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Mirrors > Home > MPE Home > Th. List > erclwwlkeqlen | Structured version Visualization version GIF version |
Description: If two classes are equivalent regarding βΌ, then they are words of the same length. (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by AV, 29-Apr-2021.) |
Ref | Expression |
---|---|
erclwwlk.r | β’ βΌ = {β¨π’, π€β© β£ (π’ β (ClWWalksβπΊ) β§ π€ β (ClWWalksβπΊ) β§ βπ β (0...(β―βπ€))π’ = (π€ cyclShift π))} |
Ref | Expression |
---|---|
erclwwlkeqlen | β’ ((π β π β§ π β π) β (π βΌ π β (β―βπ) = (β―βπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erclwwlk.r | . . 3 β’ βΌ = {β¨π’, π€β© β£ (π’ β (ClWWalksβπΊ) β§ π€ β (ClWWalksβπΊ) β§ βπ β (0...(β―βπ€))π’ = (π€ cyclShift π))} | |
2 | 1 | erclwwlkeq 29271 | . 2 β’ ((π β π β§ π β π) β (π βΌ π β (π β (ClWWalksβπΊ) β§ π β (ClWWalksβπΊ) β§ βπ β (0...(β―βπ))π = (π cyclShift π)))) |
3 | fveq2 6892 | . . . . . . . 8 β’ (π = (π cyclShift π) β (β―βπ) = (β―β(π cyclShift π))) | |
4 | eqid 2733 | . . . . . . . . . . . 12 β’ (VtxβπΊ) = (VtxβπΊ) | |
5 | 4 | clwwlkbp 29238 | . . . . . . . . . . 11 β’ (π β (ClWWalksβπΊ) β (πΊ β V β§ π β Word (VtxβπΊ) β§ π β β )) |
6 | 5 | simp2d 1144 | . . . . . . . . . 10 β’ (π β (ClWWalksβπΊ) β π β Word (VtxβπΊ)) |
7 | 6 | ad2antlr 726 | . . . . . . . . 9 β’ (((π β (ClWWalksβπΊ) β§ π β (ClWWalksβπΊ)) β§ (π β π β§ π β π)) β π β Word (VtxβπΊ)) |
8 | elfzelz 13501 | . . . . . . . . 9 β’ (π β (0...(β―βπ)) β π β β€) | |
9 | cshwlen 14749 | . . . . . . . . 9 β’ ((π β Word (VtxβπΊ) β§ π β β€) β (β―β(π cyclShift π)) = (β―βπ)) | |
10 | 7, 8, 9 | syl2an 597 | . . . . . . . 8 β’ ((((π β (ClWWalksβπΊ) β§ π β (ClWWalksβπΊ)) β§ (π β π β§ π β π)) β§ π β (0...(β―βπ))) β (β―β(π cyclShift π)) = (β―βπ)) |
11 | 3, 10 | sylan9eqr 2795 | . . . . . . 7 β’ (((((π β (ClWWalksβπΊ) β§ π β (ClWWalksβπΊ)) β§ (π β π β§ π β π)) β§ π β (0...(β―βπ))) β§ π = (π cyclShift π)) β (β―βπ) = (β―βπ)) |
12 | 11 | rexlimdva2 3158 | . . . . . 6 β’ (((π β (ClWWalksβπΊ) β§ π β (ClWWalksβπΊ)) β§ (π β π β§ π β π)) β (βπ β (0...(β―βπ))π = (π cyclShift π) β (β―βπ) = (β―βπ))) |
13 | 12 | ex 414 | . . . . 5 β’ ((π β (ClWWalksβπΊ) β§ π β (ClWWalksβπΊ)) β ((π β π β§ π β π) β (βπ β (0...(β―βπ))π = (π cyclShift π) β (β―βπ) = (β―βπ)))) |
14 | 13 | com23 86 | . . . 4 β’ ((π β (ClWWalksβπΊ) β§ π β (ClWWalksβπΊ)) β (βπ β (0...(β―βπ))π = (π cyclShift π) β ((π β π β§ π β π) β (β―βπ) = (β―βπ)))) |
15 | 14 | 3impia 1118 | . . 3 β’ ((π β (ClWWalksβπΊ) β§ π β (ClWWalksβπΊ) β§ βπ β (0...(β―βπ))π = (π cyclShift π)) β ((π β π β§ π β π) β (β―βπ) = (β―βπ))) |
16 | 15 | com12 32 | . 2 β’ ((π β π β§ π β π) β ((π β (ClWWalksβπΊ) β§ π β (ClWWalksβπΊ) β§ βπ β (0...(β―βπ))π = (π cyclShift π)) β (β―βπ) = (β―βπ))) |
17 | 2, 16 | sylbid 239 | 1 β’ ((π β π β§ π β π) β (π βΌ π β (β―βπ) = (β―βπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2941 βwrex 3071 Vcvv 3475 β c0 4323 class class class wbr 5149 {copab 5211 βcfv 6544 (class class class)co 7409 0cc0 11110 β€cz 12558 ...cfz 13484 β―chash 14290 Word cword 14464 cyclShift ccsh 14738 Vtxcvtx 28256 ClWWalkscclwwlk 29234 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-hash 14291 df-word 14465 df-concat 14521 df-substr 14591 df-pfx 14621 df-csh 14739 df-clwwlk 29235 |
This theorem is referenced by: erclwwlksym 29274 erclwwlktr 29275 |
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