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Mirrors > Home > MPE Home > Th. List > wlkd | Structured version Visualization version GIF version |
Description: Two words representing a walk in a graph. (Contributed by AV, 7-Feb-2021.) |
Ref | Expression |
---|---|
wlkd.p | β’ (π β π β Word V) |
wlkd.f | β’ (π β πΉ β Word V) |
wlkd.l | β’ (π β (β―βπ) = ((β―βπΉ) + 1)) |
wlkd.e | β’ (π β βπ β (0..^(β―βπΉ)){(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ))) |
wlkd.n | β’ (π β βπ β (0..^(β―βπΉ))(πβπ) β (πβ(π + 1))) |
wlkd.g | β’ (π β πΊ β π) |
wlkd.v | β’ π = (VtxβπΊ) |
wlkd.i | β’ πΌ = (iEdgβπΊ) |
wlkd.a | β’ (π β βπ β (0...(β―βπΉ))(πβπ) β π) |
Ref | Expression |
---|---|
wlkd | β’ (π β πΉ(WalksβπΊ)π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlkd.p | . . 3 β’ (π β π β Word V) | |
2 | wlkd.f | . . 3 β’ (π β πΉ β Word V) | |
3 | wlkd.l | . . 3 β’ (π β (β―βπ) = ((β―βπΉ) + 1)) | |
4 | wlkd.e | . . 3 β’ (π β βπ β (0..^(β―βπΉ)){(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ))) | |
5 | 1, 2, 3, 4 | wlkdlem3 29413 | . 2 β’ (π β πΉ β Word dom πΌ) |
6 | wlkd.a | . . 3 β’ (π β βπ β (0...(β―βπΉ))(πβπ) β π) | |
7 | 1, 2, 3, 6 | wlkdlem1 29411 | . 2 β’ (π β π:(0...(β―βπΉ))βΆπ) |
8 | wlkd.n | . . 3 β’ (π β βπ β (0..^(β―βπΉ))(πβπ) β (πβ(π + 1))) | |
9 | 1, 2, 3, 4, 8 | wlkdlem4 29414 | . 2 β’ (π β βπ β (0..^(β―βπΉ))if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ)))) |
10 | wlkd.g | . . 3 β’ (π β πΊ β π) | |
11 | wlkd.v | . . . 4 β’ π = (VtxβπΊ) | |
12 | wlkd.i | . . . 4 β’ πΌ = (iEdgβπΊ) | |
13 | 11, 12 | iswlk 29339 | . . 3 β’ ((πΊ β π β§ πΉ β Word V β§ π β Word V) β (πΉ(WalksβπΊ)π β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ)))))) |
14 | 10, 2, 1, 13 | syl3anc 1368 | . 2 β’ (π β (πΉ(WalksβπΊ)π β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ)))))) |
15 | 5, 7, 9, 14 | mpbir3and 1339 | 1 β’ (π β πΉ(WalksβπΊ)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 if-wif 1059 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2932 βwral 3053 Vcvv 3466 β wss 3941 {csn 4621 {cpr 4623 class class class wbr 5139 dom cdm 5667 βΆwf 6530 βcfv 6534 (class class class)co 7402 0cc0 11107 1c1 11108 + caddc 11110 ...cfz 13482 ..^cfzo 13625 β―chash 14288 Word cword 14462 Vtxcvtx 28728 iEdgciedg 28729 Walkscwlks 29325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1060 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-fzo 13626 df-hash 14289 df-word 14463 df-wlks 29328 |
This theorem is referenced by: 2wlkd 29662 3wlkd 29895 |
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