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Mirrors > Home > MPE Home > Th. List > clwwlkbp | Structured version Visualization version GIF version |
Description: Basic properties of a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.) |
Ref | Expression |
---|---|
clwwlkbp.v | β’ π = (VtxβπΊ) |
Ref | Expression |
---|---|
clwwlkbp | β’ (π β (ClWWalksβπΊ) β (πΊ β V β§ π β Word π β§ π β β )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6864 | . 2 β’ (π β (ClWWalksβπΊ) β πΊ β V) | |
2 | clwwlkbp.v | . . . 4 β’ π = (VtxβπΊ) | |
3 | eqid 2736 | . . . 4 β’ (EdgβπΊ) = (EdgβπΊ) | |
4 | 2, 3 | isclwwlk 28637 | . . 3 β’ (π β (ClWWalksβπΊ) β ((π β Word π β§ π β β ) β§ βπ β (0..^((β―βπ) β 1)){(πβπ), (πβ(π + 1))} β (EdgβπΊ) β§ {(lastSβπ), (πβ0)} β (EdgβπΊ))) |
5 | 4 | simp1bi 1144 | . 2 β’ (π β (ClWWalksβπΊ) β (π β Word π β§ π β β )) |
6 | 3anass 1094 | . 2 β’ ((πΊ β V β§ π β Word π β§ π β β ) β (πΊ β V β§ (π β Word π β§ π β β ))) | |
7 | 1, 5, 6 | sylanbrc 583 | 1 β’ (π β (ClWWalksβπΊ) β (πΊ β V β§ π β Word π β§ π β β )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2940 βwral 3061 Vcvv 3441 β c0 4270 {cpr 4576 βcfv 6480 (class class class)co 7338 0cc0 10973 1c1 10974 + caddc 10976 β cmin 11307 ..^cfzo 13484 β―chash 14146 Word cword 14318 lastSclsw 14366 Vtxcvtx 27656 Edgcedg 27707 ClWWalkscclwwlk 28634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5230 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 ax-cnex 11029 ax-resscn 11030 ax-1cn 11031 ax-icn 11032 ax-addcl 11033 ax-addrcl 11034 ax-mulcl 11035 ax-mulrcl 11036 ax-mulcom 11037 ax-addass 11038 ax-mulass 11039 ax-distr 11040 ax-i2m1 11041 ax-1ne0 11042 ax-1rid 11043 ax-rnegex 11044 ax-rrecex 11045 ax-cnre 11046 ax-pre-lttri 11047 ax-pre-lttrn 11048 ax-pre-ltadd 11049 ax-pre-mulgt0 11050 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-int 4896 df-iun 4944 df-br 5094 df-opab 5156 df-mpt 5177 df-tr 5211 df-id 5519 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5576 df-we 5578 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6239 df-ord 6306 df-on 6307 df-lim 6308 df-suc 6309 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-riota 7294 df-ov 7341 df-oprab 7342 df-mpo 7343 df-om 7782 df-1st 7900 df-2nd 7901 df-frecs 8168 df-wrecs 8199 df-recs 8273 df-rdg 8312 df-1o 8368 df-er 8570 df-map 8689 df-en 8806 df-dom 8807 df-sdom 8808 df-fin 8809 df-card 9797 df-pnf 11113 df-mnf 11114 df-xr 11115 df-ltxr 11116 df-le 11117 df-sub 11309 df-neg 11310 df-nn 12076 df-n0 12336 df-z 12422 df-uz 12685 df-fz 13342 df-fzo 13485 df-hash 14147 df-word 14319 df-clwwlk 28635 |
This theorem is referenced by: clwwlkgt0 28639 umgrclwwlkge2 28644 clwlkclwwlkfo 28662 clwwisshclwws 28668 clwwisshclwwsn 28669 erclwwlkeqlen 28672 erclwwlkref 28673 erclwwlksym 28674 erclwwlktr 28675 clwwlkn 28679 clwwlknwrd 28687 clwwlknon 28743 clwwlknonex2e 28763 |
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