Theorem cshwsiun 15148

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 2765	. . 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 2478	. . . . . . . . 9 |- ( ( W cyclShift n ) = w <-> w = ( W cyclShift n ) )
3	2	biimpi 199	. . . . . . . 8 |- ( ( W cyclShift n ) = w -> w = ( W cyclShift n ) )
4	3	reximi 2852	. . . . . . 7 |- ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w -> E. n e. ( 0..^ ( # ` W ) ) w = ( W cyclShift n ) )
5	4	adantl 473	. . . . . 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 12954	. . . . . . . . . . . 12 |- ( W e. Word V -> ( W cyclShift n ) e. Word V )
7	6	adantr 472	. . . . . . . . . . 11 |- ( ( W e. Word V /\ n e. ( 0..^ ( # ` W ) ) ) -> ( W cyclShift n ) e. Word V )
8		eleq1 2537	. . . . . . . . . . 11 |- ( w = ( W cyclShift n ) -> ( w e. Word V <-> ( W cyclShift n ) e. Word V ) )
9	7, 8	syl5ibrcom 230	. . . . . . . . . 10 |- ( ( W e. Word V /\ n e. ( 0..^ ( # ` W ) ) ) -> ( w = ( W cyclShift n ) -> w e. Word V ) )
10	9	rexlimdva 2871	. . . . . . . . 9 |- ( W e. Word V -> ( E. n e. ( 0..^ ( # ` W ) ) w = ( W cyclShift n ) -> w e. Word V ) )
11	10	imp 436	. . . . . . . 8 |- ( ( W e. Word V /\ E. n e. ( 0..^ ( # ` W ) ) w = ( W cyclShift n ) ) -> w e. Word V )
12		eqcom 2478	. . . . . . . . . . 11 |- ( w = ( W cyclShift n ) <-> ( W cyclShift n ) = w )
13	12	biimpi 199	. . . . . . . . . 10 |- ( w = ( W cyclShift n ) -> ( W cyclShift n ) = w )
14	13	reximi 2852	. . . . . . . . 9 |- ( E. n e. ( 0..^ ( # ` W ) ) w = ( W cyclShift n ) -> E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w )
15	14	adantl 473	. . . . . . . 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 541	. . . . . . 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 441	. . . . . 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 209	. . . . 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 3973	. . . . . . . 8 |- ( w e. { ( W cyclShift n ) } <-> w = ( W cyclShift n ) )
20	19	bicomi 207	. . . . . . 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 2892	. . . . 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 261	. . . 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 2589	. . 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 2517	. 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 4271	. 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 2530	1 |- ( W e. Word V -> M = U_ n e. ( 0..^ ( # ` W ) ) { ( W cyclShift n ) } )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   {cab 2457   E.wrex 2757   {crab 2760   {csn 3959   U_ciun 4269   ` cfv 5589  (class class class)co 6308   0cc0 9557  ..^cfzo 11942   #chash 12553  Word cword 12703   cyclShift ccsh 12944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-substr 12715  df-csh 12945

This theorem is referenced by:  cshwsex  15149  cshwshashnsame  15152