clwwlknclwwlkdifnum - Metamath Proof Explorer

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Theorem clwwlknclwwlkdifnum 26874

Description: In a k-regular graph, the size of the set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between k to the power of n and the size of the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.)

**Hypotheses**
Ref	Expression
clwwlknclwwlkdif.a	⊢ 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}
clwwlknclwwlkdif.b	⊢ 𝐵 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
clwwlknclwwlkdifnum.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
clwwlknclwwlkdifnum	⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (#‘𝐴) = ((𝐾↑𝑁) − (#‘𝐵)))

Distinct variable groups: 𝑤,𝐺 𝑤,𝐾 𝑤,𝑁 𝑤,𝑉 𝑤,𝑋 Allowed substitution hints: 𝐴(𝑤) 𝐵(𝑤)

**Proof of Theorem clwwlknclwwlkdifnum**
Step	Hyp	Ref	Expression
1		clwwlknclwwlkdif.a	. . . 4 ⊢ 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}
2		clwwlknclwwlkdif.b	. . . 4 ⊢ 𝐵 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
3	1, 2	clwwlknclwwlkdifs 26873	. . 3 ⊢ 𝐴 = ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)
4	3	fveq2i 6194	. 2 ⊢ (#‘𝐴) = (#‘({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵))
5		clwwlknclwwlkdifnum.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
6	5	eleq1i 2692	. . . . . . . 8 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
7	6	biimpi 206	. . . . . . 7 ⊢ (𝑉 ∈ Fin → (Vtx‘𝐺) ∈ Fin)
8	7	adantl 482	. . . . . 6 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → (Vtx‘𝐺) ∈ Fin)
9	8	adantr 481	. . . . 5 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (Vtx‘𝐺) ∈ Fin)
10		wwlksnfi 26801	. . . . 5 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)
11		rabfi 8185	. . . . 5 ⊢ ((𝑁 WWalksN 𝐺) ∈ Fin → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ Fin)
12	9, 10, 11	3syl 18	. . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ Fin)
13		simpr 477	. . . . . . 7 ⊢ ((( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋)
14	13	a1i 11	. . . . . 6 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → ((( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋))
15	14	ss2rabi 3684	. . . . 5 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} ⊆ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}
16	2, 15	eqsstri 3635	. . . 4 ⊢ 𝐵 ⊆ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}
17		hashssdif 13200	. . . 4 ⊢ (({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ Fin ∧ 𝐵 ⊆ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) → (#‘({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)))
18	12, 16, 17	sylancl 694	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (#‘({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)))
19		simpl 473	. . . . . 6 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 RegUSGraph 𝐾)
20	19	adantr 481	. . . . 5 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝐺 RegUSGraph 𝐾)
21		simpr 477	. . . . . 6 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝑉 ∈ Fin)
22	21	adantr 481	. . . . 5 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑉 ∈ Fin)
23		simpl 473	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑋 ∈ 𝑉)
24	23	adantl 482	. . . . 5 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑋 ∈ 𝑉)
25		nnnn0 11299	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
26	25	adantl 482	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
27	26	adantl 482	. . . . 5 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑁 ∈ ℕ₀)
28	5	rusgrnumwwlkg 26871	. . . . 5 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) = (𝐾↑𝑁))
29	20, 22, 24, 27, 28	syl13anc 1328	. . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) = (𝐾↑𝑁))
30	29	oveq1d 6665	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)) = ((𝐾↑𝑁) − (#‘𝐵)))
31	18, 30	eqtrd 2656	. 2 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (#‘({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((𝐾↑𝑁) − (#‘𝐵)))
32	4, 31	syl5eq 2668	1 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (#‘𝐴) = ((𝐾↑𝑁) − (#‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {crab 2916 ∖ cdif 3571 ⊆ wss 3574 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 0cc0 9936 − cmin 10266 ℕcn 11020 ℕ₀cn0 11292 ↑cexp 12860 #chash 13117 lastS clsw 13292 Vtxcvtx 25874 RegUSGraph crusgr 26452 WWalksN cwwlksn 26718

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-xadd 11947 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-vtx 25876 df-iedg 25877 df-edg 25940 df-uhgr 25953 df-ushgr 25954 df-upgr 25977 df-umgr 25978 df-uspgr 26045 df-usgr 26046 df-fusgr 26209 df-nbgr 26228 df-vtxdg 26362 df-rgr 26453 df-rusgr 26454 df-wwlks 26722 df-wwlksn 26723

This theorem is referenced by: numclwwlkqhash 27233

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