Theorem cshwsiun 14433

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 2818	. . 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 2471	. . . . . . . . 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 2927	. . . . . . 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 12721	. . . . . . . . . . . 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 2534	. . . . . . . . . . 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 2950	. . . . . . . . 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 2471	. . . . . . . . . . 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 2927	. . . . . . . . 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 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 4036	. . . . . . . 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 2968	. . . . 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 2598	. . 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 2515	. 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 4322	. 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 2528	1 |- ( W e. Word V -> M = U_ n e. ( 0..^ ( # ` W ) ) { ( W cyclShift n ) } )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1374   e. wcel 1762   {cab 2447   E.wrex 2810   {crab 2813   {csn 4022   U_ciun 4320   ` cfv 5581  (class class class)co 6277   0cc0 9483  ..^cfzo 11783   #chash 12362  Word cword 12489   cyclShift ccsh 12711

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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-fzo 11784  df-hash 12363  df-word 12497  df-concat 12499  df-substr 12501  df-csh 12712

This theorem is referenced by:  cshwsex  14434  cshwshashnsame  14437