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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 | ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 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 2737 | . . . . 5 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} | |
| 4 | 1, 2, 3 | clwwlknclwwlkdif 29998 | . . . 4 ⊢ 𝐴 = ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵) |
| 5 | 4 | fveq2i 6909 | . . 3 ⊢ (♯‘𝐴) = (♯‘({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) |
| 6 | 5 | a1i 11 | . 2 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 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 2832 | . . . . . . 7 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin) |
| 9 | 8 | biimpi 216 | . . . . . 6 ⊢ (𝑉 ∈ Fin → (Vtx‘𝐺) ∈ Fin) |
| 10 | 9 | adantl 481 | . . . . 5 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → (Vtx‘𝐺) ∈ Fin) |
| 11 | 10 | adantr 480 | . . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (Vtx‘𝐺) ∈ Fin) |
| 12 | wwlksnfi 29926 | . . . 4 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin) | |
| 13 | rabfi 9303 | . . . 4 ⊢ ((𝑁 WWalksN 𝐺) ∈ Fin → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ Fin) | |
| 14 | 11, 12, 13 | 3syl 18 | . . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ Fin) |
| 15 | 7 | iswwlksnon 29873 | . . . . . . . 8 ⊢ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)} |
| 16 | ancom 460 | . . . . . . . . 9 ⊢ (((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋) ↔ ((𝑤‘𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋)) | |
| 17 | 16 | rabbii 3442 | . . . . . . . 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 2765 | . . . . . . 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 | eqtrid 2789 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐵 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋)}) |
| 21 | simpr 484 | . . . . . . 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 4077 | . . . . 5 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘𝑁) = 𝑋 ∧ (𝑤‘0) = 𝑋)} ⊆ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} |
| 24 | 20, 23 | eqsstrdi 4028 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐵 ⊆ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) |
| 25 | 24 | adantl 481 | . . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝐵 ⊆ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) |
| 26 | hashssdif 14451 | . . 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 584 | . 2 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) − (♯‘𝐵))) |
| 28 | simpl 482 | . . . . 5 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 RegUSGraph 𝐾) | |
| 29 | 28 | adantr 480 | . . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝐺 RegUSGraph 𝐾) |
| 30 | simpr 484 | . . . . 5 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝑉 ∈ Fin) | |
| 31 | 30 | adantr 480 | . . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝑉 ∈ Fin) |
| 32 | simpl 482 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑋 ∈ 𝑉) | |
| 33 | 32 | adantl 481 | . . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝑋 ∈ 𝑉) |
| 34 | simpr 484 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
| 35 | 34 | adantl 481 | . . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝑁 ∈ ℕ0) |
| 36 | 7 | rusgrnumwwlkg 29996 | . . . 4 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) = (𝐾↑𝑁)) |
| 37 | 29, 31, 33, 35, 36 | syl13anc 1374 | . . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) = (𝐾↑𝑁)) |
| 38 | 37 | oveq1d 7446 | . 2 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) − (♯‘𝐵)) = ((𝐾↑𝑁) − (♯‘𝐵))) |
| 39 | 6, 27, 38 | 3eqtrd 2781 | 1 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘𝐴) = ((𝐾↑𝑁) − (♯‘𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {crab 3436 ∖ cdif 3948 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 0cc0 11155 − cmin 11492 ℕ0cn0 12526 ↑cexp 14102 ♯chash 14369 lastSclsw 14600 Vtxcvtx 29013 RegUSGraph crusgr 29574 WWalksN cwwlksn 29846 WWalksNOn cwwlksnon 29847 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-rp 13035 df-xadd 13155 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-vtx 29015 df-iedg 29016 df-edg 29065 df-uhgr 29075 df-ushgr 29076 df-upgr 29099 df-umgr 29100 df-uspgr 29167 df-usgr 29168 df-fusgr 29334 df-nbgr 29350 df-vtxdg 29484 df-rgr 29575 df-rusgr 29576 df-wwlks 29850 df-wwlksn 29851 df-wwlksnon 29852 |
| This theorem is referenced by: numclwwlkqhash 30394 |
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