Theorem numclwlk1lem1 30976

Description: Lemma 1 for numclwlk1 30978 (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) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
numclwlk1.f	⊢ 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = (𝑁 − 2) ∧ ((2nd ‘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 1111	. . . . . . 7 ⊢ (((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘(1st ‘𝑤)) = 2 ∧ (((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)))
2		anidm 575	. . . . . . . 8 ⊢ ((((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ↔ ((2nd ‘𝑤)‘0) = 𝑋)
3	2	anbi2i 635	. . . . . . 7 ⊢ (((♯‘(1st ‘𝑤)) = 2 ∧ (((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ↔ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋))
4	1, 3	bitri 278	. . . . . 6 ⊢ (((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋))
5	4	rabbii 3417	. . . . 5 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}
6	5	fveq2i 6889	. . . 4 ⊢ (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)})
7		simpl 488	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝑉 ∈ Fin)
8		simpr 490	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾)
9		simpl 488	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 2) → 𝑋 ∈ 𝑉)
10		numclwlk1.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
11	10	clwlknon2num 30975	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = 𝐾)
12	7, 8, 9, 11	syl2an3an 1449	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = 𝐾)
13	6, 12	eqtrid 2807	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = 𝐾)
14		rusgrusgr 30151	. . . . . . . . 9 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
15	14	anim2i 629	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
16	15	ancomd 467	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
17	10	isfusgr 29905	. . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
18	16, 17	sylibr 237	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
19		ne0i 4286	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑉 ≠ ∅)
20	19	adantr 486	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 2) → 𝑉 ≠ ∅)
21	10	frusgrnn0 30158	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
22	18, 8, 20, 21	syl2an3an 1449	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → 𝐾 ∈ ℕ0)
23	22	nn0red 12672	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → 𝐾 ∈ ℝ)
24		ax-1rid 11274	. . . 4 ⊢ (𝐾 ∈ ℝ → (𝐾 · 1) = 𝐾)
25	23, 24	syl 18	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (𝐾 · 1) = 𝐾)
26	10	wlkl0 30974	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
27	26	ad2antrl 741	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
28	27	fveq2d 6890	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}))
29		opex 5431	. . . . . 6 ⊢ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ V
30		hashsng 14517	. . . . . 6 ⊢ (⟨∅, {⟨0, 𝑋⟩}⟩ ∈ V → (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}) = 1)
31	29, 30	ax-mp 5	. . . . 5 ⊢ (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}) = 1
32	28, 31	eqtr2di 2812	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → 1 = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}))
33	32	oveq2d 7437	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (𝐾 · 1) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)})))
34	13, 25, 33	3eqtr2d 2801	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)})))
35		numclwlk1.c	. . . . . 6 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
36		eqeq2 2772	. . . . . . . 8 ⊢ (𝑁 = 2 → ((♯‘(1st ‘𝑤)) = 𝑁 ↔ (♯‘(1st ‘𝑤)) = 2))
37		oveq1 7428	. . . . . . . . . 10 ⊢ (𝑁 = 2 → (𝑁 − 2) = (2 − 2))
38		2cn 12422	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
39	38	subidi 11633	. . . . . . . . . 10 ⊢ (2 − 2) = 0
40	37, 39	eqtrdi 2811	. . . . . . . . 9 ⊢ (𝑁 = 2 → (𝑁 − 2) = 0)
41	40	fveqeq2d 6894	. . . . . . . 8 ⊢ (𝑁 = 2 → (((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2nd ‘𝑤)‘0) = 𝑋))
42	36, 41	3anbi13d 1466	. . . . . . 7 ⊢ (𝑁 = 2 → (((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋) ↔ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)))
43	42	rabbidv 3419	. . . . . 6 ⊢ (𝑁 = 2 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)})
44	35, 43	eqtrid 2807	. . . . 5 ⊢ (𝑁 = 2 → 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)})
45	44	fveq2d 6890	. . . 4 ⊢ (𝑁 = 2 → (♯‘𝐶) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}))
46		numclwlk1.f	. . . . . . 7 ⊢ 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = (𝑁 − 2) ∧ ((2nd ‘𝑤)‘0) = 𝑋)}
47	40	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑁 = 2 → ((♯‘(1st ‘𝑤)) = (𝑁 − 2) ↔ (♯‘(1st ‘𝑤)) = 0))
48	47	anbi1d 643	. . . . . . . 8 ⊢ (𝑁 = 2 → (((♯‘(1st ‘𝑤)) = (𝑁 − 2) ∧ ((2nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)))
49	48	rabbidv 3419	. . . . . . 7 ⊢ (𝑁 = 2 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = (𝑁 − 2) ∧ ((2nd ‘𝑤)‘0) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)})
50	46, 49	eqtrid 2807	. . . . . 6 ⊢ (𝑁 = 2 → 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)})
51	50	fveq2d 6890	. . . . 5 ⊢ (𝑁 = 2 → (♯‘𝐹) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}))
52	51	oveq2d 7437	. . . 4 ⊢ (𝑁 = 2 → (𝐾 · (♯‘𝐹)) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)})))
53	45, 52	eqeq12d 2776	. . 3 ⊢ (𝑁 = 2 → ((♯‘𝐶) = (𝐾 · (♯‘𝐹)) ↔ (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}))))
54	53	ad2antll 742	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → ((♯‘𝐶) = (𝐾 · (♯‘𝐹)) ↔ (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}))))
55	34, 54	mpbird 260	1 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  {crab 3412  Vcvv 3450  ∅c0 4278  {csn 4584  ⟨cop 4834   class class class wbr 5102  ‘cfv 6539  (class class class)co 7421  1^st c1st 8000  2^nd c2nd 8001  Fincfn 8977  ℝcr 11203  0cc0 11204  1c1 11205   · cmul 11209   − cmin 11545  2c2 12401  ℕ₀cn0 12610  ♯chash 14478  Vtxcvtx 29580  USGraphcusgr 29736  FinUSGraphcfusgr 29903   RegUSGraph crusgr 30143  ClWalkscclwlks 30363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8721  df-map 8853  df-pm 8854  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-dju 9986  df-card 10024  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-nn 12340  df-2 12409  df-n0 12611  df-xnn0 12684  df-z 12698  df-uz 12970  df-rp 13125  df-xadd 13246  df-fz 13644  df-fzo 13793  df-seq 14149  df-exp 14209  df-hash 14479  df-word 14663  df-lsw 14712  df-concat 14720  df-s1 14747  df-substr 14794  df-pfx 14826  df-vtx 29582  df-iedg 29583  df-edg 29632  df-uhgr 29642  df-ushgr 29643  df-upgr 29666  df-umgr 29667  df-uspgr 29737  df-usgr 29738  df-fusgr 29904  df-nbgr 29920  df-vtxdg 30053  df-rgr 30144  df-rusgr 30145  df-wlks 30186  df-clwlks 30364  df-wwlks 30425  df-wwlksn 30426  df-clwwlk 30579  df-clwwlkn 30622  df-clwwlknon 30685

This theorem is used by:  numclwlk1  30978