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Theorem cshwsiun 15749
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 2917 . . 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 3007 . . . . . . 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 13497 . . . . . . . . . . . 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 3026 . . . . . . . . 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 3007 . . . . . . . . 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 4171 . . . . . . . 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 3047 . . . . 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 4494 . 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 2909  {crab 2912  {csn 4155   ciun 4492  cfv 5857  (class class class)co 6615  0cc0 9896  ..^cfzo 12422  #chash 13073  Word cword 13246   cyclShift ccsh 13487
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 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
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 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-hash 13074  df-word 13254  df-concat 13256  df-substr 13258  df-csh 13488
This theorem is referenced by:  cshwsex  15750  cshwshashnsame  15753
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