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Mirrors > Home > MPE Home > Th. List > clwwlk1loop | Structured version Visualization version GIF version |
Description: A closed walk of length 1 is a loop. See also clwlkl1loop 27558. (Contributed by AV, 24-Apr-2021.) |
Ref | Expression |
---|---|
clwwlk1loop | ⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 1) → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2821 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | isclwwlk 27756 | . . 3 ⊢ (𝑊 ∈ (ClWWalks‘𝐺) ↔ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑊 ≠ ∅) ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺))) |
4 | lsw1 13913 | . . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → (lastS‘𝑊) = (𝑊‘0)) | |
5 | 4 | preq1d 4668 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → {(lastS‘𝑊), (𝑊‘0)} = {(𝑊‘0), (𝑊‘0)}) |
6 | 5 | eleq1d 2897 | . . . . . . . . 9 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → ({(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺) ↔ {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺))) |
7 | 6 | biimpd 231 | . . . . . . . 8 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → ({(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺) → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺))) |
8 | 7 | ex 415 | . . . . . . 7 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((♯‘𝑊) = 1 → ({(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺) → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺)))) |
9 | 8 | com23 86 | . . . . . 6 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ({(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺) → ((♯‘𝑊) = 1 → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺)))) |
10 | 9 | adantr 483 | . . . . 5 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑊 ≠ ∅) → ({(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺) → ((♯‘𝑊) = 1 → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺)))) |
11 | 10 | imp 409 | . . . 4 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑊 ≠ ∅) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) → ((♯‘𝑊) = 1 → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺))) |
12 | 11 | 3adant2 1127 | . . 3 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑊 ≠ ∅) ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) → ((♯‘𝑊) = 1 → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺))) |
13 | 3, 12 | sylbi 219 | . 2 ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → ((♯‘𝑊) = 1 → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺))) |
14 | 13 | imp 409 | 1 ⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 1) → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∅c0 4290 {cpr 4562 ‘cfv 6349 (class class class)co 7150 0cc0 10531 1c1 10532 + caddc 10534 − cmin 10864 ..^cfzo 13027 ♯chash 13684 Word cword 13855 lastSclsw 13908 Vtxcvtx 26775 Edgcedg 26826 ClWWalkscclwwlk 27753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-lsw 13909 df-clwwlk 27754 |
This theorem is referenced by: umgrclwwlkge2 27763 |
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