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Mirrors > Home > MPE Home > Th. List > clwwlknlbonbgr1 | Structured version Visualization version GIF version |
Description: The last but one vertex in a closed walk is a neighbor of the first vertex of the closed walk. (Contributed by AV, 17-Feb-2022.) |
Ref | Expression |
---|---|
clwwlknlbonbgr1 | ⊢ ((𝐺 ∈ USGraph ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑊‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx (𝑊‘0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2798 | . . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | clwwlknp 27822 | . . . 4 ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺))) |
4 | lsw 13907 | . . . . . . . . . 10 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) | |
5 | fvoveq1 7158 | . . . . . . . . . 10 ⊢ ((♯‘𝑊) = 𝑁 → (𝑊‘((♯‘𝑊) − 1)) = (𝑊‘(𝑁 − 1))) | |
6 | 4, 5 | sylan9eq 2853 | . . . . . . . . 9 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 𝑁) → (lastS‘𝑊) = (𝑊‘(𝑁 − 1))) |
7 | 6 | preq1d 4635 | . . . . . . . 8 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 𝑁) → {(lastS‘𝑊), (𝑊‘0)} = {(𝑊‘(𝑁 − 1)), (𝑊‘0)}) |
8 | 7 | eleq1d 2874 | . . . . . . 7 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 𝑁) → ({(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘0)} ∈ (Edg‘𝐺))) |
9 | 8 | biimpd 232 | . . . . . 6 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 𝑁) → ({(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺) → {(𝑊‘(𝑁 − 1)), (𝑊‘0)} ∈ (Edg‘𝐺))) |
10 | 9 | a1d 25 | . . . . 5 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 𝑁) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ({(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺) → {(𝑊‘(𝑁 − 1)), (𝑊‘0)} ∈ (Edg‘𝐺)))) |
11 | 10 | 3imp 1108 | . . . 4 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) → {(𝑊‘(𝑁 − 1)), (𝑊‘0)} ∈ (Edg‘𝐺)) |
12 | 3, 11 | syl 17 | . . 3 ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → {(𝑊‘(𝑁 − 1)), (𝑊‘0)} ∈ (Edg‘𝐺)) |
13 | 12 | adantl 485 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → {(𝑊‘(𝑁 − 1)), (𝑊‘0)} ∈ (Edg‘𝐺)) |
14 | 2 | nbusgreledg 27143 | . . 3 ⊢ (𝐺 ∈ USGraph → ((𝑊‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx (𝑊‘0)) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘0)} ∈ (Edg‘𝐺))) |
15 | 14 | adantr 484 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → ((𝑊‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx (𝑊‘0)) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘0)} ∈ (Edg‘𝐺))) |
16 | 13, 15 | mpbird 260 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑊‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx (𝑊‘0))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {cpr 4527 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 + caddc 10529 − cmin 10859 ..^cfzo 13028 ♯chash 13686 Word cword 13857 lastSclsw 13905 Vtxcvtx 26789 Edgcedg 26840 USGraphcusgr 26942 NeighbVtx cnbgr 27122 ClWWalksN cclwwlkn 27809 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-lsw 13906 df-edg 26841 df-upgr 26875 df-umgr 26876 df-usgr 26944 df-nbgr 27123 df-clwwlk 27767 df-clwwlkn 27810 |
This theorem is referenced by: extwwlkfab 28137 |
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