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Theorem numclwwlkqhash 30341

Description: In a 𝐾-regular graph, the size of the set of walks of length 𝑁 starting with a fixed vertex 𝑋 and ending not at this vertex is the difference between 𝐾 to the power of 𝑁 and the size of the set of closed walks of length 𝑁 on vertex 𝑋. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 30-May-2021.) (Revised by AV, 5-Mar-2022.) (Proof shortened by AV, 7-Jul-2022.)

**Hypotheses**
Ref	Expression
numclwwlk.v	⊢ 𝑉 = (Vtx‘𝐺)
numclwwlk.q	⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})

**Assertion**
Ref	Expression
numclwwlkqhash	⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘(𝑋𝑄𝑁)) = ((𝐾↑𝑁) − (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁))))

Distinct variable groups: 𝑛,𝐺,𝑣,𝑤 𝑛,𝑁,𝑣,𝑤 𝑛,𝑉,𝑣 𝑛,𝑋,𝑣,𝑤 𝑤,𝐾 𝑤,𝑉 Allowed substitution hints: 𝑄(𝑤,𝑣,𝑛) 𝐾(𝑣,𝑛)

Proof of Theorem numclwwlkqhash Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		numclwwlk.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		numclwwlk.q	. . . . 5 ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
3	1, 2	numclwwlkovq 30340	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
4	3	adantl 481	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
5	4	fveq2d 6891	. 2 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘(𝑋𝑄𝑁)) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}))
6		nnnn0 12517	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
7		eqid 2734	. . . . 5 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}
8		eqid 2734	. . . . 5 ⊢ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)
9	7, 8, 1	clwwlknclwwlkdifnum 29946	. . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}) = ((𝐾↑𝑁) − (♯‘(𝑋(𝑁 WWalksNOn 𝐺)𝑋))))
10	6, 9	sylanr2 683	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}) = ((𝐾↑𝑁) − (♯‘(𝑋(𝑁 WWalksNOn 𝐺)𝑋))))
11	1	iswwlksnon 29820	. . . . . . 7 ⊢ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)}
12		wwlknlsw 29814	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑤‘𝑁) = (lastS‘𝑤))
13		eqcom 2741	. . . . . . . . . . . 12 ⊢ ((𝑤‘0) = 𝑋 ↔ 𝑋 = (𝑤‘0))
14	13	biimpi 216	. . . . . . . . . . 11 ⊢ ((𝑤‘0) = 𝑋 → 𝑋 = (𝑤‘0))
15	12, 14	eqeqan12d 2748	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) → ((𝑤‘𝑁) = 𝑋 ↔ (lastS‘𝑤) = (𝑤‘0)))
16	15	pm5.32da 579	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋) ↔ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) = (𝑤‘0))))
17	16	biancomd 463	. . . . . . . 8 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋) ↔ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)))
18	17	rabbiia 3424	. . . . . . 7 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
19	11, 18	eqtri 2757	. . . . . 6 ⊢ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
20	19	fveq2i 6890	. . . . 5 ⊢ (♯‘(𝑋(𝑁 WWalksNOn 𝐺)𝑋)) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)})
21	20	a1i 11	. . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘(𝑋(𝑁 WWalksNOn 𝐺)𝑋)) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}))
22	21	oveq2d 7430	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ((𝐾↑𝑁) − (♯‘(𝑋(𝑁 WWalksNOn 𝐺)𝑋))) = ((𝐾↑𝑁) − (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)})))
23	10, 22	eqtrd 2769	. 2 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}) = ((𝐾↑𝑁) − (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)})))
24		ovex 7447	. . . . 5 ⊢ (𝑁 WWalksN 𝐺) ∈ V
25	24	rabex 5321	. . . 4 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} ∈ V
26		clwwlkvbij 30079	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
27	26	adantl 481	. . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
28		hasheqf1oi 14373	. . . 4 ⊢ ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} ∈ V → (∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁))))
29	25, 27, 28	mpsyl 68	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁)))
30	29	oveq2d 7430	. 2 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ((𝐾↑𝑁) − (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)})) = ((𝐾↑𝑁) − (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁))))
31	5, 23, 30	3eqtrd 2773	1 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘(𝑋𝑄𝑁)) = ((𝐾↑𝑁) − (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ≠ wne 2931 {crab 3420 Vcvv 3464 class class class wbr 5125 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 Fincfn 8968 0cc0 11138 − cmin 11475 ℕcn 12249 ℕ₀cn0 12510 ↑cexp 14085 ♯chash 14352 lastSclsw 14583 Vtxcvtx 28960 RegUSGraph crusgr 29521 WWalksN cwwlksn 29793 WWalksNOn cwwlksnon 29794 ClWWalksNOncclwwlknon 30053

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5261 ax-sep 5278 ax-nul 5288 ax-pow 5347 ax-pr 5414 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3773 df-csb 3882 df-dif 3936 df-un 3938 df-in 3940 df-ss 3950 df-pss 3953 df-nul 4316 df-if 4508 df-pw 4584 df-sn 4609 df-pr 4611 df-op 4615 df-uni 4890 df-int 4929 df-iun 4975 df-disj 5093 df-br 5126 df-opab 5188 df-mpt 5208 df-tr 5242 df-id 5560 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6303 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7871 df-1st 7997 df-2nd 7998 df-frecs 8289 df-wrecs 8320 df-recs 8394 df-rdg 8433 df-1o 8489 df-2o 8490 df-oadd 8493 df-er 8728 df-map 8851 df-pm 8852 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9465 df-oi 9533 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11477 df-neg 11478 df-div 11904 df-nn 12250 df-2 12312 df-3 12313 df-n0 12511 df-xnn0 12584 df-z 12598 df-uz 12862 df-rp 13018 df-xadd 13138 df-fz 13531 df-fzo 13678 df-seq 14026 df-exp 14086 df-hash 14353 df-word 14536 df-lsw 14584 df-concat 14592 df-s1 14617 df-substr 14662 df-pfx 14692 df-cj 15121 df-re 15122 df-im 15123 df-sqrt 15257 df-abs 15258 df-clim 15507 df-sum 15706 df-vtx 28962 df-iedg 28963 df-edg 29012 df-uhgr 29022 df-ushgr 29023 df-upgr 29046 df-umgr 29047 df-uspgr 29114 df-usgr 29115 df-fusgr 29281 df-nbgr 29297 df-vtxdg 29431 df-rgr 29522 df-rusgr 29523 df-wwlks 29797 df-wwlksn 29798 df-wwlksnon 29799 df-clwwlk 29948 df-clwwlkn 29991 df-clwwlknon 30054

This theorem is referenced by: numclwwlk2 30347

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