clwlknclwlkdifnum - Metamath Proof Explorer

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Theorem clwlknclwlkdifnum 26488

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.)

**Hypotheses**
Ref	Expression
clwlknclwlkdif.a	⊢ 𝐴 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}
clwlknclwlkdif.b	⊢ 𝐵 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}

**Assertion**
Ref	Expression
clwlknclwlkdifnum	⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (#‘𝐴) = ((𝐾↑𝑁) − (#‘𝐵)))

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

**Proof of Theorem clwlknclwlkdifnum**
Step	Hyp	Ref	Expression
1		clwlknclwlkdif.a	. . . 4 ⊢ 𝐴 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}
2		clwlknclwlkdif.b	. . . 4 ⊢ 𝐵 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
3	1, 2	clwlknclwlkdifs 26487	. . 3 ⊢ 𝐴 = ({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)
4	3	fveq2i 6106	. 2 ⊢ (#‘𝐴) = (#‘({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵))
5		simpr 476	. . . . . 6 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) → 𝑉 ∈ Fin)
6	5	adantr 480	. . . . 5 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑉 ∈ Fin)
7		wwlknfi 26266	. . . . 5 ⊢ (𝑉 ∈ Fin → ((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin)
8		rabfi 8070	. . . . 5 ⊢ (((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∈ Fin)
9	6, 7, 8	3syl 18	. . . 4 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∈ Fin)
10		simpr 476	. . . . . . 7 ⊢ ((( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋)
11	10	a1i 11	. . . . . 6 ⊢ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋))
12	11	ss2rabi 3647	. . . . 5 ⊢ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} ⊆ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}
13	2, 12	eqsstri 3598	. . . 4 ⊢ 𝐵 ⊆ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}
14		hashssdif 13061	. . . 4 ⊢ (({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∈ Fin ∧ 𝐵 ⊆ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) → (#‘({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)))
15	9, 13, 14	sylancl 693	. . 3 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (#‘({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)))
16		simpl 472	. . . . . 6 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) → ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾)
17	16	adantr 480	. . . . 5 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾)
18		simpl 472	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑋 ∈ 𝑉)
19	18	adantl 481	. . . . 5 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑋 ∈ 𝑉)
20		nnnn0 11176	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
21	20	adantl 481	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
22	21	adantl 481	. . . . 5 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑁 ∈ ℕ₀)
23		rusgranumwwlkg 26486	. . . . 5 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) = (𝐾↑𝑁))
24	17, 6, 19, 22, 23	syl13anc 1320	. . . 4 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) = (𝐾↑𝑁))
25	24	oveq1d 6564	. . 3 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)) = ((𝐾↑𝑁) − (#‘𝐵)))
26	15, 25	eqtrd 2644	. 2 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (#‘({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((𝐾↑𝑁) − (#‘𝐵)))
27	4, 26	syl5eq 2656	1 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (#‘𝐴) = ((𝐾↑𝑁) − (#‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 ∖ cdif 3537 ⊆ wss 3540 ⟨cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 0cc0 9815 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ↑cexp 12722 #chash 12979 lastS clsw 13147 WWalksN cwwlkn 26206 RegUSGrph crusgra 26450

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-xadd 11823 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-usgra 25862 df-nbgra 25949 df-wlk 26036 df-wwlk 26207 df-wwlkn 26208 df-vdgr 26421 df-rgra 26451 df-rusgra 26452

This theorem is referenced by: numclwwlkqhash 26627

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