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Theorem numclwwlkqhash 30445

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 30444	. . . 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 6844	. 2 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘(𝑋𝑄𝑁)) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}))
6		nnnn0 12444	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
7		eqid 2736	. . . . 5 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}
8		eqid 2736	. . . . 5 ⊢ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)
9	7, 8, 1	clwwlknclwwlkdifnum 30050	. . . 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 684	. . 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 29921	. . . . . . 7 ⊢ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)}
12		wwlknlsw 29915	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑤‘𝑁) = (lastS‘𝑤))
13		eqcom 2743	. . . . . . . . . . . 12 ⊢ ((𝑤‘0) = 𝑋 ↔ 𝑋 = (𝑤‘0))
14	13	biimpi 216	. . . . . . . . . . 11 ⊢ ((𝑤‘0) = 𝑋 → 𝑋 = (𝑤‘0))
15	12, 14	eqeqan12d 2750	. . . . . . . . . 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 3393	. . . . . . 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 2759	. . . . . 6 ⊢ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
20	19	fveq2i 6843	. . . . 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 7383	. . 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 2771	. 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 7400	. . . . 5 ⊢ (𝑁 WWalksN 𝐺) ∈ V
25	24	rabex 5280	. . . 4 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} ∈ V
26		clwwlkvbij 30183	. . . . 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 14313	. . . 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 7383	. 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 2775	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 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2932 {crab 3389 Vcvv 3429 class class class wbr 5085 –1-1-onto→wf1o 6497 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 Fincfn 8893 0cc0 11038 − cmin 11377 ℕcn 12174 ℕ₀cn0 12437 ↑cexp 14023 ♯chash 14292 lastSclsw 14524 Vtxcvtx 29065 RegUSGraph crusgr 29625 WWalksN cwwlksn 29894 WWalksNOn cwwlksnon 29895 ClWWalksNOncclwwlknon 30157

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-xnn0 12511 df-z 12525 df-uz 12789 df-rp 12943 df-xadd 13064 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-word 14476 df-lsw 14525 df-concat 14533 df-s1 14559 df-substr 14604 df-pfx 14634 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-sum 15649 df-vtx 29067 df-iedg 29068 df-edg 29117 df-uhgr 29127 df-ushgr 29128 df-upgr 29151 df-umgr 29152 df-uspgr 29219 df-usgr 29220 df-fusgr 29386 df-nbgr 29402 df-vtxdg 29535 df-rgr 29626 df-rusgr 29627 df-wwlks 29898 df-wwlksn 29899 df-wwlksnon 29900 df-clwwlk 30052 df-clwwlkn 30095 df-clwwlknon 30158

This theorem is referenced by: numclwwlk2 30451

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