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Theorem cshwsiun 16013
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 3070 . . 3 {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)}
2 eqcom 2778 . . . . . . . . 9 ((𝑊 cyclShift 𝑛) = 𝑤𝑤 = (𝑊 cyclShift 𝑛))
32biimpi 206 . . . . . . . 8 ((𝑊 cyclShift 𝑛) = 𝑤𝑤 = (𝑊 cyclShift 𝑛))
43reximi 3159 . . . . . . 7 (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 → ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛))
54adantl 467 . . . . . 6 ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤) → ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛))
6 cshwcl 13753 . . . . . . . . . . . 12 (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑛) ∈ Word 𝑉)
76adantr 466 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))) → (𝑊 cyclShift 𝑛) ∈ Word 𝑉)
8 eleq1 2838 . . . . . . . . . . 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 3179 . . . . . . . . 9 (𝑊 ∈ Word 𝑉 → (∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛) → 𝑤 ∈ Word 𝑉))
1110imp 393 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)) → 𝑤 ∈ Word 𝑉)
12 eqcom 2778 . . . . . . . . . . 11 (𝑤 = (𝑊 cyclShift 𝑛) ↔ (𝑊 cyclShift 𝑛) = 𝑤)
1312biimpi 206 . . . . . . . . . 10 (𝑤 = (𝑊 cyclShift 𝑛) → (𝑊 cyclShift 𝑛) = 𝑤)
1413reximi 3159 . . . . . . . . 9 (∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛) → ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
1514adantl 467 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)) → ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
1611, 15jca 501 . . . . . . 7 ((𝑊 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)) → (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
1716ex 397 . . . . . 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 4332 . . . . . . . 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 3200 . . . . 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 2890 . . 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 2817 . 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 4656 . 2 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)} = {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)}}
2825, 26, 273eqtr4g 2830 1 (𝑊 ∈ Word 𝑉𝑀 = 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {cab 2757  wrex 3062  {crab 3065  {csn 4316   ciun 4654  cfv 6031  (class class class)co 6793  0cc0 10138  ..^cfzo 12673  chash 13321  Word cword 13487   cyclShift ccsh 13743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-concat 13497  df-substr 13499  df-csh 13744
This theorem is referenced by:  cshwsex  16014  cshwshashnsame  16017
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