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

Step	Hyp	Ref	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 ) )
3	2	biimpi 194	. . . . . . . 8 |- ( ( W cyclShift n ) = w -> w = ( W cyclShift n ) )
4	3	reximi 2821	. . . . . . 7 |- ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w -> E. n e. ( 0..^ ( # ` W ) ) w = ( W cyclShift n ) )
5	4	adantl 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 )
7	6	adantr 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 ) )
9	7, 8	syl5ibrcom 222	. . . . . . . . . 10 |- ( ( W e. Word V /\ n e. ( 0..^ ( # ` W ) ) ) -> ( w = ( W cyclShift n ) -> w e. Word V ) )
10	9	rexlimdva 2839	. . . . . . . . 9 |- ( W e. Word V -> ( E. n e. ( 0..^ ( # ` W ) ) w = ( W cyclShift n ) -> w e. Word V ) )
11	10	imp 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 )
13	12	biimpi 194	. . . . . . . . . 10 |- ( w = ( W cyclShift n ) -> ( W cyclShift n ) = w )
14	13	reximi 2821	. . . . . . . . 9 |- ( E. n e. ( 0..^ ( # ` W ) ) w = ( W cyclShift n ) -> E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w )
15	14	adantl 463	. . . . . . . 8 |- ( ( W e. Word V /\ E. n e. ( 0..^ ( # ` W ) ) w = ( W cyclShift n ) ) -> E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w )
16	11, 15	jca 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 ) )
17	16	ex 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 ) ) )
18	5, 17	impbid2 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 ) )
20	19	bicomi 202	. . . . . . 7 |- ( w = ( W cyclShift n ) <-> w e. { ( W cyclShift n ) } )
21	20	a1i 11	. . . . . 6 |- ( W e. Word V -> ( w = ( W cyclShift n ) <-> w e. { ( W cyclShift n ) } ) )
22	21	rexbidv 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 ) } ) )
23	18, 22	bitrd 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 ) } ) )
24	23	abbidv 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 ) } } )
25	1, 24	syl5eq 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 ) } }
28	25, 26, 27	3eqtr4g 2498	1 |- ( W e. Word V -> M = U_ n e. ( 0..^ ( # ` W ) ) { ( W cyclShift n ) } )

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