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Mirrors > Home > MPE Home > Th. List > clwwlksnfi | 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.) |
Ref | Expression |
---|---|
clwwlksnfi | ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwwlksn 26881 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁}) | |
2 | 1 | adantr 481 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (Vtx‘𝐺) ∈ Fin) → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁}) |
3 | nnnn0 11299 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
4 | 3 | anim1i 592 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (Vtx‘𝐺) ∈ Fin) → (𝑁 ∈ ℕ0 ∧ (Vtx‘𝐺) ∈ Fin)) |
5 | 4 | ancomd 467 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (Vtx‘𝐺) ∈ Fin) → ((Vtx‘𝐺) ∈ Fin ∧ 𝑁 ∈ ℕ0)) |
6 | wrdnfi 13338 | . . . . . 6 ⊢ (((Vtx‘𝐺) ∈ Fin ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ Word (Vtx‘𝐺) ∣ (#‘𝑤) = 𝑁} ∈ Fin) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (Vtx‘𝐺) ∈ Fin) → {𝑤 ∈ Word (Vtx‘𝐺) ∣ (#‘𝑤) = 𝑁} ∈ Fin) |
8 | clwwlkssswrd 26911 | . . . . . 6 ⊢ (ClWWalks‘𝐺) ⊆ Word (Vtx‘𝐺) | |
9 | rabss2 3685 | . . . . . 6 ⊢ ((ClWWalks‘𝐺) ⊆ Word (Vtx‘𝐺) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁} ⊆ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (#‘𝑤) = 𝑁}) | |
10 | 8, 9 | mp1i 13 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (Vtx‘𝐺) ∈ Fin) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁} ⊆ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (#‘𝑤) = 𝑁}) |
11 | 7, 10 | ssfid 8183 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (Vtx‘𝐺) ∈ Fin) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁} ∈ Fin) |
12 | 2, 11 | eqeltrd 2701 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ (Vtx‘𝐺) ∈ Fin) → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
13 | 12 | ex 450 | . 2 ⊢ (𝑁 ∈ ℕ → ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)) |
14 | df-nel 2898 | . . . . . . 7 ⊢ (𝑁 ∉ ℕ ↔ ¬ 𝑁 ∈ ℕ) | |
15 | 14 | biimpri 218 | . . . . . 6 ⊢ (¬ 𝑁 ∈ ℕ → 𝑁 ∉ ℕ) |
16 | 15 | olcd 408 | . . . . 5 ⊢ (¬ 𝑁 ∈ ℕ → (𝐺 ∉ V ∨ 𝑁 ∉ ℕ)) |
17 | clwwlksnndef 26890 | . . . . 5 ⊢ ((𝐺 ∉ V ∨ 𝑁 ∉ ℕ) → (𝑁 ClWWalksN 𝐺) = ∅) | |
18 | 16, 17 | syl 17 | . . . 4 ⊢ (¬ 𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) = ∅) |
19 | 0fin 8188 | . . . 4 ⊢ ∅ ∈ Fin | |
20 | 18, 19 | syl6eqel 2709 | . . 3 ⊢ (¬ 𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
21 | 20 | a1d 25 | . 2 ⊢ (¬ 𝑁 ∈ ℕ → ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)) |
22 | 13, 21 | pm2.61i 176 | 1 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∉ wnel 2897 {crab 2916 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℕcn 11020 ℕ0cn0 11292 #chash 13117 Word cword 13291 Vtxcvtx 25874 ClWWalkscclwwlks 26875 ClWWalksN cclwwlksn 26876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-clwwlks 26877 df-clwwlksn 26878 |
This theorem is referenced by: qerclwwlksnfi 26950 hashclwwlksn0 26951 numclwwlkffin 27214 numclwwlk3lem 27241 numclwwlk4 27244 |
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