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Theorem cshwmodn 13338
Description: Cyclically shifting a word is invariant regarding modulo the word's length. (Contributed by AV, 26-Oct-2018.)
Assertion
Ref Expression
cshwmodn ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 mod (#‘𝑊))))

Proof of Theorem cshwmodn
StepHypRef Expression
1 0csh0 13336 . . . 4 (∅ cyclShift 𝑁) = ∅
2 oveq1 6534 . . . 4 (𝑊 = ∅ → (𝑊 cyclShift 𝑁) = (∅ cyclShift 𝑁))
3 oveq1 6534 . . . . 5 (𝑊 = ∅ → (𝑊 cyclShift (𝑁 mod (#‘𝑊))) = (∅ cyclShift (𝑁 mod (#‘𝑊))))
4 0csh0 13336 . . . . 5 (∅ cyclShift (𝑁 mod (#‘𝑊))) = ∅
53, 4syl6eq 2659 . . . 4 (𝑊 = ∅ → (𝑊 cyclShift (𝑁 mod (#‘𝑊))) = ∅)
61, 2, 53eqtr4a 2669 . . 3 (𝑊 = ∅ → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 mod (#‘𝑊))))
76a1d 25 . 2 (𝑊 = ∅ → ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 mod (#‘𝑊)))))
8 lennncl 13126 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (#‘𝑊) ∈ ℕ)
9 zre 11214 . . . . . . . . . . . 12 (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
10 nnrp 11674 . . . . . . . . . . . 12 ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℝ+)
11 modabs2 12521 . . . . . . . . . . . 12 ((𝑁 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ+) → ((𝑁 mod (#‘𝑊)) mod (#‘𝑊)) = (𝑁 mod (#‘𝑊)))
129, 10, 11syl2anr 493 . . . . . . . . . . 11 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ((𝑁 mod (#‘𝑊)) mod (#‘𝑊)) = (𝑁 mod (#‘𝑊)))
1312opeq1d 4340 . . . . . . . . . 10 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ⟨((𝑁 mod (#‘𝑊)) mod (#‘𝑊)), (#‘𝑊)⟩ = ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)
1413oveq2d 6543 . . . . . . . . 9 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑊 substr ⟨((𝑁 mod (#‘𝑊)) mod (#‘𝑊)), (#‘𝑊)⟩) = (𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩))
1512opeq2d 4341 . . . . . . . . . 10 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ⟨0, ((𝑁 mod (#‘𝑊)) mod (#‘𝑊))⟩ = ⟨0, (𝑁 mod (#‘𝑊))⟩)
1615oveq2d 6543 . . . . . . . . 9 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑊 substr ⟨0, ((𝑁 mod (#‘𝑊)) mod (#‘𝑊))⟩) = (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))
1714, 16oveq12d 6545 . . . . . . . 8 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ((𝑊 substr ⟨((𝑁 mod (#‘𝑊)) mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, ((𝑁 mod (#‘𝑊)) mod (#‘𝑊))⟩)) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)))
1817ex 448 . . . . . . 7 ((#‘𝑊) ∈ ℕ → (𝑁 ∈ ℤ → ((𝑊 substr ⟨((𝑁 mod (#‘𝑊)) mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, ((𝑁 mod (#‘𝑊)) mod (#‘𝑊))⟩)) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
198, 18syl 17 . . . . . 6 ((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (𝑁 ∈ ℤ → ((𝑊 substr ⟨((𝑁 mod (#‘𝑊)) mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, ((𝑁 mod (#‘𝑊)) mod (#‘𝑊))⟩)) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
2019impancom 454 . . . . 5 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 ≠ ∅ → ((𝑊 substr ⟨((𝑁 mod (#‘𝑊)) mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, ((𝑁 mod (#‘𝑊)) mod (#‘𝑊))⟩)) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
2120impcom 444 . . . 4 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → ((𝑊 substr ⟨((𝑁 mod (#‘𝑊)) mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, ((𝑁 mod (#‘𝑊)) mod (#‘𝑊))⟩)) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)))
22 simprl 789 . . . . 5 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → 𝑊 ∈ Word 𝑉)
23 simprr 791 . . . . . . 7 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
248ex 448 . . . . . . . . 9 (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ → (#‘𝑊) ∈ ℕ))
2524adantr 479 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 ≠ ∅ → (#‘𝑊) ∈ ℕ))
2625impcom 444 . . . . . . 7 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (#‘𝑊) ∈ ℕ)
2723, 26zmodcld 12508 . . . . . 6 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (𝑁 mod (#‘𝑊)) ∈ ℕ0)
2827nn0zd 11312 . . . . 5 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (𝑁 mod (#‘𝑊)) ∈ ℤ)
29 cshword 13334 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ (𝑁 mod (#‘𝑊)) ∈ ℤ) → (𝑊 cyclShift (𝑁 mod (#‘𝑊))) = ((𝑊 substr ⟨((𝑁 mod (#‘𝑊)) mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, ((𝑁 mod (#‘𝑊)) mod (#‘𝑊))⟩)))
3022, 28, 29syl2anc 690 . . . 4 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (𝑊 cyclShift (𝑁 mod (#‘𝑊))) = ((𝑊 substr ⟨((𝑁 mod (#‘𝑊)) mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, ((𝑁 mod (#‘𝑊)) mod (#‘𝑊))⟩)))
31 cshword 13334 . . . . 5 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)))
3231adantl 480 . . . 4 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)))
3321, 30, 323eqtr4rd 2654 . . 3 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 mod (#‘𝑊))))
3433ex 448 . 2 (𝑊 ≠ ∅ → ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 mod (#‘𝑊)))))
357, 34pm2.61ine 2864 1 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 mod (#‘𝑊))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wne 2779  c0 3873  cop 4130  cfv 5790  (class class class)co 6527  cr 9791  0cc0 9792  cn 10867  cz 11210  +crp 11664   mod cmo 12485  #chash 12934  Word cword 13092   ++ cconcat 13094   substr csubstr 13096   cyclShift ccsh 13331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-inf 8209  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-n0 11140  df-z 11211  df-uz 11520  df-rp 11665  df-fz 12153  df-fzo 12290  df-fl 12410  df-mod 12486  df-hash 12935  df-word 13100  df-concat 13102  df-substr 13104  df-csh 13332
This theorem is referenced by:  cshwsublen  13339  cshwn  13340
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