Theorem numclwlk1lem1 30527

Description: Lemma 1 for numclwlk1 30529 (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 1105	. . . . . . 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 572	. . . . . . . 8 ⊢ ((((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ↔ ((2nd ‘𝑤)‘0) = 𝑋)
3	2	anbi2i 632	. . . . . . 7 ⊢ (((♯‘(1st ‘𝑤)) = 2 ∧ (((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ↔ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋))
4	1, 3	bitri 277	. . . . . 6 ⊢ (((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋))
5	4	rabbii 3418	. . . . 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 6864	. . . 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 486	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝑉 ∈ Fin)
8		simpr 488	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾)
9		simpl 486	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 2) → 𝑋 ∈ 𝑉)
10		numclwlk1.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
11	10	clwlknon2num 30526	. . . . 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 1440	. . . 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 2808	. . 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 29721	. . . . . . . . 9 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
15	14	anim2i 626	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
16	15	ancomd 465	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
17	10	isfusgr 29475	. . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
18	16, 17	sylibr 236	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
19		ne0i 4291	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑉 ≠ ∅)
20	19	adantr 484	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 2) → 𝑉 ≠ ∅)
21	10	frusgrnn0 29728	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
22	18, 8, 20, 21	syl2an3an 1440	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → 𝐾 ∈ ℕ0)
23	22	nn0red 12536	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → 𝐾 ∈ ℝ)
24		ax-1rid 11136	. . . 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 30525	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
27	26	ad2antrl 738	. . . . . 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 6865	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}))
29		opex 5428	. . . . . 6 ⊢ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ V
30		hashsng 14375	. . . . . 6 ⊢ (⟨∅, {⟨0, 𝑋⟩}⟩ ∈ V → (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}) = 1)
31	29, 30	ax-mp 5	. . . . 5 ⊢ (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}) = 1
32	28, 31	eqtr2di 2813	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → 1 = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}))
33	32	oveq2d 7406	. . 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 2802	. 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 2773	. . . . . . . 8 ⊢ (𝑁 = 2 → ((♯‘(1st ‘𝑤)) = 𝑁 ↔ (♯‘(1st ‘𝑤)) = 2))
37		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑁 = 2 → (𝑁 − 2) = (2 − 2))
38		2cn 12286	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
39	38	subidi 11495	. . . . . . . . . 10 ⊢ (2 − 2) = 0
40	37, 39	eqtrdi 2812	. . . . . . . . 9 ⊢ (𝑁 = 2 → (𝑁 − 2) = 0)
41	40	fveqeq2d 6869	. . . . . . . 8 ⊢ (𝑁 = 2 → (((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2nd ‘𝑤)‘0) = 𝑋))
42	36, 41	3anbi13d 1458	. . . . . . 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 3420	. . . . . 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 2808	. . . . 5 ⊢ (𝑁 = 2 → 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘0) = 𝑋)})
45	44	fveq2d 6865	. . . 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 2772	. . . . . . . . 9 ⊢ (𝑁 = 2 → ((♯‘(1st ‘𝑤)) = (𝑁 − 2) ↔ (♯‘(1st ‘𝑤)) = 0))
48	47	anbi1d 640	. . . . . . . 8 ⊢ (𝑁 = 2 → (((♯‘(1st ‘𝑤)) = (𝑁 − 2) ∧ ((2nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)))
49	48	rabbidv 3420	. . . . . . 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 2808	. . . . . 6 ⊢ (𝑁 = 2 → 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)})
51	50	fveq2d 6865	. . . . 5 ⊢ (𝑁 = 2 → (♯‘𝐹) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}))
52	51	oveq2d 7406	. . . 4 ⊢ (𝑁 = 2 → (𝐾 · (♯‘𝐹)) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)})))
53	45, 52	eqeq12d 2777	. . 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 739	. 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 259	1 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  {crab 3413  Vcvv 3453  ∅c0 4283  {csn 4579  ⟨cop 4585   class class class wbr 5097  ‘cfv 6515  (class class class)co 7390  1^st c1st 7962  2^nd c2nd 7963  Fincfn 8920  ℝcr 11065  0cc0 11066  1c1 11067   · cmul 11071   − cmin 11407  2c2 12265  ℕ₀cn0 12474  ♯chash 14336  Vtxcvtx 29153  USGraphcusgr 29306  FinUSGraphcfusgr 29473   RegUSGraph crusgr 29713  ClWalkscclwlks 29926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-cnex 11122  ax-resscn 11123  ax-1cn 11124  ax-icn 11125  ax-addcl 11126  ax-addrcl 11127  ax-mulcl 11128  ax-mulrcl 11129  ax-mulcom 11130  ax-addass 11131  ax-mulass 11132  ax-distr 11133  ax-i2m1 11134  ax-1ne0 11135  ax-1rid 11136  ax-rnegex 11137  ax-rrecex 11138  ax-cnre 11139  ax-pre-lttri 11140  ax-pre-lttrn 11141  ax-pre-ltadd 11142  ax-pre-mulgt0 11143

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ifp 1074  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-2o 8431  df-oadd 8434  df-er 8671  df-map 8803  df-pm 8804  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-dju 9852  df-card 9890  df-pnf 11211  df-mnf 11212  df-xr 11213  df-ltxr 11214  df-le 11215  df-sub 11409  df-neg 11410  df-nn 12204  df-2 12273  df-n0 12475  df-xnn0 12548  df-z 12562  df-uz 12833  df-rp 12987  df-xadd 13108  df-fz 13506  df-fzo 13653  df-seq 14008  df-exp 14068  df-hash 14337  df-word 14520  df-lsw 14569  df-concat 14577  df-s1 14603  df-substr 14648  df-pfx 14678  df-vtx 29155  df-iedg 29156  df-edg 29205  df-uhgr 29215  df-ushgr 29216  df-upgr 29239  df-umgr 29240  df-uspgr 29307  df-usgr 29308  df-fusgr 29474  df-nbgr 29490  df-vtxdg 29623  df-rgr 29714  df-rusgr 29715  df-wlks 29756  df-clwlks 29927  df-wwlks 29986  df-wwlksn 29987  df-clwwlk 30140  df-clwwlkn 30183  df-clwwlknon 30246

This theorem is referenced by:  numclwlk1  30529