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Mirrors > Home > MPE Home > Th. List > erclwwlknref | Structured version Visualization version GIF version |
Description: ⌠is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 26-Mar-2018.) (Revised by AV, 30-Apr-2021.) (Proof shortened by AV, 23-Mar-2022.) |
Ref | Expression |
---|---|
erclwwlkn.w | ⢠ð = (ð ClWWalksN ðº) |
erclwwlkn.r | ⢠⌠= {âšð¡, ð¢â© ⣠(ð¡ â ð ⧠ð¢ â ð ⧠âð â (0...ð)ð¡ = (ð¢ cyclShift ð))} |
Ref | Expression |
---|---|
erclwwlknref | ⢠(ð¥ â ð â ð¥ ⌠ð¥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 1090 | . . 3 ⢠((ð¥ â ð ⧠ð¥ â ð ⧠âð â (0...ð)ð¥ = (ð¥ cyclShift ð)) â ((ð¥ â ð ⧠ð¥ â ð) ⧠âð â (0...ð)ð¥ = (ð¥ cyclShift ð))) | |
2 | anidm 566 | . . . 4 ⢠((ð¥ â ð ⧠ð¥ â ð) â ð¥ â ð) | |
3 | 2 | anbi1i 625 | . . 3 ⢠(((ð¥ â ð ⧠ð¥ â ð) ⧠âð â (0...ð)ð¥ = (ð¥ cyclShift ð)) â (ð¥ â ð ⧠âð â (0...ð)ð¥ = (ð¥ cyclShift ð))) |
4 | 1, 3 | bitri 275 | . 2 ⢠((ð¥ â ð ⧠ð¥ â ð ⧠âð â (0...ð)ð¥ = (ð¥ cyclShift ð)) â (ð¥ â ð ⧠âð â (0...ð)ð¥ = (ð¥ cyclShift ð))) |
5 | erclwwlkn.w | . . . 4 ⢠ð = (ð ClWWalksN ðº) | |
6 | erclwwlkn.r | . . . 4 ⢠⌠= {âšð¡, ð¢â© ⣠(ð¡ â ð ⧠ð¢ â ð ⧠âð â (0...ð)ð¡ = (ð¢ cyclShift ð))} | |
7 | 5, 6 | erclwwlkneq 29060 | . . 3 ⢠((ð¥ â V ⧠ð¥ â V) â (ð¥ ⌠ð¥ â (ð¥ â ð ⧠ð¥ â ð ⧠âð â (0...ð)ð¥ = (ð¥ cyclShift ð)))) |
8 | 7 | el2v 3455 | . 2 ⢠(ð¥ ⌠ð¥ â (ð¥ â ð ⧠ð¥ â ð ⧠âð â (0...ð)ð¥ = (ð¥ cyclShift ð))) |
9 | eqid 2733 | . . . . . 6 ⢠(Vtxâðº) = (Vtxâðº) | |
10 | 9 | clwwlknwrd 29027 | . . . . 5 ⢠(ð¥ â (ð ClWWalksN ðº) â ð¥ â Word (Vtxâðº)) |
11 | clwwlknnn 29026 | . . . . 5 ⢠(ð¥ â (ð ClWWalksN ðº) â ð â â) | |
12 | cshw0 14691 | . . . . . 6 ⢠(ð¥ â Word (Vtxâðº) â (ð¥ cyclShift 0) = ð¥) | |
13 | nnnn0 12428 | . . . . . . . . 9 ⢠(ð â â â ð â â0) | |
14 | 0elfz 13547 | . . . . . . . . 9 ⢠(ð â â0 â 0 â (0...ð)) | |
15 | 13, 14 | syl 17 | . . . . . . . 8 ⢠(ð â â â 0 â (0...ð)) |
16 | eqcom 2740 | . . . . . . . . 9 ⢠((ð¥ cyclShift 0) = ð¥ â ð¥ = (ð¥ cyclShift 0)) | |
17 | 16 | biimpi 215 | . . . . . . . 8 ⢠((ð¥ cyclShift 0) = ð¥ â ð¥ = (ð¥ cyclShift 0)) |
18 | oveq2 7369 | . . . . . . . . 9 ⢠(ð = 0 â (ð¥ cyclShift ð) = (ð¥ cyclShift 0)) | |
19 | 18 | rspceeqv 3599 | . . . . . . . 8 ⢠((0 â (0...ð) ⧠ð¥ = (ð¥ cyclShift 0)) â âð â (0...ð)ð¥ = (ð¥ cyclShift ð)) |
20 | 15, 17, 19 | syl2anr 598 | . . . . . . 7 ⢠(((ð¥ cyclShift 0) = ð¥ ⧠ð â â) â âð â (0...ð)ð¥ = (ð¥ cyclShift ð)) |
21 | 20 | ex 414 | . . . . . 6 ⢠((ð¥ cyclShift 0) = ð¥ â (ð â â â âð â (0...ð)ð¥ = (ð¥ cyclShift ð))) |
22 | 12, 21 | syl 17 | . . . . 5 ⢠(ð¥ â Word (Vtxâðº) â (ð â â â âð â (0...ð)ð¥ = (ð¥ cyclShift ð))) |
23 | 10, 11, 22 | sylc 65 | . . . 4 ⢠(ð¥ â (ð ClWWalksN ðº) â âð â (0...ð)ð¥ = (ð¥ cyclShift ð)) |
24 | 23, 5 | eleq2s 2852 | . . 3 ⢠(ð¥ â ð â âð â (0...ð)ð¥ = (ð¥ cyclShift ð)) |
25 | 24 | pm4.71i 561 | . 2 ⢠(ð¥ â ð â (ð¥ â ð ⧠âð â (0...ð)ð¥ = (ð¥ cyclShift ð))) |
26 | 4, 8, 25 | 3bitr4ri 304 | 1 ⢠(ð¥ â ð â ð¥ ⌠ð¥) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â wb 205 ⧠wa 397 ⧠w3a 1088 = wceq 1542 â wcel 2107 âwrex 3070 Vcvv 3447 class class class wbr 5109 {copab 5171 âcfv 6500 (class class class)co 7361 0cc0 11059 âcn 12161 â0cn0 12421 ...cfz 13433 Word cword 14411 cyclShift ccsh 14685 Vtxcvtx 27996 ClWWalksN cclwwlkn 29017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-oadd 8420 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-inf 9387 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-n0 12422 df-xnn0 12494 df-z 12508 df-uz 12772 df-rp 12924 df-fz 13434 df-fzo 13577 df-fl 13706 df-mod 13784 df-hash 14240 df-word 14412 df-concat 14468 df-substr 14538 df-pfx 14568 df-csh 14686 df-clwwlk 28975 df-clwwlkn 29018 |
This theorem is referenced by: erclwwlkn 29065 |
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