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Mirrors > Home > MPE Home > Th. List > clwwlknclwwlkdifnum | Structured version Visualization version GIF version |
Description: In a 𝐾-regular graph, the size of the set 𝐴 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 𝐵 of closed walks of length 𝑁 anchored at this vertex 𝑋. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.) (Revised by AV, 8-Mar-2022.) (Proof shortened by AV, 16-Mar-2022.) |
Ref | Expression |
---|---|
clwwlknclwwlkdif.a | ⊢ 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)} |
clwwlknclwwlkdif.b | ⊢ 𝐵 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋) |
clwwlknclwwlkdifnum.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
clwwlknclwwlkdifnum | ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (#‘𝐴) = ((𝐾↑𝑁) − (#‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwwlknclwwlkdif.a | . . . . 5 ⊢ 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)} | |
2 | clwwlknclwwlkdif.b | . . . . 5 ⊢ 𝐵 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋) | |
3 | eqid 2651 | . . . . 5 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} | |
4 | 1, 2, 3 | clwwlknclwwlkdif 26945 | . . . 4 ⊢ 𝐴 = ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵) |
5 | 4 | fveq2i 6232 | . . 3 ⊢ (#‘𝐴) = (#‘({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) |
6 | 5 | a1i 11 | . 2 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (#‘𝐴) = (#‘({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵))) |
7 | clwwlknclwwlkdifnum.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | 7 | eleq1i 2721 | . . . . . . 7 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin) |
9 | 8 | biimpi 206 | . . . . . 6 ⊢ (𝑉 ∈ Fin → (Vtx‘𝐺) ∈ Fin) |
10 | 9 | adantl 481 | . . . . 5 ⊢ ((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) → (Vtx‘𝐺) ∈ Fin) |
11 | 10 | adantr 480 | . . . 4 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (Vtx‘𝐺) ∈ Fin) |
12 | wwlksnfi 26869 | . . . 4 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin) | |
13 | rabfi 8226 | . . . 4 ⊢ ((𝑁 WWalksN 𝐺) ∈ Fin → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ Fin) | |
14 | 11, 12, 13 | 3syl 18 | . . 3 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ Fin) |
15 | 7 | iswwlksnon 26802 | . . . . . . . 8 ⊢ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)} |
16 | ancom 465 | . . . . . . . . 9 ⊢ (((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋) ↔ ((𝑤‘𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋)) | |
17 | 16 | rabbii 3216 | . . . . . . . 8 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋)} |
18 | 15, 17 | eqtri 2673 | . . . . . . 7 ⊢ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋)} |
19 | 18 | a1i 11 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋)}) |
20 | 2, 19 | syl5eq 2697 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐵 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋)}) |
21 | simpr 476 | . . . . . . 7 ⊢ (((𝑤‘𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋) | |
22 | 21 | a1i 11 | . . . . . 6 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋)) |
23 | 22 | ss2rabi 3717 | . . . . 5 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋)} ⊆ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} |
24 | 20, 23 | syl6eqss 3688 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐵 ⊆ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) |
25 | 24 | adantl 481 | . . 3 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝐵 ⊆ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) |
26 | hashssdif 13238 | . . 3 ⊢ (({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))) | |
27 | 14, 25, 26 | syl2anc 694 | . 2 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (#‘({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵))) |
28 | simpl 472 | . . . . 5 ⊢ ((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) → 𝐺RegUSGraph𝐾) | |
29 | 28 | adantr 480 | . . . 4 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝐺RegUSGraph𝐾) |
30 | simpr 476 | . . . . 5 ⊢ ((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) → 𝑉 ∈ Fin) | |
31 | 30 | adantr 480 | . . . 4 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝑉 ∈ Fin) |
32 | simpl 472 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑋 ∈ 𝑉) | |
33 | 32 | adantl 481 | . . . 4 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝑋 ∈ 𝑉) |
34 | simpr 476 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
35 | 34 | adantl 481 | . . . 4 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝑁 ∈ ℕ0) |
36 | 7 | rusgrnumwwlkg 26943 | . . . 4 ⊢ ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) = (𝐾↑𝑁)) |
37 | 29, 31, 33, 35, 36 | syl13anc 1368 | . . 3 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) = (𝐾↑𝑁)) |
38 | 37 | oveq1d 6705 | . 2 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)) = ((𝐾↑𝑁) − (#‘𝐵))) |
39 | 6, 27, 38 | 3eqtrd 2689 | 1 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (#‘𝐴) = ((𝐾↑𝑁) − (#‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 {crab 2945 ∖ cdif 3604 ⊆ wss 3607 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 Fincfn 7997 0cc0 9974 − cmin 10304 ℕ0cn0 11330 ↑cexp 12900 #chash 13157 lastS clsw 13324 Vtxcvtx 25919 RegUSGraphcrusgr 26508 WWalksN cwwlksn 26774 WWalksNOn cwwlksnon 26775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-disj 4653 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-xnn0 11402 df-z 11416 df-uz 11726 df-rp 11871 df-xadd 11985 df-fz 12365 df-fzo 12505 df-seq 12842 df-exp 12901 df-hash 13158 df-word 13331 df-lsw 13332 df-concat 13333 df-s1 13334 df-substr 13335 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-sum 14461 df-vtx 25921 df-iedg 25922 df-edg 25985 df-uhgr 25998 df-ushgr 25999 df-upgr 26022 df-umgr 26023 df-uspgr 26090 df-usgr 26091 df-fusgr 26254 df-nbgr 26270 df-vtxdg 26418 df-rgr 26509 df-rusgr 26510 df-wwlks 26778 df-wwlksn 26779 df-wwlksnon 26780 |
This theorem is referenced by: numclwwlkqhash 27355 |
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