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Mirrors > Home > MPE Home > Th. List > clwwlknwrd | Structured version Visualization version GIF version |
Description: A closed walk of a fixed length as word is a word over the vertices. (Contributed by AV, 30-Apr-2021.) |
Ref | Expression |
---|---|
clwwlknwrd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
clwwlknwrd | ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ Word 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isclwwlkn 27173 | . 2 ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁)) | |
2 | clwwlknwrd.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | clwwlkbp 27128 | . . . 4 ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅)) |
4 | 3 | simp2d 1137 | . . 3 ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → 𝑊 ∈ Word 𝑉) |
5 | 4 | adantr 466 | . 2 ⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁) → 𝑊 ∈ Word 𝑉) |
6 | 1, 5 | sylbi 207 | 1 ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ Word 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 Vcvv 3351 ∅c0 4063 ‘cfv 6029 (class class class)co 6791 ♯chash 13314 Word cword 13480 Vtxcvtx 26088 ClWWalkscclwwlk 27124 ClWWalksN cclwwlkn 27167 |
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 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 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-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-oadd 7715 df-er 7894 df-map 8009 df-pm 8010 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-card 8963 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-nn 11221 df-n0 11493 df-z 11578 df-uz 11887 df-fz 12527 df-fzo 12667 df-hash 13315 df-word 13488 df-clwwlk 27125 df-clwwlkn 27169 |
This theorem is referenced by: clwwlknbp 27183 clwwnisshclwwsn 27210 clwwlknscsh 27213 clwwlknccat 27214 erclwwlkneqlen 27219 erclwwlknref 27220 erclwwlknsym 27221 clwwlknonccat 27264 2clwwlklem 27520 numclwwlk1lem2f1 27536 |
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