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Mirrors > Home > MPE Home > Th. List > crctcshtrl | Structured version Visualization version GIF version |
Description: Cyclically shifting the indices of a circuit β¨πΉ, πβ© results in a trail β¨π», πβ©. (Contributed by AV, 14-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
crctcsh.v | β’ π = (VtxβπΊ) |
crctcsh.i | β’ πΌ = (iEdgβπΊ) |
crctcsh.d | β’ (π β πΉ(CircuitsβπΊ)π) |
crctcsh.n | β’ π = (β―βπΉ) |
crctcsh.s | β’ (π β π β (0..^π)) |
crctcsh.h | β’ π» = (πΉ cyclShift π) |
crctcsh.q | β’ π = (π₯ β (0...π) β¦ if(π₯ β€ (π β π), (πβ(π₯ + π)), (πβ((π₯ + π) β π)))) |
Ref | Expression |
---|---|
crctcshtrl | β’ (π β π»(TrailsβπΊ)π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crctcsh.v | . . 3 β’ π = (VtxβπΊ) | |
2 | crctcsh.i | . . 3 β’ πΌ = (iEdgβπΊ) | |
3 | crctcsh.d | . . 3 β’ (π β πΉ(CircuitsβπΊ)π) | |
4 | crctcsh.n | . . 3 β’ π = (β―βπΉ) | |
5 | crctcsh.s | . . 3 β’ (π β π β (0..^π)) | |
6 | crctcsh.h | . . 3 β’ π» = (πΉ cyclShift π) | |
7 | crctcsh.q | . . 3 β’ π = (π₯ β (0...π) β¦ if(π₯ β€ (π β π), (πβ(π₯ + π)), (πβ((π₯ + π) β π)))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | crctcshwlk 29585 | . 2 β’ (π β π»(WalksβπΊ)π) |
9 | crctistrl 29561 | . . . . 5 β’ (πΉ(CircuitsβπΊ)π β πΉ(TrailsβπΊ)π) | |
10 | 2 | trlf1 29464 | . . . . . 6 β’ (πΉ(TrailsβπΊ)π β πΉ:(0..^(β―βπΉ))β1-1βdom πΌ) |
11 | df-f1 6542 | . . . . . . 7 β’ (πΉ:(0..^(β―βπΉ))β1-1βdom πΌ β (πΉ:(0..^(β―βπΉ))βΆdom πΌ β§ Fun β‘πΉ)) | |
12 | iswrdi 14474 | . . . . . . . 8 β’ (πΉ:(0..^(β―βπΉ))βΆdom πΌ β πΉ β Word dom πΌ) | |
13 | 12 | anim1i 614 | . . . . . . 7 β’ ((πΉ:(0..^(β―βπΉ))βΆdom πΌ β§ Fun β‘πΉ) β (πΉ β Word dom πΌ β§ Fun β‘πΉ)) |
14 | 11, 13 | sylbi 216 | . . . . . 6 β’ (πΉ:(0..^(β―βπΉ))β1-1βdom πΌ β (πΉ β Word dom πΌ β§ Fun β‘πΉ)) |
15 | 10, 14 | syl 17 | . . . . 5 β’ (πΉ(TrailsβπΊ)π β (πΉ β Word dom πΌ β§ Fun β‘πΉ)) |
16 | 3, 9, 15 | 3syl 18 | . . . 4 β’ (π β (πΉ β Word dom πΌ β§ Fun β‘πΉ)) |
17 | elfzoelz 13638 | . . . . 5 β’ (π β (0..^π) β π β β€) | |
18 | 5, 17 | syl 17 | . . . 4 β’ (π β π β β€) |
19 | df-3an 1086 | . . . 4 β’ ((πΉ β Word dom πΌ β§ Fun β‘πΉ β§ π β β€) β ((πΉ β Word dom πΌ β§ Fun β‘πΉ) β§ π β β€)) | |
20 | 16, 18, 19 | sylanbrc 582 | . . 3 β’ (π β (πΉ β Word dom πΌ β§ Fun β‘πΉ β§ π β β€)) |
21 | cshinj 14767 | . . 3 β’ ((πΉ β Word dom πΌ β§ Fun β‘πΉ β§ π β β€) β (π» = (πΉ cyclShift π) β Fun β‘π»)) | |
22 | 20, 6, 21 | mpisyl 21 | . 2 β’ (π β Fun β‘π») |
23 | istrl 29462 | . 2 β’ (π»(TrailsβπΊ)π β (π»(WalksβπΊ)π β§ Fun β‘π»)) | |
24 | 8, 22, 23 | sylanbrc 582 | 1 β’ (π β π»(TrailsβπΊ)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 ifcif 4523 class class class wbr 5141 β¦ cmpt 5224 β‘ccnv 5668 dom cdm 5669 Fun wfun 6531 βΆwf 6533 β1-1βwf1 6534 βcfv 6537 (class class class)co 7405 0cc0 11112 + caddc 11115 β€ cle 11253 β cmin 11448 β€cz 12562 ...cfz 13490 ..^cfzo 13633 β―chash 14295 Word cword 14470 cyclShift ccsh 14744 Vtxcvtx 28764 iEdgciedg 28765 Walkscwlks 29362 Trailsctrls 29456 Circuitsccrcts 29550 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-hash 14296 df-word 14471 df-concat 14527 df-substr 14597 df-pfx 14627 df-csh 14745 df-wlks 29365 df-trls 29458 df-crcts 29552 |
This theorem is referenced by: crctcsh 29587 eucrctshift 30005 |
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