Theorem cshwsiun 14795

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 2413	. . . . . . . . 9 |- ( ( W cyclShift n ) = w <-> w = ( W cyclShift n ) )
3	2	biimpi 196	. . . . . . . 8 |- ( ( W cyclShift n ) = w -> w = ( W cyclShift n ) )
4	3	reximi 2874	. . . . . . 7 |- ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w -> E. n e. ( 0..^ ( # ` W ) ) w = ( W cyclShift n ) )
5	4	adantl 466	. . . . . 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 12827	. . . . . . . . . . . 12 |- ( W e. Word V -> ( W cyclShift n ) e. Word V )
7	6	adantr 465	. . . . . . . . . . 11 |- ( ( W e. Word V /\ n e. ( 0..^ ( # ` W ) ) ) -> ( W cyclShift n ) e. Word V )
8		eleq1 2476	. . . . . . . . . . 11 |- ( w = ( W cyclShift n ) -> ( w e. Word V <-> ( W cyclShift n ) e. Word V ) )
9	7, 8	syl5ibrcom 224	. . . . . . . . . 10 |- ( ( W e. Word V /\ n e. ( 0..^ ( # ` W ) ) ) -> ( w = ( W cyclShift n ) -> w e. Word V ) )
10	9	rexlimdva 2898	. . . . . . . . 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 2413	. . . . . . . . . . 11 |- ( w = ( W cyclShift n ) <-> ( W cyclShift n ) = w )
13	12	biimpi 196	. . . . . . . . . 10 |- ( w = ( W cyclShift n ) -> ( W cyclShift n ) = w )
14	13	reximi 2874	. . . . . . . . 9 |- ( E. n e. ( 0..^ ( # ` W ) ) w = ( W cyclShift n ) -> E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w )
15	14	adantl 466	. . . . . . . 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 532	. . . . . . 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 206	. . . . 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 3988	. . . . . . . 8 |- ( w e. { ( W cyclShift n ) } <-> w = ( W cyclShift n ) )
20	19	bicomi 204	. . . . . . 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 2920	. . . . 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 255	. . . 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 2540	. . 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 2457	. 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 4275	. 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 2470	1 |- ( W e. Word V -> M = U_ n e. ( 0..^ ( # ` W ) ) { ( W cyclShift n ) } )

Syntax hints:   -> wi 4   <-> wb 186   /\ wa 369   = wceq 1407   e. wcel 1844   {cab 2389   E.wrex 2757   {crab 2760   {csn 3974   U_ciun 4273   ` cfv 5571  (class class class)co 6280   0cc0 9524  ..^cfzo 11856   #chash 12454  Word cword 12585   cyclShift ccsh 12817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-fzo 11857  df-hash 12455  df-word 12593  df-concat 12595  df-substr 12597  df-csh 12818

This theorem is referenced by:  cshwsex  14796  cshwshashnsame  14799