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Mirrors > Home > MPE Home > Th. List > clwwlknfi | Structured version Visualization version GIF version |
Description: If there is only a finite number of vertices, the number of closed walks of fixed length (as words) is also finite. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.) |
Ref | Expression |
---|---|
clwwlknfi | ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 11502 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | clwwlkn 27179 | . . . . 5 ⊢ (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} | |
3 | wrdnfi 13535 | . . . . . 6 ⊢ (((Vtx‘𝐺) ∈ Fin ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 𝑁} ∈ Fin) | |
4 | clwwlksswrd 27138 | . . . . . . 7 ⊢ (ClWWalks‘𝐺) ⊆ Word (Vtx‘𝐺) | |
5 | rabss2 3835 | . . . . . . 7 ⊢ ((ClWWalks‘𝐺) ⊆ Word (Vtx‘𝐺) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ⊆ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 𝑁}) | |
6 | 4, 5 | mp1i 13 | . . . . . 6 ⊢ (((Vtx‘𝐺) ∈ Fin ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ⊆ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 𝑁}) |
7 | 3, 6 | ssfid 8340 | . . . . 5 ⊢ (((Vtx‘𝐺) ∈ Fin ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ∈ Fin) |
8 | 2, 7 | syl5eqel 2854 | . . . 4 ⊢ (((Vtx‘𝐺) ∈ Fin ∧ 𝑁 ∈ ℕ0) → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
9 | 8 | expcom 398 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)) |
10 | 1, 9 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)) |
11 | df-nel 3047 | . . . . . . 7 ⊢ (𝑁 ∉ ℕ ↔ ¬ 𝑁 ∈ ℕ) | |
12 | 11 | biimpri 218 | . . . . . 6 ⊢ (¬ 𝑁 ∈ ℕ → 𝑁 ∉ ℕ) |
13 | 12 | olcd 855 | . . . . 5 ⊢ (¬ 𝑁 ∈ ℕ → (𝐺 ∉ V ∨ 𝑁 ∉ ℕ)) |
14 | clwwlkneq0 27184 | . . . . 5 ⊢ ((𝐺 ∉ V ∨ 𝑁 ∉ ℕ) → (𝑁 ClWWalksN 𝐺) = ∅) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (¬ 𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) = ∅) |
16 | 0fin 8345 | . . . 4 ⊢ ∅ ∈ Fin | |
17 | 15, 16 | syl6eqel 2858 | . . 3 ⊢ (¬ 𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
18 | 17 | a1d 25 | . 2 ⊢ (¬ 𝑁 ∈ ℕ → ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)) |
19 | 10, 18 | pm2.61i 176 | 1 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 ∨ wo 828 = wceq 1631 ∈ wcel 2145 ∉ wnel 3046 {crab 3065 Vcvv 3351 ⊆ wss 3724 ∅c0 4064 ‘cfv 6032 (class class class)co 6794 Fincfn 8110 ℕcn 11223 ℕ0cn0 11495 ♯chash 13322 Word cword 13488 Vtxcvtx 26096 ClWWalkscclwwlk 27132 ClWWalksN cclwwlkn 27175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-cnex 10195 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 ax-pre-mulgt0 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-om 7214 df-1st 7316 df-2nd 7317 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-1o 7714 df-2o 7715 df-oadd 7718 df-er 7897 df-map 8012 df-pm 8013 df-en 8111 df-dom 8112 df-sdom 8113 df-fin 8114 df-card 8966 df-cda 9193 df-pnf 10279 df-mnf 10280 df-xr 10281 df-ltxr 10282 df-le 10283 df-sub 10471 df-neg 10472 df-nn 11224 df-n0 11496 df-xnn0 11567 df-z 11581 df-uz 11890 df-fz 12535 df-fzo 12675 df-seq 13010 df-exp 13069 df-hash 13323 df-word 13496 df-clwwlk 27133 df-clwwlkn 27177 |
This theorem is referenced by: qerclwwlknfi 27232 hashclwwlkn0 27233 clwwlknonfin 27269 numclwwlk3lemOLD 27581 |
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