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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 28930 | . 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 28928 | . 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 28931 | . 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 28856 | . . 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 1371 | . 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 1342 | 1 β’ (π β πΉ(WalksβπΊ)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 if-wif 1061 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 βwral 3061 Vcvv 3474 β wss 3947 {csn 4627 {cpr 4629 class class class wbr 5147 dom cdm 5675 βΆwf 6536 βcfv 6540 (class class class)co 7405 0cc0 11106 1c1 11107 + caddc 11109 ...cfz 13480 ..^cfzo 13623 β―chash 14286 Word cword 14460 Vtxcvtx 28245 iEdgciedg 28246 Walkscwlks 28842 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-wlks 28845 |
This theorem is referenced by: 2wlkd 29179 3wlkd 29412 |
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