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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tocycfvres2 | Structured version Visualization version GIF version |
Description: A cyclic permutation is the identity outside of its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.) |
Ref | Expression |
---|---|
tocycval.1 | β’ πΆ = (toCycβπ·) |
tocycfv.d | β’ (π β π· β π) |
tocycfv.w | β’ (π β π β Word π·) |
tocycfv.1 | β’ (π β π:dom πβ1-1βπ·) |
Ref | Expression |
---|---|
tocycfvres2 | β’ (π β ((πΆβπ) βΎ (π· β ran π)) = ( I βΎ (π· β ran π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tocycval.1 | . . . 4 β’ πΆ = (toCycβπ·) | |
2 | tocycfv.d | . . . 4 β’ (π β π· β π) | |
3 | tocycfv.w | . . . 4 β’ (π β π β Word π·) | |
4 | tocycfv.1 | . . . 4 β’ (π β π:dom πβ1-1βπ·) | |
5 | 1, 2, 3, 4 | tocycfv 32873 | . . 3 β’ (π β (πΆβπ) = (( I βΎ (π· β ran π)) βͺ ((π cyclShift 1) β β‘π))) |
6 | 5 | reseq1d 5978 | . 2 β’ (π β ((πΆβπ) βΎ (π· β ran π)) = ((( I βΎ (π· β ran π)) βͺ ((π cyclShift 1) β β‘π)) βΎ (π· β ran π))) |
7 | fnresi 6678 | . . . 4 β’ ( I βΎ (π· β ran π)) Fn (π· β ran π) | |
8 | 7 | a1i 11 | . . 3 β’ (π β ( I βΎ (π· β ran π)) Fn (π· β ran π)) |
9 | 1zzd 12621 | . . . . 5 β’ (π β 1 β β€) | |
10 | cshwfn 14781 | . . . . 5 β’ ((π β Word π· β§ 1 β β€) β (π cyclShift 1) Fn (0..^(β―βπ))) | |
11 | 3, 9, 10 | syl2anc 582 | . . . 4 β’ (π β (π cyclShift 1) Fn (0..^(β―βπ))) |
12 | f1f1orn 6844 | . . . . 5 β’ (π:dom πβ1-1βπ· β π:dom πβ1-1-ontoβran π) | |
13 | f1ocnv 6845 | . . . . 5 β’ (π:dom πβ1-1-ontoβran π β β‘π:ran πβ1-1-ontoβdom π) | |
14 | f1ofn 6834 | . . . . 5 β’ (β‘π:ran πβ1-1-ontoβdom π β β‘π Fn ran π) | |
15 | 4, 12, 13, 14 | 4syl 19 | . . . 4 β’ (π β β‘π Fn ran π) |
16 | dfdm4 5892 | . . . . 5 β’ dom π = ran β‘π | |
17 | wrddm 14501 | . . . . . . 7 β’ (π β Word π· β dom π = (0..^(β―βπ))) | |
18 | 3, 17 | syl 17 | . . . . . 6 β’ (π β dom π = (0..^(β―βπ))) |
19 | ssidd 3996 | . . . . . 6 β’ (π β (0..^(β―βπ)) β (0..^(β―βπ))) | |
20 | 18, 19 | eqsstrd 4011 | . . . . 5 β’ (π β dom π β (0..^(β―βπ))) |
21 | 16, 20 | eqsstrrid 4022 | . . . 4 β’ (π β ran β‘π β (0..^(β―βπ))) |
22 | fnco 6666 | . . . 4 β’ (((π cyclShift 1) Fn (0..^(β―βπ)) β§ β‘π Fn ran π β§ ran β‘π β (0..^(β―βπ))) β ((π cyclShift 1) β β‘π) Fn ran π) | |
23 | 11, 15, 21, 22 | syl3anc 1368 | . . 3 β’ (π β ((π cyclShift 1) β β‘π) Fn ran π) |
24 | disjdifr 4468 | . . . 4 β’ ((π· β ran π) β© ran π) = β | |
25 | 24 | a1i 11 | . . 3 β’ (π β ((π· β ran π) β© ran π) = β ) |
26 | fnunres1 6660 | . . 3 β’ ((( I βΎ (π· β ran π)) Fn (π· β ran π) β§ ((π cyclShift 1) β β‘π) Fn ran π β§ ((π· β ran π) β© ran π) = β ) β ((( I βΎ (π· β ran π)) βͺ ((π cyclShift 1) β β‘π)) βΎ (π· β ran π)) = ( I βΎ (π· β ran π))) | |
27 | 8, 23, 25, 26 | syl3anc 1368 | . 2 β’ (π β ((( I βΎ (π· β ran π)) βͺ ((π cyclShift 1) β β‘π)) βΎ (π· β ran π)) = ( I βΎ (π· β ran π))) |
28 | 6, 27 | eqtrd 2765 | 1 β’ (π β ((πΆβπ) βΎ (π· β ran π)) = ( I βΎ (π· β ran π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β cdif 3937 βͺ cun 3938 β© cin 3939 β wss 3940 β c0 4318 I cid 5569 β‘ccnv 5671 dom cdm 5672 ran crn 5673 βΎ cres 5674 β ccom 5676 Fn wfn 6537 β1-1βwf1 6539 β1-1-ontoβwf1o 6541 βcfv 6542 (class class class)co 7415 0cc0 11136 1c1 11137 β€cz 12586 ..^cfzo 13657 β―chash 14319 Word cword 14494 cyclShift ccsh 14768 toCycctocyc 32870 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-inf 9464 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-n0 12501 df-z 12587 df-uz 12851 df-rp 13005 df-fz 13515 df-fzo 13658 df-fl 13787 df-mod 13865 df-hash 14320 df-word 14495 df-concat 14551 df-substr 14621 df-pfx 14651 df-csh 14769 df-tocyc 32871 |
This theorem is referenced by: cycpmconjslem2 32919 cyc3conja 32921 |
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