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Theorem cshword2 41762
Description: Perform a cyclical shift for a word. (Contributed by AV, 11-May-2020.)
Assertion
Ref Expression
cshword2 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (#‘𝑊)))))

Proof of Theorem cshword2
Dummy variables 𝑙 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iswrd 13339 . . . . 5 (𝑊 ∈ Word 𝑉 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑉)
2 ffn 6083 . . . . . 6 (𝑊:(0..^𝑙)⟶𝑉𝑊 Fn (0..^𝑙))
32reximi 3040 . . . . 5 (∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑉 → ∃𝑙 ∈ ℕ0 𝑊 Fn (0..^𝑙))
41, 3sylbi 207 . . . 4 (𝑊 ∈ Word 𝑉 → ∃𝑙 ∈ ℕ0 𝑊 Fn (0..^𝑙))
5 fneq1 6017 . . . . . 6 (𝑤 = 𝑊 → (𝑤 Fn (0..^𝑙) ↔ 𝑊 Fn (0..^𝑙)))
65rexbidv 3081 . . . . 5 (𝑤 = 𝑊 → (∃𝑙 ∈ ℕ0 𝑤 Fn (0..^𝑙) ↔ ∃𝑙 ∈ ℕ0 𝑊 Fn (0..^𝑙)))
76elabg 3383 . . . 4 (𝑊 ∈ Word 𝑉 → (𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤 Fn (0..^𝑙)} ↔ ∃𝑙 ∈ ℕ0 𝑊 Fn (0..^𝑙)))
84, 7mpbird 247 . . 3 (𝑊 ∈ Word 𝑉𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤 Fn (0..^𝑙)})
9 cshfn 13582 . . 3 ((𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
108, 9sylan 487 . 2 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
11 iftrue 4125 . . . . 5 (𝑊 = ∅ → if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ∅)
1211adantr 480 . . . 4 ((𝑊 = ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ∅)
13 oveq1 6697 . . . . . . . 8 (𝑊 = ∅ → (𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) = (∅ substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩))
14 swrd0 13480 . . . . . . . 8 (∅ substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) = ∅
1513, 14syl6eq 2701 . . . . . . 7 (𝑊 = ∅ → (𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) = ∅)
16 oveq1 6697 . . . . . . . 8 (𝑊 = ∅ → (𝑊 prefix (𝑁 mod (#‘𝑊))) = (∅ prefix (𝑁 mod (#‘𝑊))))
17 pfx0 41710 . . . . . . . 8 (∅ prefix (𝑁 mod (#‘𝑊))) = ∅
1816, 17syl6eq 2701 . . . . . . 7 (𝑊 = ∅ → (𝑊 prefix (𝑁 mod (#‘𝑊))) = ∅)
1915, 18oveq12d 6708 . . . . . 6 (𝑊 = ∅ → ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (#‘𝑊)))) = (∅ ++ ∅))
2019adantr 480 . . . . 5 ((𝑊 = ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (#‘𝑊)))) = (∅ ++ ∅))
21 wrd0 13362 . . . . . 6 ∅ ∈ Word V
22 ccatrid 13405 . . . . . 6 (∅ ∈ Word V → (∅ ++ ∅) = ∅)
2321, 22ax-mp 5 . . . . 5 (∅ ++ ∅) = ∅
2420, 23syl6req 2702 . . . 4 ((𝑊 = ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → ∅ = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (#‘𝑊)))))
2512, 24eqtrd 2685 . . 3 ((𝑊 = ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (#‘𝑊)))))
26 iffalse 4128 . . . . 5 𝑊 = ∅ → if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)))
2726adantr 480 . . . 4 ((¬ 𝑊 = ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)))
28 simprl 809 . . . . . . 7 ((¬ 𝑊 = ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → 𝑊 ∈ Word 𝑉)
29 simprr 811 . . . . . . . 8 ((¬ 𝑊 = ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
30 df-ne 2824 . . . . . . . . . 10 (𝑊 ≠ ∅ ↔ ¬ 𝑊 = ∅)
31 lennncl 13357 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (#‘𝑊) ∈ ℕ)
3231ex 449 . . . . . . . . . . 11 (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ → (#‘𝑊) ∈ ℕ))
3332adantr 480 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 ≠ ∅ → (#‘𝑊) ∈ ℕ))
3430, 33syl5bir 233 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (¬ 𝑊 = ∅ → (#‘𝑊) ∈ ℕ))
3534impcom 445 . . . . . . . 8 ((¬ 𝑊 = ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (#‘𝑊) ∈ ℕ)
3629, 35zmodcld 12731 . . . . . . 7 ((¬ 𝑊 = ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (𝑁 mod (#‘𝑊)) ∈ ℕ0)
37 pfxval 41708 . . . . . . 7 ((𝑊 ∈ Word 𝑉 ∧ (𝑁 mod (#‘𝑊)) ∈ ℕ0) → (𝑊 prefix (𝑁 mod (#‘𝑊))) = (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))
3828, 36, 37syl2anc 694 . . . . . 6 ((¬ 𝑊 = ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (𝑊 prefix (𝑁 mod (#‘𝑊))) = (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))
3938eqcomd 2657 . . . . 5 ((¬ 𝑊 = ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩) = (𝑊 prefix (𝑁 mod (#‘𝑊))))
4039oveq2d 6706 . . . 4 ((¬ 𝑊 = ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (#‘𝑊)))))
4127, 40eqtrd 2685 . . 3 ((¬ 𝑊 = ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (#‘𝑊)))))
4225, 41pm2.61ian 848 . 2 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (#‘𝑊)))))
4310, 42eqtrd 2685 1 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (#‘𝑊)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wrex 2942  Vcvv 3231  c0 3948  ifcif 4119  cop 4216   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  0cc0 9974  cn 11058  0cn0 11330  cz 11415  ..^cfzo 12504   mod cmo 12708  #chash 13157  Word cword 13323   ++ cconcat 13325   substr csubstr 13327   cyclShift ccsh 13580   prefix cpfx 41706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-hash 13158  df-word 13331  df-concat 13333  df-substr 13335  df-csh 13581  df-pfx 41707
This theorem is referenced by: (None)
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