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Theorem cshwmodn 13392
 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 13390 . . . 4 (∅ cyclShift 𝑁) = ∅
2 oveq1 6556 . . . 4 (𝑊 = ∅ → (𝑊 cyclShift 𝑁) = (∅ cyclShift 𝑁))
3 oveq1 6556 . . . . 5 (𝑊 = ∅ → (𝑊 cyclShift (𝑁 mod (#‘𝑊))) = (∅ cyclShift (𝑁 mod (#‘𝑊))))
4 0csh0 13390 . . . . 5 (∅ cyclShift (𝑁 mod (#‘𝑊))) = ∅
53, 4syl6eq 2660 . . . 4 (𝑊 = ∅ → (𝑊 cyclShift (𝑁 mod (#‘𝑊))) = ∅)
61, 2, 53eqtr4a 2670 . . 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 13180 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (#‘𝑊) ∈ ℕ)
9 zre 11258 . . . . . . . . . . . 12 (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
10 nnrp 11718 . . . . . . . . . . . 12 ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℝ+)
11 modabs2 12566 . . . . . . . . . . . 12 ((𝑁 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ+) → ((𝑁 mod (#‘𝑊)) mod (#‘𝑊)) = (𝑁 mod (#‘𝑊)))
129, 10, 11syl2anr 494 . . . . . . . . . . 11 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ((𝑁 mod (#‘𝑊)) mod (#‘𝑊)) = (𝑁 mod (#‘𝑊)))
1312opeq1d 4346 . . . . . . . . . 10 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ⟨((𝑁 mod (#‘𝑊)) mod (#‘𝑊)), (#‘𝑊)⟩ = ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)
1413oveq2d 6565 . . . . . . . . 9 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑊 substr ⟨((𝑁 mod (#‘𝑊)) mod (#‘𝑊)), (#‘𝑊)⟩) = (𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩))
1512opeq2d 4347 . . . . . . . . . 10 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ⟨0, ((𝑁 mod (#‘𝑊)) mod (#‘𝑊))⟩ = ⟨0, (𝑁 mod (#‘𝑊))⟩)
1615oveq2d 6565 . . . . . . . . 9 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑊 substr ⟨0, ((𝑁 mod (#‘𝑊)) mod (#‘𝑊))⟩) = (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))
1714, 16oveq12d 6567 . . . . . . . 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 449 . . . . . . 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 455 . . . . 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 445 . . . 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 790 . . . . 5 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → 𝑊 ∈ Word 𝑉)
23 simprr 792 . . . . . . 7 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
248ex 449 . . . . . . . . 9 (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ → (#‘𝑊) ∈ ℕ))
2524adantr 480 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 ≠ ∅ → (#‘𝑊) ∈ ℕ))
2625impcom 445 . . . . . . 7 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (#‘𝑊) ∈ ℕ)
2723, 26zmodcld 12553 . . . . . 6 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (𝑁 mod (#‘𝑊)) ∈ ℕ0)
2827nn0zd 11356 . . . . 5 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (𝑁 mod (#‘𝑊)) ∈ ℤ)
29 cshword 13388 . . . . 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 691 . . . 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 13388 . . . . 5 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)))
3231adantl 481 . . . 4 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)))
3321, 30, 323eqtr4rd 2655 . . 3 ((𝑊 ≠ ∅ ∧ (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ)) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 mod (#‘𝑊))))
3433ex 449 . 2 (𝑊 ≠ ∅ → ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 mod (#‘𝑊)))))
357, 34pm2.61ine 2865 1 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 mod (#‘𝑊))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  ℕcn 10897  ℤcz 11254  ℝ+crp 11708   mod cmo 12530  #chash 12979  Word cword 13146   ++ cconcat 13148   substr csubstr 13150   cyclShift ccsh 13385 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386 This theorem is referenced by:  cshwsublen  13393  cshwn  13394
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