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Mirrors > Home > MPE Home > Th. List > clwwisshclwwsn | Structured version Visualization version GIF version |
Description: Cyclically shifting a closed walk as word results in a closed walk as word (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jun-2018.) (Revised by AV, 29-Apr-2021.) |
Ref | Expression |
---|---|
clwwisshclwwsn | ⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6809 | . . . 4 ⊢ (𝑁 = (♯‘𝑊) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (♯‘𝑊))) | |
2 | eqid 2748 | . . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
3 | 2 | clwwlkbp 27079 | . . . . . . 7 ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑊 ≠ ∅)) |
4 | 3 | simp2d 1135 | . . . . . 6 ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → 𝑊 ∈ Word (Vtx‘𝐺)) |
5 | cshwn 13714 | . . . . . 6 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (𝑊 cyclShift (♯‘𝑊)) = 𝑊) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → (𝑊 cyclShift (♯‘𝑊)) = 𝑊) |
7 | 6 | adantr 472 | . . . 4 ⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 cyclShift (♯‘𝑊)) = 𝑊) |
8 | 1, 7 | sylan9eq 2802 | . . 3 ⊢ ((𝑁 = (♯‘𝑊) ∧ (𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))) → (𝑊 cyclShift 𝑁) = 𝑊) |
9 | simprl 811 | . . 3 ⊢ ((𝑁 = (♯‘𝑊) ∧ (𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))) → 𝑊 ∈ (ClWWalks‘𝐺)) | |
10 | 8, 9 | eqeltrd 2827 | . 2 ⊢ ((𝑁 = (♯‘𝑊) ∧ (𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺)) |
11 | simprl 811 | . . 3 ⊢ ((¬ 𝑁 = (♯‘𝑊) ∧ (𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))) → 𝑊 ∈ (ClWWalks‘𝐺)) | |
12 | df-ne 2921 | . . . . . 6 ⊢ (𝑁 ≠ (♯‘𝑊) ↔ ¬ 𝑁 = (♯‘𝑊)) | |
13 | fzofzim 12680 | . . . . . . 7 ⊢ ((𝑁 ≠ (♯‘𝑊) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → 𝑁 ∈ (0..^(♯‘𝑊))) | |
14 | 13 | expcom 450 | . . . . . 6 ⊢ (𝑁 ∈ (0...(♯‘𝑊)) → (𝑁 ≠ (♯‘𝑊) → 𝑁 ∈ (0..^(♯‘𝑊)))) |
15 | 12, 14 | syl5bir 233 | . . . . 5 ⊢ (𝑁 ∈ (0...(♯‘𝑊)) → (¬ 𝑁 = (♯‘𝑊) → 𝑁 ∈ (0..^(♯‘𝑊)))) |
16 | 15 | adantl 473 | . . . 4 ⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (¬ 𝑁 = (♯‘𝑊) → 𝑁 ∈ (0..^(♯‘𝑊)))) |
17 | 16 | impcom 445 | . . 3 ⊢ ((¬ 𝑁 = (♯‘𝑊) ∧ (𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))) → 𝑁 ∈ (0..^(♯‘𝑊))) |
18 | clwwisshclwws 27109 | . . 3 ⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺)) | |
19 | 11, 17, 18 | syl2anc 696 | . 2 ⊢ ((¬ 𝑁 = (♯‘𝑊) ∧ (𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺)) |
20 | 10, 19 | pm2.61ian 866 | 1 ⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1620 ∈ wcel 2127 ≠ wne 2920 Vcvv 3328 ∅c0 4046 ‘cfv 6037 (class class class)co 6801 0cc0 10099 ...cfz 12490 ..^cfzo 12630 ♯chash 13282 Word cword 13448 cyclShift ccsh 13705 Vtxcvtx 26044 ClWWalkscclwwlk 27075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 ax-pre-sup 10177 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rmo 3046 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-int 4616 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-om 7219 df-1st 7321 df-2nd 7322 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7899 df-map 8013 df-pm 8014 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8501 df-inf 8502 df-card 8926 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-div 10848 df-nn 11184 df-2 11242 df-n0 11456 df-z 11541 df-uz 11851 df-rp 11997 df-ico 12345 df-fz 12491 df-fzo 12631 df-fl 12758 df-mod 12834 df-hash 13283 df-word 13456 df-lsw 13457 df-concat 13458 df-substr 13460 df-csh 13706 df-clwwlk 27076 |
This theorem is referenced by: clwwnisshclwwsn 27161 |
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