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Theorem cshwsiun 14122
Description: The set of (different!) words resulting by cyclically shifting a given word is an indexed union. (Contributed by AV, 19-May-2018.) (Revised by AV, 8-Jun-2018.) (Proof shortened by AV, 8-Nov-2018.)
Hypothesis
Ref Expression
cshwrepswhash1.m  |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W
) ) ( W cyclShift  n )  =  w }
Assertion
Ref Expression
cshwsiun  |-  ( W  e. Word  V  ->  M  =  U_ n  e.  ( 0..^ ( # `  W
) ) { ( W cyclShift  n ) } )
Distinct variable groups:    n, V, w    n, W, w
Allowed substitution hints:    M( w, n)

Proof of Theorem cshwsiun
StepHypRef Expression
1 df-rab 2722 . . 3  |-  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }  =  { w  |  ( w  e. Word  V  /\  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w ) }
2 eqcom 2443 . . . . . . . . 9  |-  ( ( W cyclShift  n )  =  w  <-> 
w  =  ( W cyclShift  n ) )
32biimpi 194 . . . . . . . 8  |-  ( ( W cyclShift  n )  =  w  ->  w  =  ( W cyclShift  n ) )
43reximi 2821 . . . . . . 7  |-  ( E. n  e.  ( 0..^ ( # `  W
) ) ( W cyclShift  n )  =  w  ->  E. n  e.  ( 0..^ ( # `  W
) ) w  =  ( W cyclShift  n )
)
54adantl 463 . . . . . 6  |-  ( ( w  e. Word  V  /\  E. n  e.  ( 0..^ ( # `  W
) ) ( W cyclShift  n )  =  w )  ->  E. n  e.  ( 0..^ ( # `  W ) ) w  =  ( W cyclShift  n ) )
6 cshwcl 12431 . . . . . . . . . . . 12  |-  ( W  e. Word  V  ->  ( W cyclShift  n )  e. Word  V
)
76adantr 462 . . . . . . . . . . 11  |-  ( ( W  e. Word  V  /\  n  e.  ( 0..^ ( # `  W
) ) )  -> 
( W cyclShift  n )  e. Word  V )
8 eleq1 2501 . . . . . . . . . . 11  |-  ( w  =  ( W cyclShift  n )  ->  ( w  e. Word  V 
<->  ( W cyclShift  n )  e. Word  V ) )
97, 8syl5ibrcom 222 . . . . . . . . . 10  |-  ( ( W  e. Word  V  /\  n  e.  ( 0..^ ( # `  W
) ) )  -> 
( w  =  ( W cyclShift  n )  ->  w  e. Word  V ) )
109rexlimdva 2839 . . . . . . . . 9  |-  ( W  e. Word  V  ->  ( E. n  e.  (
0..^ ( # `  W
) ) w  =  ( W cyclShift  n )  ->  w  e. Word  V ) )
1110imp 429 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  E. n  e.  ( 0..^ ( # `  W
) ) w  =  ( W cyclShift  n )
)  ->  w  e. Word  V )
12 eqcom 2443 . . . . . . . . . . 11  |-  ( w  =  ( W cyclShift  n )  <-> 
( W cyclShift  n )  =  w )
1312biimpi 194 . . . . . . . . . 10  |-  ( w  =  ( W cyclShift  n )  ->  ( W cyclShift  n )  =  w )
1413reximi 2821 . . . . . . . . 9  |-  ( E. n  e.  ( 0..^ ( # `  W
) ) w  =  ( W cyclShift  n )  ->  E. n  e.  ( 0..^ ( # `  W
) ) ( W cyclShift  n )  =  w )
1514adantl 463 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  E. n  e.  ( 0..^ ( # `  W
) ) w  =  ( W cyclShift  n )
)  ->  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w )
1611, 15jca 529 . . . . . . 7  |-  ( ( W  e. Word  V  /\  E. n  e.  ( 0..^ ( # `  W
) ) w  =  ( W cyclShift  n )
)  ->  ( w  e. Word  V  /\  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w ) )
1716ex 434 . . . . . 6  |-  ( W  e. Word  V  ->  ( E. n  e.  (
0..^ ( # `  W
) ) w  =  ( W cyclShift  n )  ->  ( w  e. Word  V  /\  E. n  e.  ( 0..^ ( # `  W
) ) ( W cyclShift  n )  =  w ) ) )
185, 17impbid2 204 . . . . 5  |-  ( W  e. Word  V  ->  (
( w  e. Word  V  /\  E. n  e.  ( 0..^ ( # `  W
) ) ( W cyclShift  n )  =  w )  <->  E. n  e.  ( 0..^ ( # `  W
) ) w  =  ( W cyclShift  n )
) )
19 elsn 3888 . . . . . . . 8  |-  ( w  e.  { ( W cyclShift  n ) }  <->  w  =  ( W cyclShift  n ) )
2019bicomi 202 . . . . . . 7  |-  ( w  =  ( W cyclShift  n )  <-> 
w  e.  { ( W cyclShift  n ) } )
2120a1i 11 . . . . . 6  |-  ( W  e. Word  V  ->  (
w  =  ( W cyclShift  n )  <->  w  e.  { ( W cyclShift  n ) } ) )
2221rexbidv 2734 . . . . 5  |-  ( W  e. Word  V  ->  ( E. n  e.  (
0..^ ( # `  W
) ) w  =  ( W cyclShift  n )  <->  E. n  e.  ( 0..^ ( # `  W
) ) w  e. 
{ ( W cyclShift  n ) } ) )
2318, 22bitrd 253 . . . 4  |-  ( W  e. Word  V  ->  (
( w  e. Word  V  /\  E. n  e.  ( 0..^ ( # `  W
) ) ( W cyclShift  n )  =  w )  <->  E. n  e.  ( 0..^ ( # `  W
) ) w  e. 
{ ( W cyclShift  n ) } ) )
2423abbidv 2555 . . 3  |-  ( W  e. Word  V  ->  { w  |  ( w  e. Word  V  /\  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w ) }  =  {
w  |  E. n  e.  ( 0..^ ( # `  W ) ) w  e.  { ( W cyclShift  n ) } }
)
251, 24syl5eq 2485 . 2  |-  ( W  e. Word  V  ->  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }  =  { w  |  E. n  e.  ( 0..^ ( # `  W
) ) w  e. 
{ ( W cyclShift  n ) } } )
26 cshwrepswhash1.m . 2  |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W
) ) ( W cyclShift  n )  =  w }
27 df-iun 4170 . 2  |-  U_ n  e.  ( 0..^ ( # `  W ) ) { ( W cyclShift  n ) }  =  { w  |  E. n  e.  ( 0..^ ( # `  W
) ) w  e. 
{ ( W cyclShift  n ) } }
2825, 26, 273eqtr4g 2498 1  |-  ( W  e. Word  V  ->  M  =  U_ n  e.  ( 0..^ ( # `  W
) ) { ( W cyclShift  n ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   {cab 2427   E.wrex 2714   {crab 2717   {csn 3874   U_ciun 4168   ` cfv 5415  (class class class)co 6090   0cc0 9278  ..^cfzo 11544   #chash 12099  Word cword 12217   cyclShift ccsh 12421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-concat 12227  df-substr 12229  df-csh 12422
This theorem is referenced by:  cshwsex  14123  cshwshashnsame  14126
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