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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 ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2))) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzp1 12092 | . . . 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 27938 | . . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))) |
6 | 5 | expcom 406 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 2) → ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))) |
7 | 6 | expcom 406 | . . . . 5 ⊢ (𝑁 = 2 → (𝑋 ∈ 𝑉 → ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))))) |
8 | 2, 3, 4 | numclwlk1lem2 27939 | . . . . . . . 8 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))) |
9 | 8 | expcom 406 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))) |
10 | 9 | expcom 406 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑋 ∈ 𝑉 → ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))))) |
11 | 2p1e3 11588 | . . . . . . 7 ⊢ (2 + 1) = 3 | |
12 | 11 | fveq2i 6500 | . . . . . 6 ⊢ (ℤ≥‘(2 + 1)) = (ℤ≥‘3) |
13 | 10, 12 | eleq2s 2879 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘(2 + 1)) → (𝑋 ∈ 𝑉 → ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))))) |
14 | 7, 13 | jaoi 844 | . . . 4 ⊢ ((𝑁 = 2 ∨ 𝑁 ∈ (ℤ≥‘(2 + 1))) → (𝑋 ∈ 𝑉 → ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))))) |
15 | 1, 14 | syl 17 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑋 ∈ 𝑉 → ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))))) |
16 | 15 | impcom 399 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))) |
17 | 16 | impcom 399 | 1 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2))) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∨ wo 834 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 {crab 3087 class class class wbr 4926 ‘cfv 6186 (class class class)co 6975 1st c1st 7498 2nd c2nd 7499 Fincfn 8305 0cc0 10334 1c1 10335 + caddc 10337 · cmul 10339 − cmin 10669 2c2 11494 3c3 11495 ℤ≥cuz 12057 ♯chash 13504 Vtxcvtx 26500 RegUSGraphcrusgr 27057 ClWalkscclwlks 27275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-ifp 1045 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-2o 7905 df-oadd 7908 df-er 8088 df-map 8207 df-pm 8208 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-dju 9123 df-card 9161 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-3 11503 df-n0 11707 df-xnn0 11779 df-z 11793 df-uz 12058 df-rp 12204 df-xadd 12324 df-fz 12708 df-fzo 12849 df-seq 13184 df-exp 13244 df-hash 13505 df-word 13672 df-lsw 13725 df-concat 13733 df-s1 13758 df-substr 13803 df-pfx 13852 df-s2 14071 df-vtx 26502 df-iedg 26503 df-edg 26552 df-uhgr 26562 df-ushgr 26563 df-upgr 26586 df-umgr 26587 df-uspgr 26654 df-usgr 26655 df-fusgr 26818 df-nbgr 26834 df-vtxdg 26967 df-rgr 27058 df-rusgr 27059 df-wlks 27100 df-clwlks 27276 df-wwlks 27332 df-wwlksn 27333 df-clwwlk 27504 df-clwwlkn 27556 df-clwwlknon 27632 |
This theorem is referenced by: (None) |
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