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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 7276 | . . . 4 ⊢ (𝑁 = (♯‘𝑊) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (♯‘𝑊))) | |
2 | eqid 2738 | . . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
3 | 2 | clwwlkbp 28335 | . . . . . . 7 ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑊 ≠ ∅)) |
4 | 3 | simp2d 1142 | . . . . . 6 ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → 𝑊 ∈ Word (Vtx‘𝐺)) |
5 | cshwn 14498 | . . . . . 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 481 | . . . 4 ⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 cyclShift (♯‘𝑊)) = 𝑊) |
8 | 1, 7 | sylan9eq 2798 | . . 3 ⊢ ((𝑁 = (♯‘𝑊) ∧ (𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))) → (𝑊 cyclShift 𝑁) = 𝑊) |
9 | simprl 768 | . . 3 ⊢ ((𝑁 = (♯‘𝑊) ∧ (𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))) → 𝑊 ∈ (ClWWalks‘𝐺)) | |
10 | 8, 9 | eqeltrd 2839 | . 2 ⊢ ((𝑁 = (♯‘𝑊) ∧ (𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺)) |
11 | simprl 768 | . . 3 ⊢ ((¬ 𝑁 = (♯‘𝑊) ∧ (𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))) → 𝑊 ∈ (ClWWalks‘𝐺)) | |
12 | df-ne 2944 | . . . . . 6 ⊢ (𝑁 ≠ (♯‘𝑊) ↔ ¬ 𝑁 = (♯‘𝑊)) | |
13 | fzofzim 13422 | . . . . . . 7 ⊢ ((𝑁 ≠ (♯‘𝑊) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → 𝑁 ∈ (0..^(♯‘𝑊))) | |
14 | 13 | expcom 414 | . . . . . 6 ⊢ (𝑁 ∈ (0...(♯‘𝑊)) → (𝑁 ≠ (♯‘𝑊) → 𝑁 ∈ (0..^(♯‘𝑊)))) |
15 | 12, 14 | syl5bir 242 | . . . . 5 ⊢ (𝑁 ∈ (0...(♯‘𝑊)) → (¬ 𝑁 = (♯‘𝑊) → 𝑁 ∈ (0..^(♯‘𝑊)))) |
16 | 15 | adantl 482 | . . . 4 ⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (¬ 𝑁 = (♯‘𝑊) → 𝑁 ∈ (0..^(♯‘𝑊)))) |
17 | 16 | impcom 408 | . . 3 ⊢ ((¬ 𝑁 = (♯‘𝑊) ∧ (𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))) → 𝑁 ∈ (0..^(♯‘𝑊))) |
18 | clwwisshclwws 28365 | . . 3 ⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺)) | |
19 | 11, 17, 18 | syl2anc 584 | . 2 ⊢ ((¬ 𝑁 = (♯‘𝑊) ∧ (𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺)) |
20 | 10, 19 | pm2.61ian 809 | 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 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 Vcvv 3430 ∅c0 4257 ‘cfv 6427 (class class class)co 7268 0cc0 10859 ...cfz 13227 ..^cfzo 13370 ♯chash 14032 Word cword 14205 cyclShift ccsh 14489 Vtxcvtx 27354 ClWWalkscclwwlk 28331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-pre-sup 10937 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-map 8605 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-sup 9189 df-inf 9190 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-n0 12222 df-z 12308 df-uz 12571 df-rp 12719 df-ico 13073 df-fz 13228 df-fzo 13371 df-fl 13500 df-mod 13578 df-hash 14033 df-word 14206 df-lsw 14254 df-concat 14262 df-substr 14342 df-pfx 14372 df-csh 14490 df-clwwlk 28332 |
This theorem is referenced by: clwwnisshclwwsn 28409 |
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