![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > tocycfvres1 | Structured version Visualization version GIF version |
Description: A cyclic permutation is a cyclic shift on 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 |
---|---|
tocycfvres1 | β’ (π β ((πΆβπ) βΎ ran π) = ((π cyclShift 1) β β‘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 32007 | . . 3 β’ (π β (πΆβπ) = (( I βΎ (π· β ran π)) βͺ ((π cyclShift 1) β β‘π))) |
6 | 5 | reseq1d 5937 | . 2 β’ (π β ((πΆβπ) βΎ ran π) = ((( I βΎ (π· β ran π)) βͺ ((π cyclShift 1) β β‘π)) βΎ ran π)) |
7 | fnresi 6631 | . . . 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 12539 | . . . . 5 β’ (π β 1 β β€) | |
10 | cshwfn 14695 | . . . . 5 β’ ((π β Word π· β§ 1 β β€) β (π cyclShift 1) Fn (0..^(β―βπ))) | |
11 | 3, 9, 10 | syl2anc 585 | . . . 4 β’ (π β (π cyclShift 1) Fn (0..^(β―βπ))) |
12 | f1f1orn 6796 | . . . . 5 β’ (π:dom πβ1-1βπ· β π:dom πβ1-1-ontoβran π) | |
13 | f1ocnv 6797 | . . . . 5 β’ (π:dom πβ1-1-ontoβran π β β‘π:ran πβ1-1-ontoβdom π) | |
14 | f1ofn 6786 | . . . . 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 5852 | . . . . 5 β’ dom π = ran β‘π | |
17 | wrddm 14415 | . . . . . . 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 3968 | . . . . . 6 β’ (π β (0..^(β―βπ)) β (0..^(β―βπ))) | |
20 | 18, 19 | eqsstrd 3983 | . . . . 5 β’ (π β dom π β (0..^(β―βπ))) |
21 | 16, 20 | eqsstrrid 3994 | . . . 4 β’ (π β ran β‘π β (0..^(β―βπ))) |
22 | fnco 6619 | . . . 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 1372 | . . 3 β’ (π β ((π cyclShift 1) β β‘π) Fn ran π) |
24 | disjdifr 4433 | . . . 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 | fnunres2 31640 | . . 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 ran π) = ((π cyclShift 1) β β‘π)) | |
27 | 8, 23, 25, 26 | syl3anc 1372 | . 2 β’ (π β ((( I βΎ (π· β ran π)) βͺ ((π cyclShift 1) β β‘π)) βΎ ran π) = ((π cyclShift 1) β β‘π)) |
28 | 6, 27 | eqtrd 2773 | 1 β’ (π β ((πΆβπ) βΎ ran π) = ((π cyclShift 1) β β‘π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β cdif 3908 βͺ cun 3909 β© cin 3910 β wss 3911 β c0 4283 I cid 5531 β‘ccnv 5633 dom cdm 5634 ran crn 5635 βΎ cres 5636 β ccom 5638 Fn wfn 6492 β1-1βwf1 6494 β1-1-ontoβwf1o 6496 βcfv 6497 (class class class)co 7358 0cc0 11056 1c1 11057 β€cz 12504 ..^cfzo 13573 β―chash 14236 Word cword 14408 cyclShift ccsh 14682 toCycctocyc 32004 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-fl 13703 df-mod 13781 df-hash 14237 df-word 14409 df-concat 14465 df-substr 14535 df-pfx 14565 df-csh 14683 df-tocyc 32005 |
This theorem is referenced by: cycpmconjslem1 32052 cycpmconjslem2 32053 |
Copyright terms: Public domain | W3C validator |