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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 29492 | . 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 29490 | . 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 29493 | . 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 29418 | . . 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 1369 | . 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 1340 | 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 1085 = wceq 1534 β wcel 2099 β wne 2936 βwral 3057 Vcvv 3470 β wss 3945 {csn 4625 {cpr 4627 class class class wbr 5143 dom cdm 5673 βΆwf 6539 βcfv 6543 (class class class)co 7415 0cc0 11133 1c1 11134 + caddc 11136 ...cfz 13511 ..^cfzo 13654 β―chash 14316 Word cword 14491 Vtxcvtx 28803 iEdgciedg 28804 Walkscwlks 29404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-fzo 13655 df-hash 14317 df-word 14492 df-wlks 29407 |
This theorem is referenced by: 2wlkd 29741 3wlkd 29974 |
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