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Theorem cshwsiun 15730
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 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}
Assertion
Ref Expression
cshwsiun (𝑊 ∈ Word 𝑉𝑀 = 𝑛 ∈ (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 2916 . . 3 {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)}
2 eqcom 2628 . . . . . . . . 9 ((𝑊 cyclShift 𝑛) = 𝑤𝑤 = (𝑊 cyclShift 𝑛))
32biimpi 206 . . . . . . . 8 ((𝑊 cyclShift 𝑛) = 𝑤𝑤 = (𝑊 cyclShift 𝑛))
43reximi 3005 . . . . . . 7 (∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 → ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛))
54adantl 482 . . . . . 6 ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤) → ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛))
6 cshwcl 13481 . . . . . . . . . . . 12 (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑛) ∈ Word 𝑉)
76adantr 481 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))) → (𝑊 cyclShift 𝑛) ∈ Word 𝑉)
8 eleq1 2686 . . . . . . . . . . 11 (𝑤 = (𝑊 cyclShift 𝑛) → (𝑤 ∈ Word 𝑉 ↔ (𝑊 cyclShift 𝑛) ∈ Word 𝑉))
97, 8syl5ibrcom 237 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))) → (𝑤 = (𝑊 cyclShift 𝑛) → 𝑤 ∈ Word 𝑉))
109rexlimdva 3024 . . . . . . . . 9 (𝑊 ∈ Word 𝑉 → (∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛) → 𝑤 ∈ Word 𝑉))
1110imp 445 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)) → 𝑤 ∈ Word 𝑉)
12 eqcom 2628 . . . . . . . . . . 11 (𝑤 = (𝑊 cyclShift 𝑛) ↔ (𝑊 cyclShift 𝑛) = 𝑤)
1312biimpi 206 . . . . . . . . . 10 (𝑤 = (𝑊 cyclShift 𝑛) → (𝑊 cyclShift 𝑛) = 𝑤)
1413reximi 3005 . . . . . . . . 9 (∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛) → ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
1514adantl 482 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)) → ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
1611, 15jca 554 . . . . . . 7 ((𝑊 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)) → (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
1716ex 450 . . . . . 6 (𝑊 ∈ Word 𝑉 → (∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛) → (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)))
185, 17impbid2 216 . . . . 5 (𝑊 ∈ Word 𝑉 → ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤) ↔ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)))
19 velsn 4164 . . . . . . . 8 (𝑤 ∈ {(𝑊 cyclShift 𝑛)} ↔ 𝑤 = (𝑊 cyclShift 𝑛))
2019bicomi 214 . . . . . . 7 (𝑤 = (𝑊 cyclShift 𝑛) ↔ 𝑤 ∈ {(𝑊 cyclShift 𝑛)})
2120a1i 11 . . . . . 6 (𝑊 ∈ Word 𝑉 → (𝑤 = (𝑊 cyclShift 𝑛) ↔ 𝑤 ∈ {(𝑊 cyclShift 𝑛)}))
2221rexbidv 3045 . . . . 5 (𝑊 ∈ Word 𝑉 → (∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)}))
2318, 22bitrd 268 . . . 4 (𝑊 ∈ Word 𝑉 → ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤) ↔ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)}))
2423abbidv 2738 . . 3 (𝑊 ∈ Word 𝑉 → {𝑤 ∣ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)} = {𝑤 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)}})
251, 24syl5eq 2667 . 2 (𝑊 ∈ Word 𝑉 → {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)}})
26 cshwrepswhash1.m . 2 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}
27 df-iun 4487 . 2 𝑛 ∈ (0..^(#‘𝑊)){(𝑊 cyclShift 𝑛)} = {𝑤 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)}}
2825, 26, 273eqtr4g 2680 1 (𝑊 ∈ Word 𝑉𝑀 = 𝑛 ∈ (0..^(#‘𝑊)){(𝑊 cyclShift 𝑛)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wrex 2908  {crab 2911  {csn 4148   ciun 4485  cfv 5847  (class class class)co 6604  0cc0 9880  ..^cfzo 12406  #chash 13057  Word cword 13230   cyclShift ccsh 13471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-concat 13240  df-substr 13242  df-csh 13472
This theorem is referenced by:  cshwsex  15731  cshwshashnsame  15734
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