 12477 ♯chash 14295 Vtxcvtx 28520 USGraphcusgr 28673 FinUSGraphcfusgr 28837 RegUSGraph crusgr 29077 ClWalkscclwlks 29291

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-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

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1061 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-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-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-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-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-nn 12218 df-2 12280 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-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-wlks 29120 df-clwlks 29292 df-wwlks 29348 df-wwlksn 29349 df-clwwlk 29499 df-clwwlkn 29542 df-clwwlknon 29605

This theorem is referenced by: numclwlk1 29888

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