Theorem numclwlk1lem1 30355

Description: Lemma 1 for numclwlk1 30357 (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 1094	. . . . . . 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 564	. . . . . . . 8 ⊢ ((((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ↔ ((2nd ‘𝑤)‘0) = 𝑋)
3	2	anbi2i 623	. . . . . . 7 ⊢ (((♯‘(1st ‘𝑤)) = 2 ∧ (((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ↔ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋))
4	1, 3	bitri 275	. . . . . 6 ⊢ (((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋))
5	4	rabbii 3426	. . . . 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 6884	. . . 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 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 30354	. . . . 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 1424	. . . 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 2783	. . 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 29549	. . . . . . . . 9 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
15	14	anim2i 617	. . . . . . . 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 29302	. . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
18	16, 17	sylibr 234	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
19		ne0i 4321	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑉 ≠ ∅)
20	19	adantr 480	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 2) → 𝑉 ≠ ∅)
21	10	frusgrnn0 29556	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
22	18, 8, 20, 21	syl2an3an 1424	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → 𝐾 ∈ ℕ0)
23	22	nn0red 12568	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → 𝐾 ∈ ℝ)
24		ax-1rid 11204	. . . 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 30353	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
27	26	ad2antrl 728	. . . . . 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 6885	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}))
29		opex 5444	. . . . . 6 ⊢ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ V
30		hashsng 14392	. . . . . 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) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}))
33	32	oveq2d 7426	. . 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 2777	. 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 2748	. . . . . . . 8 ⊢ (𝑁 = 2 → ((♯‘(1st ‘𝑤)) = 𝑁 ↔ (♯‘(1st ‘𝑤)) = 2))
37		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑁 = 2 → (𝑁 − 2) = (2 − 2))
38		2cn 12320	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
39	38	subidi 11559	. . . . . . . . . 10 ⊢ (2 − 2) = 0
40	37, 39	eqtrdi 2787	. . . . . . . . 9 ⊢ (𝑁 = 2 → (𝑁 − 2) = 0)
41	40	fveqeq2d 6889	. . . . . . . 8 ⊢ (𝑁 = 2 → (((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2nd ‘𝑤)‘0) = 𝑋))
42	36, 41	3anbi13d 1440	. . . . . . 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 3428	. . . . . 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 2783	. . . . 5 ⊢ (𝑁 = 2 → 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)})
45	44	fveq2d 6885	. . . 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 2747	. . . . . . . . 9 ⊢ (𝑁 = 2 → ((♯‘(1st ‘𝑤)) = (𝑁 − 2) ↔ (♯‘(1st ‘𝑤)) = 0))
48	47	anbi1d 631	. . . . . . . 8 ⊢ (𝑁 = 2 → (((♯‘(1st ‘𝑤)) = (𝑁 − 2) ∧ ((2nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)))
49	48	rabbidv 3428	. . . . . . 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 2783	. . . . . 6 ⊢ (𝑁 = 2 → 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)})
51	50	fveq2d 6885	. . . . 5 ⊢ (𝑁 = 2 → (♯‘𝐹) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}))
52	51	oveq2d 7426	. . . 4 ⊢ (𝑁 = 2 → (𝐾 · (♯‘𝐹)) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)})))
53	45, 52	eqeq12d 2752	. . 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 729	. 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 257	1 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  {crab 3420  Vcvv 3464  ∅c0 4313  {csn 4606  ⟨cop 4612   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  1^st c1st 7991  2^nd c2nd 7992  Fincfn 8964  ℝcr 11133  0cc0 11134  1c1 11135   · cmul 11139   − cmin 11471  2c2 12300  ℕ₀cn0 12506  ♯chash 14353  Vtxcvtx 28980  USGraphcusgr 29133  FinUSGraphcfusgr 29300   RegUSGraph crusgr 29541  ClWalkscclwlks 29757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-er 8724  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-dju 9920  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-n0 12507  df-xnn0 12580  df-z 12594  df-uz 12858  df-rp 13014  df-xadd 13134  df-fz 13530  df-fzo 13677  df-seq 14025  df-exp 14085  df-hash 14354  df-word 14537  df-lsw 14586  df-concat 14594  df-s1 14619  df-substr 14664  df-pfx 14694  df-vtx 28982  df-iedg 28983  df-edg 29032  df-uhgr 29042  df-ushgr 29043  df-upgr 29066  df-umgr 29067  df-uspgr 29134  df-usgr 29135  df-fusgr 29301  df-nbgr 29317  df-vtxdg 29451  df-rgr 29542  df-rusgr 29543  df-wlks 29584  df-clwlks 29758  df-wwlks 29817  df-wwlksn 29818  df-clwwlk 29968  df-clwwlkn 30011  df-clwwlknon 30074

This theorem is referenced by:  numclwlk1  30357