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Mirrors > Home > MPE Home > Th. List > clwwnonrepclwwnon | Structured version Visualization version GIF version |
Description: If the initial vertex of a closed walk occurs another time in the walk, the walk starts with a closed walk on this vertex. See also the remarks in clwwnrepclwwn 27725. (Contributed by AV, 24-Apr-2022.) (Revised by AV, 10-May-2022.) (Revised by AV, 30-Oct-2022.) |
Ref | Expression |
---|---|
clwwnonrepclwwnon | ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋) → (𝑊 prefix (𝑁 − 2)) ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1172 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋) → 𝑁 ∈ (ℤ≥‘3)) | |
2 | isclwwlknon 27462 | . . . . 5 ⊢ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) | |
3 | 2 | simplbi 493 | . . . 4 ⊢ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) → 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) |
4 | 3 | 3ad2ant2 1170 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋) → 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) |
5 | simpr 479 | . . . . . . . 8 ⊢ ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (𝑊‘0) = 𝑋) | |
6 | 5 | eqcomd 2830 | . . . . . . 7 ⊢ ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → 𝑋 = (𝑊‘0)) |
7 | 2, 6 | sylbi 209 | . . . . . 6 ⊢ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) → 𝑋 = (𝑊‘0)) |
8 | 7 | eqeq2d 2834 | . . . . 5 ⊢ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) → ((𝑊‘(𝑁 − 2)) = 𝑋 ↔ (𝑊‘(𝑁 − 2)) = (𝑊‘0))) |
9 | 8 | biimpa 470 | . . . 4 ⊢ ((𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋) → (𝑊‘(𝑁 − 2)) = (𝑊‘0)) |
10 | 9 | 3adant1 1166 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋) → (𝑊‘(𝑁 − 2)) = (𝑊‘0)) |
11 | clwwnrepclwwn 27725 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘(𝑁 − 2)) = (𝑊‘0)) → (𝑊 prefix (𝑁 − 2)) ∈ ((𝑁 − 2) ClWWalksN 𝐺)) | |
12 | 1, 4, 10, 11 | syl3anc 1496 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋) → (𝑊 prefix (𝑁 − 2)) ∈ ((𝑁 − 2) ClWWalksN 𝐺)) |
13 | 2clwwlklem 27723 | . . . . . 6 ⊢ ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑁 ∈ (ℤ≥‘3)) → ((𝑊 prefix (𝑁 − 2))‘0) = (𝑊‘0)) | |
14 | 3, 13 | sylan 577 | . . . . 5 ⊢ ((𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ 𝑁 ∈ (ℤ≥‘3)) → ((𝑊 prefix (𝑁 − 2))‘0) = (𝑊‘0)) |
15 | 14 | ancoms 452 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁)) → ((𝑊 prefix (𝑁 − 2))‘0) = (𝑊‘0)) |
16 | 15 | 3adant3 1168 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋) → ((𝑊 prefix (𝑁 − 2))‘0) = (𝑊‘0)) |
17 | 2 | simprbi 492 | . . . 4 ⊢ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) → (𝑊‘0) = 𝑋) |
18 | 17 | 3ad2ant2 1170 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋) → (𝑊‘0) = 𝑋) |
19 | 16, 18 | eqtrd 2860 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋) → ((𝑊 prefix (𝑁 − 2))‘0) = 𝑋) |
20 | isclwwlknon 27462 | . 2 ⊢ ((𝑊 prefix (𝑁 − 2)) ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ↔ ((𝑊 prefix (𝑁 − 2)) ∈ ((𝑁 − 2) ClWWalksN 𝐺) ∧ ((𝑊 prefix (𝑁 − 2))‘0) = 𝑋)) | |
21 | 12, 19, 20 | sylanbrc 580 | 1 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋) → (𝑊 prefix (𝑁 − 2)) ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ‘cfv 6122 (class class class)co 6904 0cc0 10251 − cmin 10584 2c2 11405 3c3 11406 ℤ≥cuz 11967 prefix cpfx 13748 ClWWalksN cclwwlkn 27361 ClWWalksNOncclwwlknon 27457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-map 8123 df-pm 8124 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-card 9077 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-3 11414 df-n0 11618 df-xnn0 11690 df-z 11704 df-uz 11968 df-fz 12619 df-fzo 12760 df-hash 13410 df-word 13574 df-lsw 13622 df-substr 13700 df-pfx 13749 df-wwlks 27128 df-wwlksn 27129 df-clwwlk 27310 df-clwwlkn 27363 df-clwwlknon 27458 |
This theorem is referenced by: 2clwwlk2clwwlk 27733 |
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