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Mirrors > Home > MPE Home > Th. List > numclwlk1 | Structured version Visualization version GIF version |
Description: Statement 9 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v is kf(n-2)". Since 𝐺 is k-regular, the vertex v(n-2) = v has k neighbors v(n-1), so there are k walks from v(n-2) = v to v(n) = v (via each of v's neighbors) completing each of the f(n-2) walks from v=v(0) to v(n-2)=v. This theorem holds even for k=0. (Contributed by AV, 23-May-2022.) |
Ref | Expression |
---|---|
numclwlk1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
numclwlk1.c | ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)} |
numclwlk1.f | ⊢ 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = (𝑁 − 2) ∧ ((2nd ‘𝑤)‘0) = 𝑋)} |
Ref | Expression |
---|---|
numclwlk1 | ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2))) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzp1 12267 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 = 2 ∨ 𝑁 ∈ (ℤ≥‘(2 + 1)))) | |
2 | numclwlk1.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | numclwlk1.c | . . . . . . . 8 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)} | |
4 | numclwlk1.f | . . . . . . . 8 ⊢ 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = (𝑁 − 2) ∧ ((2nd ‘𝑤)‘0) = 𝑋)} | |
5 | 2, 3, 4 | numclwlk1lem1 28154 | . . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))) |
6 | 5 | expcom 417 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 2) → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))) |
7 | 6 | expcom 417 | . . . . 5 ⊢ (𝑁 = 2 → (𝑋 ∈ 𝑉 → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))))) |
8 | 2, 3, 4 | numclwlk1lem2 28155 | . . . . . . . 8 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))) |
9 | 8 | expcom 417 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))) |
10 | 9 | expcom 417 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑋 ∈ 𝑉 → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))))) |
11 | 2p1e3 11767 | . . . . . . 7 ⊢ (2 + 1) = 3 | |
12 | 11 | fveq2i 6648 | . . . . . 6 ⊢ (ℤ≥‘(2 + 1)) = (ℤ≥‘3) |
13 | 10, 12 | eleq2s 2908 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘(2 + 1)) → (𝑋 ∈ 𝑉 → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))))) |
14 | 7, 13 | jaoi 854 | . . . 4 ⊢ ((𝑁 = 2 ∨ 𝑁 ∈ (ℤ≥‘(2 + 1))) → (𝑋 ∈ 𝑉 → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))))) |
15 | 1, 14 | syl 17 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑋 ∈ 𝑉 → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))))) |
16 | 15 | impcom 411 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))) |
17 | 16 | impcom 411 | 1 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2))) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {crab 3110 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 Fincfn 8492 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 − cmin 10859 2c2 11680 3c3 11681 ℤ≥cuz 12231 ♯chash 13686 Vtxcvtx 26789 RegUSGraph crusgr 27346 ClWalkscclwlks 27559 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ifp 1059 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-rp 12378 df-xadd 12496 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-word 13858 df-lsw 13906 df-concat 13914 df-s1 13941 df-substr 13994 df-pfx 14024 df-s2 14201 df-vtx 26791 df-iedg 26792 df-edg 26841 df-uhgr 26851 df-ushgr 26852 df-upgr 26875 df-umgr 26876 df-uspgr 26943 df-usgr 26944 df-fusgr 27107 df-nbgr 27123 df-vtxdg 27256 df-rgr 27347 df-rusgr 27348 df-wlks 27389 df-clwlks 27560 df-wwlks 27616 df-wwlksn 27617 df-clwwlk 27767 df-clwwlkn 27810 df-clwwlknon 27873 |
This theorem is referenced by: (None) |
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